%------------------------------------------------------------------------------
% File     : ITP207_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Heap_Time_Monad 00386_012441
% 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    : 0018_Heap_Time_Monad_00386_012441 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 11229 (2972 unt;2108 typ;   0 def)
%            Number of atoms       : 22589 (7638 equ)
%            Maximal formula atoms :   18 (   2 avg)
%            Number of connectives : 15190 (1722   ~; 279   |;1480   &)
%                                         (1915 <=>;9794  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   6 avg)
%            Maximal term depth    :   20 (   2 avg)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  : 1882 (1497   >; 385   *;   0   +;   0  <<)
%            Number of predicates  :  281 ( 278 usr;   2 prp; 0-9 aty)
%            Number of functors    : 1817 (1817 usr;  69 con; 0-8 aty)
%            Number of variables   : 34959 (31099   !; 460   ?;34959   :)
%                                         (3400  !>;   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 13:57:15.675
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
tff(ty_t_Record_Otuple__isomorphism,type,
    tuple_isomorphism: ( $tType * $tType * $tType ) > $tType ).

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_Old__Datatype_Onode,type,
    old_node: ( $tType * $tType ) > $tType ).

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

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

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

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

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

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

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

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

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

tff(ty_t_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 (2081)
tff(sy_cl_Enum_Oenum,type,
    enum: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
    ordere166539214618696060dd_abs: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
    ordere6658533253407199908up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__no__zero__divisors,type,
    semiri3467727345109120633visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
    boolea8198339166811842893lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
    quickc658316121487927005ustive: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring__strict,type,
    linord8928482502909563296strict: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

tff(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
    quickc3360725361186068524ustive: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
    euclid4440199948858584721cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
    euclid3128863361964157862miring: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ata____,type,
    aTP_Lamp_ata: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rc____,type,
    aTP_Lamp_rc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rd____,type,
    aTP_Lamp_rd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__re____,type,
    aTP_Lamp_re: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rg____,type,
    aTP_Lamp_rg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rh____,type,
    aTP_Lamp_rh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ri____,type,
    aTP_Lamp_ri: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rj____,type,
    aTP_Lamp_rj: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rm____,type,
    aTP_Lamp_rm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rn____,type,
    aTP_Lamp_rn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ro____,type,
    aTP_Lamp_ro: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rt____,type,
    aTP_Lamp_rt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ru____,type,
    aTP_Lamp_ru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rv____,type,
    aTP_Lamp_rv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rw____,type,
    aTP_Lamp_rw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rx____,type,
    aTP_Lamp_rx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ry____,type,
    aTP_Lamp_ry: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rz____,type,
    aTP_Lamp_rz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sa____,type,
    aTP_Lamp_sa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sb____,type,
    aTP_Lamp_sb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sc____,type,
    aTP_Lamp_sc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sr____,type,
    aTP_Lamp_sr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ss____,type,
    aTP_Lamp_ss: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__st____,type,
    aTP_Lamp_st: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__su____,type,
    aTP_Lamp_su: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sv____,type,
    aTP_Lamp_sv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sw____,type,
    aTP_Lamp_sw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__sx____,type,
    aTP_Lamp_sx: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
    bNF_Cardinal_Csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
    bNF_Cardinal_cexp: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
    bNF_Cardinal_cfinite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
    bNF_Ca4139267488887388095finite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
    bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
    bNF_Cardinal_cprod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
    bNF_Cardinal_csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
    bNF_Cardinal_ctwo: set(product_prod(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) > fun(set(product_prod(A,A)),bool) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
    bNF_Ca7293521722713021262ofinal: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
    bNF_Ca6246979054910435723ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(set(A),set(A))),bool) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
    bNF_Ca7133664381575040944arCard: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
    bNF_Ca3754400796208372196lChain: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_BNF__Composition_Oid__bnf,type,
    bNF_id_bnf: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_BNF__Def_OGr,type,
    bNF_Gr: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Def_OGrp,type,
    bNF_Grp: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,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_Oconvol,type,
    bNF_convol: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C)))) ).

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_OfstOp,type,
    bNF_fstOp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) ) > fun(product_prod(A,C),product_prod(A,B)) ) ).

tff(sy_c_BNF__Def_Opick__middlep,type,
    bNF_pick_middlep: 
      !>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,bool)) * fun(A,fun(C,bool)) * B * C ) > A ) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).

tff(sy_c_BNF__Def_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__Def_Orel__sum,type,
    bNF_rel_sum: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(sum_sum(A,B),fun(sum_sum(C,D),bool)) ) ).

tff(sy_c_BNF__Def_OsndOp,type,
    bNF_sndOp: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,bool)) * fun(A,fun(B,bool)) ) > fun(product_prod(C,B),product_prod(A,B)) ) ).

tff(sy_c_BNF__Def_Ovimage2p,type,
    bNF_vimage2p: 
      !>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) ) > fun(fun(D,fun(E,C)),fun(A,fun(B,C))) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OShift,type,
    bNF_Greatest_Shift: 
      !>[A: $tType] : ( ( set(list(A)) * A ) > set(list(A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
    bNF_Greatest_Succ: 
      !>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
    bNF_Gr5436034075474128252omCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
    bNF_Greatest_image2: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
    bNF_Greatest_image2p: 
      !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,bool)) * A * B ) > $o ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
    bNF_Gr4221423524335903396lImage: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
    bNF_Gr7122648621184425601vImage: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
    bNF_Greatest_toCard: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
    bNF_Gr1419584066657907630d_pred: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(fun(A,B),bool) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
    bNF_Greatest_univ: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
    bNF_Wellorder_Func: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
    bNF_Wellorder_bsqr: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
    bNF_Wellorder_curr: 
      !>[A: $tType,B: $tType,C: $tType] : ( set(A) > fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
    bNF_We2720479622203943262_image: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
    bNF_We413866401316099525erIncl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
    bNF_Wellorder_ordIso: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
    bNF_Wellorder_ordLeq: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
    bNF_We4044943003108391690rdLess: 
      !>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).

tff(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
    bNF_We8469521843155493636filter: 
      !>[A: $tType] : ( set(product_prod(A,A)) > fun(set(product_prod(A,A)),set(A)) ) ).

tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
    bNF_Wellorder_compat: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).

tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
    bNF_Wellorder_embed: 
      !>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) ) > fun(fun(A,A2),bool) ) ).

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_Octor__rec,type,
    basic_BNF_ctor_rec: 
      !>[A: $tType] : ( 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) ) > fun(product_prod(A,B),nat) ) ).

tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
    basic_BNF_size_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) ) > fun(sum_sum(A,B),nat) ) ).

tff(sy_c_Basic__BNF__LFPs_Oxtor,type,
    basic_BNF_xtor: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Basic__BNFs_Ofsts,type,
    basic_fsts: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),set(A)) ).

tff(sy_c_Basic__BNFs_Ofstsp,type,
    basic_fstsp: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > fun(A,bool) ) ).

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_Basic__BNFs_Opred__prod,type,
    basic_pred_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(product_prod(A,B),bool) ) ).

tff(sy_c_Basic__BNFs_Opred__sum,type,
    basic_pred_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) * sum_sum(A,B) ) > $o ) ).

tff(sy_c_Basic__BNFs_Orel__prod,type,
    basic_rel_prod: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).

tff(sy_c_Basic__BNFs_Osetl,type,
    basic_setl: 
      !>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(A)) ).

tff(sy_c_Basic__BNFs_Osetlp,type,
    basic_setlp: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > fun(A,bool) ) ).

tff(sy_c_Basic__BNFs_Osetr,type,
    basic_setr: 
      !>[A: $tType,B: $tType] : fun(sum_sum(A,B),set(B)) ).

tff(sy_c_Basic__BNFs_Osetrp,type,
    basic_setrp: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > fun(B,bool) ) ).

tff(sy_c_Basic__BNFs_Osnds,type,
    basic_snds: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).

tff(sy_c_Basic__BNFs_Osndsp,type,
    basic_sndsp: 
      !>[A: $tType,B: $tType] : ( product_prod(A,B) > fun(B,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_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,bool) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
    boolea2506097494486148201lgebra: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A ) > $o ) ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).

tff(sy_c_Code__Numeral_Ocr__integer,type,
    code_cr_integer: ( int * code_integer ) > $o ).

tff(sy_c_Code__Numeral_Ocr__natural,type,
    code_cr_natural: ( nat * code_natural ) > $o ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: fun(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: fun(code_integer,int) ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: fun(int,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: num > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
    code_case_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,T) ) > fun(code_natural,T) ) ).

tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: fun(code_natural,nat) ).

tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: fun(nat,code_natural) ).

tff(sy_c_Code__Numeral_Onatural_Orec__natural,type,
    code_rec_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) ) > fun(code_natural,T) ) ).

tff(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
    code_rec_set_natural: 
      !>[T: $tType] : ( ( T * fun(code_natural,fun(T,T)) * code_natural ) > fun(T,bool) ) ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: fun(code_integer,num) ).

tff(sy_c_Code__Numeral_Opcr__integer,type,
    code_pcr_integer: fun(int,fun(code_integer,bool)) ).

tff(sy_c_Code__Numeral_Osub,type,
    code_sub: fun(num,fun(num,code_integer)) ).

tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
    complete_Inf_Inf: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
    complete_Sup_Sup: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Complete__Partial__Order_Occpo_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__class_Obdd__above,type,
    condit941137186595557371_above: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
    condit1013018076250108175_below: 
      !>[A: $tType] : fun(set(A),bool) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
    condit622319405099724424ng_bdd: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
    condit16957441358409770ng_bdd: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),bool) ) ).

tff(sy_c_Countable_Ofrom__nat,type,
    from_nat: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Countable_Onat__to__rat__surj,type,
    nat_to_rat_surj: fun(nat,rat) ).

tff(sy_c_Countable_Onth__item,type,
    nth_item: 
      !>[A: $tType] : ( nat > set(old_node(A,product_unit)) ) ).

tff(sy_c_Countable_Onth__item__rel,type,
    nth_item_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_Countable_Oto__nat,type,
    to_nat: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Equiv__Relations_Ocongruent,type,
    equiv_congruent: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_Equiv__Relations_Ocongruent2,type,
    equiv_congruent2: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).

tff(sy_c_Equiv__Relations_Oequiv,type,
    equiv_equiv: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Equiv__Relations_Oequivp,type,
    equiv_equivp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Equiv__Relations_Opart__equivp,type,
    equiv_part_equivp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Equiv__Relations_Oproj,type,
    equiv_proj: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > 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] : fun(A,nat) ).

tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
    euclid7384307370059645450egment: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Ocofinite,type,
    cofinite: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,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_Ofiltermap,type,
    filtermap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Ofrequently,type,
    frequently: 
      !>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

tff(sy_c_Filter_Oprod__filter,type,
    prod_filter: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).

tff(sy_c_Filter_Orel__filter,type,
    rel_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * filter(A) * filter(B) ) > $o ) ).

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__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_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : fun(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_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).

tff(sy_c_Fun__Def_Ois__measure,type,
    fun_is_measure: 
      !>[A: $tType] : ( fun(A,nat) > $o ) ).

tff(sy_c_Fun__Def_Omax__strict,type,
    fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omax__weak,type,
    fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__strict,type,
    fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Omin__weak,type,
    fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).

tff(sy_c_Fun__Def_Opair__leq,type,
    fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Opair__less,type,
    fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).

tff(sy_c_Fun__Def_Oreduction__pair,type,
    fun_reduction_pair: 
      !>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).

tff(sy_c_Fun__Def_Orp__inv__image,type,
    fun_rp_inv_image: 
      !>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_OGcd__class_OLcm,type,
    gcd_Lcm: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_GCD_Obounded__quasi__semilattice,type,
    bounde8507323023520639062attice: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__axioms,type,
    bounde5338082073960204281axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__class_Olcm,type,
    gcd_lcm: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
    semiring_gcd_Lcm_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Groups_Oabel__semigroup,type,
    abel_semigroup: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Groups_Oabel__semigroup__axioms,type,
    abel_s757365448890700780axioms: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ocomm__monoid,type,
    comm_monoid: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups_Ocomm__monoid__axioms,type,
    comm_monoid_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups_Ogroup,type,
    group: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).

tff(sy_c_Groups_Ogroup__axioms,type,
    group_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,A) ) > $o ) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Omonoid,type,
    monoid: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups_Omonoid__axioms,type,
    monoid_axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osemigroup,type,
    semigroup: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : fun(fun(C,A),fun(set(C),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
    groups778175481326437816id_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
    groups_comm_monoid_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
    groups_comm_monoid_G: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list,type,
    groups1828464146339083142d_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__monoid__list__set,type,
    groups4802862169904069756st_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_Groups__List_Omonoid__list,type,
    groups_monoid_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Groups__List_Omonoid__list_OF,type,
    groups_monoid_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(list(A),A) ) ).

tff(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
    groups5270119922927024881d_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,bool) > $o ) ).

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__Time__Monad_OHeap_OHeap,type,
    heap_Time_Heap2: 
      !>[A: $tType] : fun(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_Ocase__Heap,type,
    heap_Time_case_Heap: 
      !>[A: $tType,B: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B)) ).

tff(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
    heap_Time_map_Heap: 
      !>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(heap_Time_Heap(A),heap_Time_Heap(Aa))) ).

tff(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
    heap_Time_pred_Heap: 
      !>[A: $tType] : fun(fun(A,bool),fun(heap_Time_Heap(A),bool)) ).

tff(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
    heap_Time_rec_Heap: 
      !>[A: $tType,C: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)) ).

tff(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
    heap_Time_rel_Heap: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))) ).

tff(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
    heap_Time_set_Heap: 
      !>[A: $tType] : fun(heap_Time_Heap(A),set(A)) ).

tff(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
    heap_Time_size_Heap: 
      !>[A: $tType] : ( fun(A,nat) > fun(heap_Time_Heap(A),nat) ) ).

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_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] : fun(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_Hilbert__Choice_Obijection,type,
    hilbert_bijection: 
      !>[A: $tType] : ( fun(A,A) > $o ) ).

tff(sy_c_Hilbert__Choice_Oinv__into,type,
    hilbert_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : fun(bool,fun(A,fun(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_Int_OAbs__Integ,type,
    abs_Integ: fun(product_prod(nat,nat),int) ).

tff(sy_c_Int_ONeg,type,
    neg: num > int ).

tff(sy_c_Int_OPos,type,
    pos: fun(num,int) ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: fun(int,product_prod(nat,nat)) ).

tff(sy_c_Int_Ocr__int,type,
    cr_int: fun(product_prod(nat,nat),fun(int,bool)) ).

tff(sy_c_Int_Odup,type,
    dup: int > int ).

tff(sy_c_Int_Oint_OAbs__int,type,
    abs_int: fun(set(product_prod(nat,nat)),int) ).

tff(sy_c_Int_Oint_ORep__int,type,
    rep_int: fun(int,set(product_prod(nat,nat))) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),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_Int_Osub,type,
    sub: ( num * num ) > int ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr,type,
    semilattice_neutr: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osemilattice__order,type,
    semilattice_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > fun(B,bool) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
    lattic5214292709420241887eutr_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set,type,
    lattic149705377957585745ce_set: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
    lattic1715443433743089157tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Lifting_OQuotient,type,
    quotient: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) * fun(A,fun(B,bool)) ) > $o ) ).

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) * 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) * 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)) * B * list(A) ) > B ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).

tff(sy_c_List_Ogen__length,type,
    gen_length: 
      !>[A: $tType] : ( nat > fun(list(A),nat) ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( set(product_prod(A,A)) > 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 ) > fun(list(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) * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : fun(set(A),list(A)) ).

tff(sy_c_List_Olinorder__class_Ostable__sort__key,type,
    linord3483353639454293061rt_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_List_Olist_Olist__all,type,
    list_all: 
      !>[A: $tType] : ( fun(A,bool) > fun(list(A),bool) ) ).

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) > fun(list(A),nat) ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Olist__ex1,type,
    list_ex1: 
      !>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).

tff(sy_c_List_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_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_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_Olexordp__eq,type,
    lexordp_eq: 
      !>[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] : ( 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_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_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_Othose,type,
    those: 
      !>[A: $tType] : ( list(option(A)) > option(list(A)) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).

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__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__comp,type,
    map_comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,option(C)) * fun(A,option(B)) * A ) > option(C) ) ).

tff(sy_c_Map_Omap__le,type,
    map_le: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > $o ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : fun(A,fun(fun(nat,A),fun(nat,A))) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred2: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,bool) ) ).

tff(sy_c_Nat_Osemiring__1__class_ONats,type,
    semiring_1_Nats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oint__decode,type,
    nat_int_decode: nat > int ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list(nat) ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun(nat,fun(nat,bool)) ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list(nat) > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: nat > fun(nat,product_prod(nat,nat)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__encode,type,
    nat_prod_encode: fun(product_prod(nat,nat),nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: set(nat) > nat ).

tff(sy_c_Nat__Bijection_Osum__decode,type,
    nat_sum_decode: nat > sum_sum(nat,nat) ).

tff(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: sum_sum(nat,nat) > nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Onat__of__num,type,
    nat_of_num: num > nat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Ois__num,type,
    neg_numeral_is_num: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun(nat,num) ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Num_Oring__1__class_Oiszero,type,
    ring_1_iszero: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Old__Datatype_OAtom,type,
    old_Atom: 
      !>[A: $tType,B: $tType] : fun(sum_sum(A,nat),set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OCase,type,
    old_Case: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(set(old_node(A,B)),C) * fun(set(old_node(A,B)),C) * set(old_node(A,B)) ) > C ) ).

tff(sy_c_Old__Datatype_OIn0,type,
    old_In0: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OIn1,type,
    old_In1: 
      !>[A: $tType,B: $tType] : ( set(old_node(A,B)) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OLeaf,type,
    old_Leaf: 
      !>[A: $tType,B: $tType] : fun(A,set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OLim,type,
    old_Lim: 
      !>[B: $tType,A: $tType] : ( fun(B,set(old_node(A,B))) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_ONode,type,
    old_Node: 
      !>[B: $tType,A: $tType] : set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))) ).

tff(sy_c_Old__Datatype_ONumb,type,
    old_Numb: 
      !>[A: $tType,B: $tType] : fun(nat,set(old_node(A,B))) ).

tff(sy_c_Old__Datatype_OPush,type,
    old_Push: 
      !>[B: $tType] : fun(sum_sum(B,nat),fun(fun(nat,sum_sum(B,nat)),fun(nat,sum_sum(B,nat)))) ).

tff(sy_c_Old__Datatype_OPush__Node,type,
    old_Push_Node: 
      !>[B: $tType,A: $tType] : ( sum_sum(B,nat) > fun(old_node(A,B),old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OScons,type,
    old_Scons: 
      !>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).

tff(sy_c_Old__Datatype_OSplit,type,
    old_Split: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)) > fun(set(old_node(A,B)),C) ) ).

tff(sy_c_Old__Datatype_Odprod,type,
    old_dprod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).

tff(sy_c_Old__Datatype_Odsum,type,
    old_dsum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).

tff(sy_c_Old__Datatype_Ondepth,type,
    old_ndepth: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).

tff(sy_c_Old__Datatype_Onode_OAbs__Node,type,
    old_Abs_Node: 
      !>[B: $tType,A: $tType] : ( product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) > old_node(A,B) ) ).

tff(sy_c_Old__Datatype_Onode_ORep__Node,type,
    old_Rep_Node: 
      !>[A: $tType,B: $tType] : ( old_node(A,B) > product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) ) ).

tff(sy_c_Old__Datatype_Ontrunc,type,
    old_ntrunc: 
      !>[A: $tType,B: $tType] : ( nat > fun(set(old_node(A,B)),set(old_node(A,B))) ) ).

tff(sy_c_Old__Datatype_Ouprod,type,
    old_uprod: 
      !>[A: $tType,B: $tType] : ( ( set(set(old_node(A,B))) * set(set(old_node(A,B))) ) > set(set(old_node(A,B))) ) ).

tff(sy_c_Old__Datatype_Ousum,type,
    old_usum: 
      !>[A: $tType,B: $tType] : ( ( set(set(old_node(A,B))) * set(set(old_node(A,B))) ) > set(set(old_node(A,B))) ) ).

tff(sy_c_Option_Obind,type,
    bind2: 
      !>[A: $tType,B: $tType] : fun(option(A),fun(fun(A,option(B)),option(B))) ).

tff(sy_c_Option_Ocombine__options,type,
    combine_options: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * option(A) * option(A) ) > option(A) ) ).

tff(sy_c_Option_Ois__none,type,
    is_none: 
      !>[A: $tType] : ( option(A) > $o ) ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : fun(B,fun(fun(A,B),fun(option(A),B))) ).

tff(sy_c_Option_Ooption_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(option(A),option(Aa))) ).

tff(sy_c_Option_Ooption_Opred__option,type,
    pred_option: 
      !>[A: $tType] : fun(fun(A,bool),fun(option(A),bool)) ).

tff(sy_c_Option_Ooption_Orec__option,type,
    rec_option: 
      !>[C: $tType,A: $tType] : fun(C,fun(fun(A,C),fun(option(A),C))) ).

tff(sy_c_Option_Ooption_Orel__option,type,
    rel_option: 
      !>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))) ).

tff(sy_c_Option_Ooption_Oset__option,type,
    set_option: 
      !>[A: $tType] : fun(option(A),set(A)) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( fun(A,nat) > fun(option(A),nat) ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__Relation_OAbove,type,
    order_Above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OAboveS,type,
    order_AboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnder,type,
    order_Under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_OUnderS,type,
    order_UnderS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_Oabove,type,
    order_above: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_OaboveS,type,
    order_aboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Opreorder__on,type,
    order_preorder_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Orelation__of,type,
    order_relation_of: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( set(product_prod(A,A)) > 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(A,bool) ) > A ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oordering,type,
    ordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * A ) > $o ) ).

tff(sy_c_Orderings_Opreordering,type,
    preordering: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( ( A * set(A) ) > A ) ).

tff(sy_c_Partial__Function_Oimg__ord,type,
    partial_img_ord: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(C,fun(C,B)) * A * A ) > B ) ).

tff(sy_c_Partial__Function_Omk__less,type,
    partial_mk_less: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * A * A ) > $o ) ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,A)) ).

tff(sy_c_Predicate_OSeq,type,
    seq2: 
      !>[A: $tType] : ( fun(product_unit,seq(A)) > pred(A) ) ).

tff(sy_c_Predicate_Oadjunct,type,
    adjunct: 
      !>[A: $tType] : ( ( pred(A) * seq(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oapply,type,
    apply: 
      !>[A: $tType,B: $tType] : ( ( fun(A,pred(B)) * seq(A) ) > seq(B) ) ).

tff(sy_c_Predicate_Obind,type,
    bind3: 
      !>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).

tff(sy_c_Predicate_Ocontained,type,
    contained: 
      !>[A: $tType] : ( ( seq(A) * pred(A) ) > $o ) ).

tff(sy_c_Predicate_Oiterate__upto,type,
    iterate_upto: 
      !>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > pred(A) ) ).

tff(sy_c_Predicate_Oiterate__upto__rel,type,
    iterate_upto_rel: 
      !>[A: $tType] : fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),fun(product_prod(fun(code_natural,A),product_prod(code_natural,code_natural)),bool)) ).

tff(sy_c_Predicate_Omap,type,
    map2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(pred(A),pred(B)) ) ).

tff(sy_c_Predicate_Opred_Ocase__pred,type,
    case_pred: 
      !>[A: $tType,B: $tType] : ( ( fun(fun(A,bool),B) * pred(A) ) > B ) ).

tff(sy_c_Predicate_Opred_Oeval,type,
    eval: 
      !>[A: $tType] : fun(pred(A),fun(A,bool)) ).

tff(sy_c_Predicate_Opred__of__seq,type,
    pred_of_seq: 
      !>[A: $tType] : fun(seq(A),pred(A)) ).

tff(sy_c_Predicate_Oseq_OEmpty,type,
    empty: 
      !>[A: $tType] : seq(A) ).

tff(sy_c_Predicate_Oseq_OInsert,type,
    insert: 
      !>[A: $tType] : ( ( A * pred(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oseq_OJoin,type,
    join: 
      !>[A: $tType] : ( ( pred(A) * seq(A) ) > seq(A) ) ).

tff(sy_c_Predicate_Oseq_Ocase__seq,type,
    case_seq: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(pred(A),B)) * fun(pred(A),fun(seq(A),B)) * seq(A) ) > B ) ).

tff(sy_c_Predicate_Oset__of__pred,type,
    set_of_pred: 
      !>[A: $tType] : fun(pred(A),set(A)) ).

tff(sy_c_Predicate_Oset__of__seq,type,
    set_of_seq: 
      !>[A: $tType] : fun(seq(A),set(A)) ).

tff(sy_c_Predicate_Osingle,type,
    single: 
      !>[A: $tType] : fun(A,pred(A)) ).

tff(sy_c_Predicate_Othe,type,
    the3: 
      !>[A: $tType] : ( pred(A) > A ) ).

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

tff(sy_c_Product__Type_OSigma,type,
    product_Sigma: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

tff(sy_c_Product__Type_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Omap__prod,type,
    product_map_prod: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).

tff(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
    product_rec_bool: 
      !>[T: $tType] : ( ( T * T * bool ) > T ) ).

tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
    product_rec_set_bool: 
      !>[T: $tType] : ( ( T * T * bool ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
    product_rec_set_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).

tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
    product_rec_unit: 
      !>[T: $tType] : ( ( T * product_unit ) > T ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oprod_Oswap,type,
    product_swap: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Product__Type_Oscomp,type,
    product_scomp: 
      !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).

tff(sy_c_Product__Type_Ounit_Ocase__unit,type,
    product_case_unit: 
      !>[A: $tType] : ( ( A * product_unit ) > A ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quotient_OQuotient3,type,
    quotient3: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) ) > $o ) ).

tff(sy_c_Quotient_Oquot__type,type,
    quot_type: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),B) * fun(B,set(A)) ) > $o ) ).

tff(sy_c_Quotient_Oquot__type_Oabs,type,
    quot_abs: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),B) ) > fun(A,B) ) ).

tff(sy_c_Quotient_Oquot__type_Orep,type,
    quot_rep: 
      !>[B: $tType,A: $tType] : ( fun(B,set(A)) > fun(B,A) ) ).

tff(sy_c_Random_Oiterate,type,
    iterate: 
      !>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).

tff(sy_c_Random_Oiterate__rel,type,
    iterate_rel: 
      !>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).

tff(sy_c_Random_Olog,type,
    log: ( code_natural * code_natural ) > code_natural ).

tff(sy_c_Random_Olog__rel,type,
    log_rel: fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),bool)) ).

tff(sy_c_Random_Opick,type,
    pick: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(code_natural,A) ) ).

tff(sy_c_Random_Orange,type,
    range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).

tff(sy_c_Random_Oselect,type,
    select: 
      !>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Oselect__weight,type,
    select_weight: 
      !>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random_Osplit__seed,type,
    split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).

tff(sy_c_Random__Pred_ORandom,type,
    random_Random: 
      !>[A: $tType] : ( fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Obind,type,
    random_bind: 
      !>[A: $tType,B: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))) ) > fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Oempty,type,
    random_empty: 
      !>[A: $tType] : fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ).

tff(sy_c_Random__Pred_Oiterate__upto,type,
    random_iterate_upto: 
      !>[A: $tType] : ( ( fun(code_natural,A) * code_natural * code_natural ) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Omap,type,
    random_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) > fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))) ) ).

tff(sy_c_Random__Pred_Onot__randompred,type,
    random6974930770145893639ompred: ( fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(product_unit),product_prod(code_natural,code_natural)) ).

tff(sy_c_Random__Pred_Osingle,type,
    random_single: 
      !>[A: $tType] : fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))) ).

tff(sy_c_Random__Pred_Ounion,type,
    random_union: 
      !>[A: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(A),product_prod(code_natural,code_natural)) ) ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun(product_prod(int,int),rat) ).

tff(sy_c_Rat_OFract,type,
    fract: fun(int,fun(int,rat)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun(rat,product_prod(int,int)) ).

tff(sy_c_Rat_Ocr__rat,type,
    cr_rat: ( product_prod(int,int) * rat ) > $o ).

tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
    field_char_0_Rats: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
    field_char_0_of_rat: 
      !>[A: $tType] : fun(rat,A) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,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_Orat_OAbs__rat,type,
    abs_rat: fun(set(product_prod(int,int)),rat) ).

tff(sy_c_Rat_Orat_ORep__rat,type,
    rep_rat: fun(rat,set(product_prod(int,int))) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),bool)) ).

tff(sy_c_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
    tuple_1188178415141063261rphism: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,product_prod(B,C)) * fun(product_prod(B,C),A) ) > tuple_isomorphism(A,B,C) ) ).

tff(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
    tuple_9181185373184732606rphism: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,nat) * fun(B,nat) * fun(C,nat) * tuple_isomorphism(A,B,C) ) > nat ) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(A)) ).

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] : ( set(product_prod(A,A)) > set(A) ) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_OImage,type,
    image: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > fun(set(A),set(B)) ) ).

tff(sy_c_Relation_OPowp,type,
    powp: 
      !>[A: $tType] : ( ( fun(A,bool) * set(A) ) > $o ) ).

tff(sy_c_Relation_ORange,type,
    range2: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(B)) ).

tff(sy_c_Relation_ORangep,type,
    rangep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oantisymp,type,
    antisymp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Oasym,type,
    asym: 
      !>[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_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Oreflp,type,
    reflp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Orelcompp,type,
    relcompp: 
      !>[A: $tType,B: $tType,C: $tType] : fun(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_Osingle__valuedp,type,
    single_valuedp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Otrans,type,
    trans: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_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_Onormalization__semidom__class_Onormalize,type,
    normal6383669964737779283malize: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
    unit_f5069060285200089521factor: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,A) ).

tff(sy_c_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,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Obind,type,
    bind4: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),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,
    insert2: 
      !>[A: $tType] : ( A > fun(set(A),set(A)) ) ).

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_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__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_OLiteral,type,
    literal2: fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))))) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: fun(char,char) ).

tff(sy_c_String_Oasciis__of__literal,type,
    asciis_of_literal: literal > list(code_integer) ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

tff(sy_c_String_Ochar_Ocase__char,type,
    case_char: 
      !>[A: $tType] : ( ( fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))) * char ) > A ) ).

tff(sy_c_String_Ochar_Odigit0,type,
    digit0: char > $o ).

tff(sy_c_String_Ochar_Odigit1,type,
    digit1: char > $o ).

tff(sy_c_String_Ochar_Odigit2,type,
    digit2: char > $o ).

tff(sy_c_String_Ochar_Odigit3,type,
    digit3: char > $o ).

tff(sy_c_String_Ochar_Odigit4,type,
    digit4: char > $o ).

tff(sy_c_String_Ochar_Odigit5,type,
    digit5: char > $o ).

tff(sy_c_String_Ochar_Odigit6,type,
    digit6: char > $o ).

tff(sy_c_String_Ochar_Odigit7,type,
    digit7: char > $o ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ocr__literal,type,
    cr_literal: fun(list(char),fun(literal,bool)) ).

tff(sy_c_String_Oliteral_OAbs__literal,type,
    abs_literal: fun(list(char),literal) ).

tff(sy_c_String_Oliteral_Oexplode,type,
    explode: fun(literal,list(char)) ).

tff(sy_c_String_Oliteral__of__asciis,type,
    literal_of_asciis: fun(list(code_integer),literal) ).

tff(sy_c_String_Opcr__literal,type,
    pcr_literal: fun(list(char),fun(literal,bool)) ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Sum__Type_OInl,type,
    sum_Inl: 
      !>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OInr,type,
    sum_Inr: 
      !>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_Sum__Type_Omap__sum,type,
    sum_map_sum: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(sum_sum(A,B),sum_sum(C,D)) ) ).

tff(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
    sum_rec_set_sum: 
      !>[A: $tType,T: $tType,B: $tType] : ( ( fun(A,T) * fun(B,T) * sum_sum(A,B) ) > fun(T,bool) ) ).

tff(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
    sum_rec_sum: 
      !>[A: $tType,T: $tType,B: $tType] : ( ( fun(A,T) * fun(B,T) * sum_sum(A,B) ) > T ) ).

tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
    sum_case_sum: 
      !>[A: $tType,C: $tType,B: $tType] : fun(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C))) ).

tff(sy_c_Sum__Type_Osum_Oprojl,type,
    sum_projl: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > A ) ).

tff(sy_c_Sum__Type_Osum_Oprojr,type,
    sum_projr: 
      !>[A: $tType,B: $tType] : ( sum_sum(A,B) > B ) ).

tff(sy_c_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_Oright__total,type,
    right_total: 
      !>[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_Oacyclic,type,
    transitive_acyclic: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortranclp,type,
    transitive_rtranclp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Otranclp,type,
    transitive_tranclp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).

tff(sy_c_Typedef_Otype__definition,type,
    type_definition: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: ( literal * list(typerep) ) > typerep ).

tff(sy_c_Typerep_Otyperep_Orec__typerep,type,
    rec_typerep: 
      !>[A: $tType] : ( ( fun(literal,fun(list(product_prod(typerep,A)),A)) * typerep ) > A ) ).

tff(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: fun(typerep,nat) ).

tff(sy_c_Wellfounded_Oacc,type,
    acc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).

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_Oadm__wf,type,
    adm_wf: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $o ) ).

tff(sy_c_Wfrec_Ocut,type,
    cut: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,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_Zorn_Opred__on_Omaxchain,type,
    pred_maxchain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > $o ) ).

tff(sy_c_Zorn_Opred__on_Osuc,type,
    pred_suc: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) * set(A) ) > set(A) ) ).

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_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( A > fun(set(A),bool) ) ).

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_P,type,
    p: fun(a,bool) ).

tff(sy_v_h,type,
    h: heap_ext(product_unit) ).

tff(sy_v_x,type,
    x: a ).

% Relevant facts (8413)
tff(fact_0_success__ifI,axiom,
    ! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H: heap_ext(product_unit),E2: heap_Time_Heap(A)] :
      ( ( pp(C2)
       => heap_Time_success(A,T2,H) )
     => ( ( ~ pp(C2)
         => heap_Time_success(A,E2,H) )
       => heap_Time_success(A,aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),aa(bool,fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A))),if(heap_Time_Heap(A)),C2),T2),E2),H) ) ) ).

% success_ifI
tff(fact_1_success__tapI,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_tap(A,F),H) ).

% success_tapI
tff(fact_2_success__guardI,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( pp(aa(heap_ext(product_unit),bool,P,H))
     => heap_Time_success(A,heap_Time_guard(A,P,F),H) ) ).

% success_guardI
tff(fact_3_success__ureturnI,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H) ).

% success_ureturnI
tff(fact_4_success__heapI,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit)] : heap_Time_success(A,heap_Time_heap(A,F),H) ).

% success_heapI
tff(fact_5_success__returnI,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H) ).

% success_returnI
tff(fact_6_effect__success,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R,N)
     => heap_Time_success(A,C2,H) ) ).

% effect_success
tff(fact_7_success__effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,C2,H)
     => ~ ! [R2: A,H3: heap_ext(product_unit),N2: nat] : ~ heap_Time_effect(A,C2,H,H3,R2,N2) ) ).

% success_effectE
tff(fact_8_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_9_success__bind__effectI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),X: A,N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_effect(A,F,H,H2,X,N)
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X),H2)
       => heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).

% success_bind_effectI
tff(fact_10_successI,axiom,
    ! [A: $tType,F: 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,F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => heap_Time_success(A,F,H) ) ).

% successI
tff(fact_11_success__def,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% success_def
tff(fact_12_success__LetI,axiom,
    ! [A: $tType,B: $tType,X: A,T2: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( ( X = T2 )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F,X),H)
       => heap_Time_success(B,aa(A,heap_Time_Heap(B),F,T2),H) ) ) ).

% success_LetI
tff(fact_13_Heap__Time__Monad_Obind__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: heap_Time_Heap(A),G: fun(A,heap_Time_Heap(C)),K: fun(C,heap_Time_Heap(B))] : heap_Time_bind(C,B,heap_Time_bind(A,C,F,G),K) = heap_Time_bind(A,B,F,aa(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aTP_Lamp_a(fun(A,heap_Time_Heap(C)),fun(fun(C,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_14_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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,F,G)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_15_execute__guard_I1_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: 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,F)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_guard(1)
tff(fact_16_effect__deterministic_I3_J,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N3: nat] :
      ( heap_Time_effect(A,F,H,H2,A3,N)
     => ( heap_Time_effect(A,F,H,H4,B2,N3)
       => ( N = N3 ) ) ) ).

% effect_deterministic(3)
tff(fact_17_effect__deterministic_I2_J,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N3: nat] :
      ( heap_Time_effect(A,F,H,H2,A3,N)
     => ( heap_Time_effect(A,F,H,H4,B2,N3)
       => ( H2 = H4 ) ) ) ).

% effect_deterministic(2)
tff(fact_18_effect__deterministic_I1_J,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H4: heap_ext(product_unit),B2: A,N3: nat] :
      ( heap_Time_effect(A,F,H,H2,A3,N)
     => ( heap_Time_effect(A,F,H,H4,B2,N3)
       => ( A3 = B2 ) ) ) ).

% effect_deterministic(1)
tff(fact_19_Heap__eqI,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),G: heap_Time_Heap(A)] :
      ( ! [H5: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H5) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,G),H5)
     => ( F = G ) ) ).

% Heap_eqI
tff(fact_20_effect__ifE,axiom,
    ! [A: $tType,C2: bool,T2: heap_Time_Heap(A),E2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),aa(bool,fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A))),if(heap_Time_Heap(A)),C2),T2),E2),H,H2,R,N)
     => ( ( pp(C2)
         => ~ heap_Time_effect(A,T2,H,H2,R,N) )
       => ~ ( ~ pp(C2)
           => ~ heap_Time_effect(A,E2,H,H2,R,N) ) ) ) ).

% effect_ifE
tff(fact_21_effect__ifI,axiom,
    ! [A: $tType,C2: bool,T2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat,E2: heap_Time_Heap(A)] :
      ( ( pp(C2)
       => heap_Time_effect(A,T2,H,H2,R,N) )
     => ( ( ~ pp(C2)
         => heap_Time_effect(A,E2,H,H2,R,N) )
       => heap_Time_effect(A,aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),aa(bool,fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),heap_Time_Heap(A))),if(heap_Time_Heap(A)),C2),T2),E2),H,H2,R,N) ) ) ).

% effect_ifI
tff(fact_22_effect__LetI,axiom,
    ! [B: $tType,A: $tType,X: A,T2: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: B,N: nat] :
      ( ( X = T2 )
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,X),H,H2,R,N)
       => heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,T2),H,H2,R,N) ) ) ).

% effect_LetI
tff(fact_23_bind__ureturn,axiom,
    ! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_ureturn(A)) = F ).

% bind_ureturn
tff(fact_24_ureturn__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),X),F) = aa(B,heap_Time_Heap(A),F,X) ).

% ureturn_bind
tff(fact_25_execute__raise,axiom,
    ! [A: $tType,S: list(char),X2: 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,S)),X2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% execute_raise
tff(fact_26_raise__bind,axiom,
    ! [B: $tType,A: $tType,E2: list(char),F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_raise(B,E2),F) = heap_Time_raise(A,E2) ).

% raise_bind
tff(fact_27_effect__tapE,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_tap(A,F),H,H2,R,N)
     => ~ ( ( H2 = H )
         => ( ( R = aa(heap_ext(product_unit),A,F,H) )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_tapE
tff(fact_28_effect__tapI,axiom,
    ! [A: $tType,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,F: fun(heap_ext(product_unit),A)] :
      ( ( H2 = H )
     => ( ( R = aa(heap_ext(product_unit),A,F,H) )
       => heap_Time_effect(A,heap_Time_tap(A,F),H,H2,R,one_one(nat)) ) ) ).

% effect_tapI
tff(fact_29_effect__raiseE,axiom,
    ! [A: $tType,X: list(char),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] : ~ heap_Time_effect(A,heap_Time_raise(A,X),H,H2,R,N) ).

% effect_raiseE
tff(fact_30_effect__ureturnE,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H,H2,R,N)
     => ~ ( ( R = X )
         => ( ( H2 = H )
           => ( N != zero_zero(nat) ) ) ) ) ).

% effect_ureturnE
tff(fact_31_effect__ureturnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),X: A] :
      ( ( H = H2 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X),H,H2,X,zero_zero(nat)) ) ).

% effect_ureturnI
tff(fact_32_effect__returnE,axiom,
    ! [A: $tType,X: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H,H2,R,N)
     => ~ ( ( R = X )
         => ( ( H2 = H )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_returnE
tff(fact_33_effect__returnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),X: A] :
      ( ( H = H2 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),X),H,H2,X,one_one(nat)) ) ).

% effect_returnI
tff(fact_34_execute__guard_I2_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: 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,F)),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)),F,H)) ) ) ).

% execute_guard(2)
tff(fact_35_effect__bindE,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit),H4: heap_ext(product_unit),R3: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_bind(B,A,F,G),H,H4,R3,N)
     => ~ ! [H3: heap_ext(product_unit),R2: B,N1: nat] :
            ( heap_Time_effect(B,F,H,H3,R2,N1)
           => ! [N22: nat] :
                ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G,R2),H3,H4,R3,N22)
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_36_option_Oinject,axiom,
    ! [A: $tType,X22: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X22) = aa(A,option(A),some(A),Y2) )
    <=> ( X22 = Y2 ) ) ).

% option.inject
tff(fact_37_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y: A] : X != aa(A,option(A),some(A),Y)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_38_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y: A] : X = aa(A,option(A),some(A),Y) ) ).

% not_None_eq
tff(fact_39_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y3: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y3)) )
     => ( ( ( Y3 = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y3)) )
       => ( ! [A4: A,B3: B] :
              ( ( X = aa(A,option(A),some(A),A4) )
             => ( ( Y3 = aa(B,option(B),some(B),B3) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y3)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y3)) ) ) ) ).

% combine_options_cases
tff(fact_40_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X3: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X3))) ) ) ).

% split_option_all
tff(fact_41_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X3: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X3))) ) ) ).

% split_option_ex
tff(fact_42_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_43_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_aa(set(A),fun(A,bool)),A5)) = A5 ).

% Collect_mem_eq
tff(fact_44_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
        <=> pp(aa(A,bool,Q,X4)) )
     => ( aa(fun(A,bool),set(A),collect(A),P) = aa(fun(A,bool),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_45_ext,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
      ( ! [X4: A] : aa(A,B,F,X4) = aa(A,B,G,X4)
     => ( F = G ) ) ).

% ext
tff(fact_46_option_Oexhaust,axiom,
    ! [A: $tType,Y3: option(A)] :
      ( ( Y3 != none(A) )
     => ~ ! [X23: A] : Y3 != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_47_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X22: A] :
      ( ( Option = aa(A,option(A),some(A),X22) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_48_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).

% option.distinct(1)
tff(fact_49_effect__bindI,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat,G: fun(A,heap_Time_Heap(B)),H4: heap_ext(product_unit),R3: B,N3: nat] :
      ( heap_Time_effect(A,F,H,H2,R,N)
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G,R),H2,H4,R3,N3)
       => heap_Time_effect(B,heap_Time_bind(A,B,F,G),H,H4,R3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3)) ) ) ).

% effect_bindI
tff(fact_50_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_51_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_52_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_53_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_54_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) )
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_55_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_56_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_57_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_58_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_59_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = A3 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_60_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_61_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_62_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_63_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_64_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_65_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_66_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_67_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: A,K: A,A3: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_68_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_69_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_70_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_71_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_72_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) ) ).

% add.commute
tff(fact_73_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.left_commute
tff(fact_74_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_75_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_76_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_77_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_78_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_79_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_80_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_81_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_82_bind__return,axiom,
    ! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_ab(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F)) ).

% bind_return
tff(fact_83_return__bind,axiom,
    ! [A: $tType,B: $tType,X: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),X),F) = 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_ac(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),X),F)) ).

% return_bind
tff(fact_84_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A3: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B3),A4)) )
     => ( ! [A4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),zero_zero(nat)))
       => ( ! [A4: nat,B3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),B3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A3),B2)) ) ) ) ).

% Euclid_induct
tff(fact_85_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_86_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_87_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_88_guard__def,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),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_ad(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),F)) ).

% guard_def
tff(fact_89_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F: fun(fun(A,B),C),G: C] :
          ( ! [X4: fun(A,B)] : aa(fun(A,B),C,F,X4) = G
         => ( aa(fun(A,B),C,F,aTP_Lamp_ae(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_90_raise__def,axiom,
    ! [A: $tType,S: list(char)] : heap_Time_raise(A,S) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_af(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% raise_def
tff(fact_91_successE,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F,H)
     => ~ ! [R2: A,H3: 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,F),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)),R2),H3)) ) ).

% successE
tff(fact_92_Heap__cases,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ! [X4: A,H3: 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,F),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)),X4),H3))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_93_execute__heap,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_execute(A,heap_Time_heap(A,F)) = 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)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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))))),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)))),F) ).

% execute_heap
tff(fact_94_Heap_Oinject,axiom,
    ! [A: $tType,X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Ya: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ( aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Ya) )
    <=> ( X = Ya ) ) ).

% Heap.inject
tff(fact_95_Heap__execute,axiom,
    ! [A: $tType,F: heap_Time_Heap(A)] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),heap_Time_execute(A,F)) = F ).

% Heap_execute
tff(fact_96_rewriteR__comp__comp2,axiom,
    ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R1: fun(D,B),R22: fun(A,D),F: fun(B,E),L: fun(D,E)] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),R1),R22) )
     => ( ( aa(fun(D,B),fun(D,E),aa(fun(B,E),fun(fun(D,B),fun(D,E)),comp(B,E,D),F),R1) = L )
       => ( aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),F),G)),H) = aa(fun(A,D),fun(A,E),aa(fun(D,E),fun(fun(A,D),fun(A,E)),comp(D,E,A),L),R22) ) ) ) ).

% rewriteR_comp_comp2
tff(fact_97_rewriteL__comp__comp2,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: fun(C,B),G: fun(A,C),L1: fun(D,B),L2: fun(A,D),H: fun(E,A),R: fun(E,D)] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),L1),L2) )
     => ( ( aa(fun(E,A),fun(E,D),aa(fun(A,D),fun(fun(E,A),fun(E,D)),comp(A,D,E),L2),H) = R )
       => ( aa(fun(E,C),fun(E,B),aa(fun(C,B),fun(fun(E,C),fun(E,B)),comp(C,B,E),F),aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),G),H)) = aa(fun(E,D),fun(E,B),aa(fun(D,B),fun(fun(E,D),fun(E,B)),comp(D,B,E),L1),R) ) ) ) ).

% rewriteL_comp_comp2
tff(fact_98_rewriteR__comp__comp,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R: fun(A,B),F: fun(B,D)] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = R )
     => ( aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),F),G)),H) = aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),R) ) ) ).

% rewriteR_comp_comp
tff(fact_99_rewriteL__comp__comp,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,B),G: fun(A,C),L: fun(A,B),H: fun(D,A)] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = L )
     => ( aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),G),H)) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),L),H) ) ) ).

% rewriteL_comp_comp
tff(fact_100_Heap_Oexhaust,axiom,
    ! [A: $tType,Y3: heap_Time_Heap(A)] :
      ~ ! [X4: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : Y3 != aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X4) ).

% Heap.exhaust
tff(fact_101_heap__def,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),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)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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))))),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)))),F)) ).

% heap_def
tff(fact_102_execute_Osimps,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_execute(A,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),F)) = F ).

% execute.simps
tff(fact_103_comp__apply,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X: C] : aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X) = aa(B,A,F,aa(C,B,G,X)) ).

% comp_apply
tff(fact_104_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: 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),A6),B5) )
    <=> ( ( A3 = A6 )
        & ( B2 = B5 ) ) ) ).

% old.prod.inject
tff(fact_105_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
tff(fact_106_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)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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))))),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_ag(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_return
tff(fact_107_execute__ureturn,axiom,
    ! [A: $tType,X: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(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)))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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))))),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_ah(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% execute_ureturn
tff(fact_108_prod__induct7,axiom,
    ! [G2: $tType,F2: $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(F2,G2)))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))] :
      ( ! [A4: A,B3: B,C3: C,D2: D,E3: E,F3: F2,G3: G2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C3),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E3),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3))))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),bool,P,X)) ) ).

% prod_induct7
tff(fact_109_prod__induct6,axiom,
    ! [F2: $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,F2))))),bool),X: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
      ( ! [A4: A,B3: B,C3: C,D2: D,E3: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E3),F3)))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P,X)) ) ).

% prod_induct6
tff(fact_110_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))))] :
      ( ! [A4: A,B3: B,C3: C,D2: D,E3: 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)))),A4),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))),B3),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),E3))))))
     => pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P,X)) ) ).

% prod_induct5
tff(fact_111_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)))] :
      ( ! [A4: A,B3: 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))),A4),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)),B3),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_112_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))] :
      ( ! [A4: A,B3: 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)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3))))
     => pp(aa(product_prod(A,product_prod(B,C)),bool,P,X)) ) ).

% prod_induct3
tff(fact_113_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))] :
      ~ ! [A4: A,B3: B,C3: C,D2: D,E3: E,F3: F2,G3: G2] : Y3 != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C3),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E3),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3)))))) ).

% prod_cases7
tff(fact_114_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
      ~ ! [A4: A,B3: B,C3: C,D2: D,E3: E,F3: F2] : Y3 != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E3),F3))))) ).

% prod_cases6
tff(fact_115_timeFrame_Ocases,axiom,
    ! [A: $tType,X: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N2: nat,R2: A,H5: 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)))),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)),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),H5),N4))))
     => ~ ! [N2: 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)))),N2),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

% timeFrame.cases
tff(fact_116_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y3: product_prod(A,B)] :
      ~ ! [A4: A,B3: B] : Y3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) ).

% old.prod.exhaust
tff(fact_117_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X4: A,Y4: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) ).

% surj_pair
tff(fact_118_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool),P2: product_prod(A,B)] :
      ( ! [A4: A,B3: 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),A4),B3)))
     => pp(aa(product_prod(A,B),bool,P,P2)) ) ).

% prod_cases
tff(fact_119_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: 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),A6),B5) )
     => ~ ( ( A3 = A6 )
         => ( B2 != B5 ) ) ) ).

% Pair_inject
tff(fact_120_comp__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B),X2: A] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G),X2) = aa(B,C,F,aa(A,B,G,X2)) ).

% comp_def
tff(fact_121_comp__assoc,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,F: fun(D,B),G: fun(C,D),H: fun(A,C)] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aa(fun(C,D),fun(C,B),aa(fun(D,B),fun(fun(C,D),fun(C,B)),comp(D,B,C),F),G)),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),G),H)) ).

% comp_assoc
tff(fact_122_comp__eq__dest,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(D,B),D3: fun(A,D),V: A] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C2),D3) )
     => ( aa(C,B,A3,aa(A,C,B2,V)) = aa(D,B,C2,aa(A,D,D3,V)) ) ) ).

% comp_eq_dest
tff(fact_123_comp__eq__elim,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(D,B),D3: fun(A,D)] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C2),D3) )
     => ! [V2: A] : aa(C,B,A3,aa(A,C,B2,V2)) = aa(D,B,C2,aa(A,D,D3,V2)) ) ).

% comp_eq_elim
tff(fact_124_comp__eq__dest__lhs,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(A,B),V: A] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = C2 )
     => ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C2,V) ) ) ).

% comp_eq_dest_lhs
tff(fact_125_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),R: A,H2: heap_ext(product_unit),N: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),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),H2),N))) )
     => heap_Time_effect(A,C2,H,H2,R,N) ) ).

% effectI
tff(fact_126_effect__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R,N)
    <=> ( 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),H2),N))) ) ) ).

% effect_def
tff(fact_127_ureturn__def,axiom,
    ! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),X) = heap_Time_heap(A,aTP_Lamp_ah(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% ureturn_def
tff(fact_128_return__def,axiom,
    ! [A: $tType,X: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),X) = heap_Time_heap(A,aTP_Lamp_ag(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),X)) ).

% return_def
tff(fact_129_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y3: B,H4: heap_ext(product_unit),N3: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H2),N))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,X)),H2) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y3),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),N3))) )
       => ( 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,F,G)),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)),Y3),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),N),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_130_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H2),N))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,X),H2)
       => heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).

% success_bind_executeI
tff(fact_131_execute__tap,axiom,
    ! [A: $tType,F: 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,F)),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,F,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_132_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_133_tap__def,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_ai(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F)) ).

% tap_def
tff(fact_134_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y3: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B3: B,C3: C] : Y3 != 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)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).

% prod_cases3
tff(fact_135_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B3: B,C3: C,D2: D] : Y3 != 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))),A4),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)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).

% prod_cases4
tff(fact_136_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B3: B,C3: C,D2: D,E3: E] : Y3 != 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)))),A4),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))),B3),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),E3)))) ).

% prod_cases5
tff(fact_137_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_138_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),X: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,F),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),H2),N))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,F,G)),H) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,X)),H2)) ) ) ).

% execute_bind(1)
tff(fact_139_timeFrame_Oelims,axiom,
    ! [A: $tType,X: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y3: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,X,Xa) = Y3 )
     => ( ! [R2: A,H5: heap_ext(product_unit),N4: nat] :
            ( ( Xa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H5),N4))) )
           => ( Y3 != 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),H5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),N4)))) ) )
       => ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
           => ( Y3 != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).

% timeFrame.elims
tff(fact_140_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,N: nat,R: A,H: heap_ext(product_unit),N3: nat] : heap_Time_timeFrame(A,N,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H),N3)))) = 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),H),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3)))) ).

% timeFrame.simps(1)
tff(fact_141_K__record__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,C2: B,F: fun(A,C),X2: A] : aa(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aTP_Lamp_aj(B,fun(C,B),C2)),F),X2) = C2 ).

% K_record_comp
tff(fact_142_fun_Omap__ident,axiom,
    ! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),aTP_Lamp_ak(A,A)),T2) = T2 ).

% fun.map_ident
tff(fact_143_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X3: A,Xa2: 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),Xa2)),R4))
        <=> 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),Xa2)),S2)) )
    <=> ( R4 = S2 ) ) ).

% pred_equals_eq2
tff(fact_144_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_145_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R))
        <=> ( R = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_146_wait__def,axiom,
    ! [N: nat] : heap_Time_wait(N) = aa(fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(product_unit),heap_Time_Heap2(product_unit),aTP_Lamp_al(nat,fun(heap_ext(product_unit),option(product_prod(product_unit,product_prod(heap_ext(product_unit),nat)))),N)) ).

% wait_def
tff(fact_147_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_148_fun_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,G: fun(B,C),F: fun(A,B),V: fun(D,A)] : aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)) = aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).

% fun.map_comp
tff(fact_149_unit__abs__eta__conv,axiom,
    ! [A: $tType,F: fun(product_unit,A)] : aTP_Lamp_am(fun(product_unit,A),fun(product_unit,A),F) = F ).

% unit_abs_eta_conv
tff(fact_150_timeFrame__zero,axiom,
    ! [A: $tType,H: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,zero_zero(nat),H) = H ).

% timeFrame_zero
tff(fact_151_timeFrame__assoc,axiom,
    ! [A: $tType,N: nat,N3: nat,F: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,N,heap_Time_timeFrame(A,N3,F)) = heap_Time_timeFrame(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N3),F) ).

% timeFrame_assoc
tff(fact_152_old_Ounit_Oexhaust,axiom,
    ! [Y3: product_unit] : Y3 = product_Unity ).

% old.unit.exhaust
tff(fact_153_unit__all__impI,axiom,
    ! [P: fun(product_unit,bool),Q: fun(product_unit,bool)] :
      ( ( pp(aa(product_unit,bool,P,product_Unity))
       => pp(aa(product_unit,bool,Q,product_Unity)) )
     => ! [X2: product_unit] :
          ( pp(aa(product_unit,bool,P,X2))
         => pp(aa(product_unit,bool,Q,X2)) ) ) ).

% unit_all_impI
tff(fact_154_eq__sym__Unity__conv,axiom,
    ! [X: bool] :
      ( ( pp(X)
      <=> ( product_Unity = product_Unity ) )
    <=> pp(X) ) ).

% eq_sym_Unity_conv
tff(fact_155_timeFrame_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat] : heap_Time_timeFrame(A,N,none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% timeFrame.simps(2)
tff(fact_156_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_157_old_Ounit_Orec,axiom,
    ! [T: $tType,F1: T] : product_rec_unit(T,F1,product_Unity) = F1 ).

% old.unit.rec
tff(fact_158_default__unit__def,axiom,
    default_default(product_unit) = product_Unity ).

% default_unit_def
tff(fact_159_timeFrame_Opelims,axiom,
    ! [A: $tType,X: nat,Xa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y3: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,X,Xa) = Y3 )
     => ( 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),Xa)))
       => ( ! [R2: A,H5: heap_ext(product_unit),N4: nat] :
              ( ( Xa = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),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),H5),N4))) )
             => ( ( Y3 = 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),H5),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)),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),H5),N4)))))) ) )
         => ~ ( ( Xa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y3 = 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_160_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F3: fun(nat,fun(A,A)),A4: nat,B3: 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))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_161_type__copy__map__cong0,axiom,
    ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M2: fun(B,A),G: fun(C,B),X: C,N5: fun(D,A),H: fun(C,D),F: fun(A,E)] :
      ( ( aa(B,A,M2,aa(C,B,G,X)) = aa(D,A,N5,aa(C,D,H,X)) )
     => ( aa(C,E,aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),aa(fun(B,A),fun(B,E),aa(fun(A,E),fun(fun(B,A),fun(B,E)),comp(A,E,B),F),M2)),G),X) = aa(C,E,aa(fun(C,D),fun(C,E),aa(fun(D,E),fun(fun(C,D),fun(C,E)),comp(D,E,C),aa(fun(D,A),fun(D,E),aa(fun(A,E),fun(fun(D,A),fun(D,E)),comp(A,E,D),F),N5)),H),X) ) ) ).

% type_copy_map_cong0
tff(fact_162_comp__apply__eq,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X: C,H: fun(D,A),K: fun(C,D)] :
      ( ( aa(B,A,F,aa(C,B,G,X)) = aa(D,A,H,aa(C,D,K,X)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X) = aa(C,A,aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),H),K),X) ) ) ).

% comp_apply_eq
tff(fact_163_comp__cong,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: fun(B,A),G: fun(C,B),X: C,F4: fun(D,A),G4: fun(E,D),X5: E] :
      ( ( aa(B,A,F,aa(C,B,G,X)) = aa(D,A,F4,aa(E,D,G4,X5)) )
     => ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X) = aa(E,A,aa(fun(E,D),fun(E,A),aa(fun(D,A),fun(fun(E,D),fun(E,A)),comp(D,A,E),F4),G4),X5) ) ) ).

% comp_cong
tff(fact_164_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_165_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_166_old_Ounit_Ocase,axiom,
    ! [A: $tType,F: A] : product_case_unit(A,F,product_Unity) = F ).

% old.unit.case
tff(fact_167_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R: A,S: B,R4: set(product_prod(A,B)),S3: 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),R),S)),R4))
     => ( ( S3 = 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),R),S3)),R4)) ) ) ).

% ssubst_Pair_rhs
tff(fact_168_unit_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F: A,Unit: product_unit] : aa(A,B,H,product_case_unit(A,F,Unit)) = product_case_unit(B,aa(A,B,H,F),Unit) ).

% unit.case_distrib
tff(fact_169_case__unit__Unity,axiom,
    ! [A: $tType,F: A,U: product_unit] : product_case_unit(A,F,U) = F ).

% case_unit_Unity
tff(fact_170_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,member(A,zero_zero(A)),ring_1_Ints(A))) ) ).

% Ints_0
tff(fact_171_Ints__add,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),plus_plus(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_172_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,member(A,one_one(A)),ring_1_Ints(A))) ) ).

% Ints_1
tff(fact_173_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_174_dbl__inc__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,zero_zero(A)) = one_one(A) ) ) ).

% dbl_inc_simps(2)
tff(fact_175_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_176_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_177_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_178_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_179_dbl__dec__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).

% dbl_dec_simps(3)
tff(fact_180_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A3),B2) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).

% curry_conv
tff(fact_181_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P),Q),X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y4: A] :
              ( ( X = aa(A,option(A),some(A),Y4) )
             => ~ pp(aa(A,bool,Q,Y4)) ) ) ) ).

% case_optionE
tff(fact_182_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).

% dbl_simps(2)
tff(fact_183_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_184_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_185_curryI,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,F,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,F),A3),B2)) ) ).

% curryI
tff(fact_186_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).

% add.inverse_inverse
tff(fact_187_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A3 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_188_verit__minus__simplify_I4_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,uminus_uminus(B),aa(B,B,uminus_uminus(B),B2)) = B2 ) ).

% verit_minus_simplify(4)
tff(fact_189_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_190_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_191_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_192_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_193_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_194_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_195_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_196_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_197_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_198_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_199_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_200_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_201_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_202_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_203_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_204_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_205_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
    <=> ( 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)),N)) ) ) ).

% add_gr_0
tff(fact_206_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_207_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_208_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_209_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_210_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_211_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_212_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_213_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_214_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_215_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_216_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_217_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_218_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_219_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_inc_simps(4)
tff(fact_220_dbl__dec__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_dec_simps(2)
tff(fact_221_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_222_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_223_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_224_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_225_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y5)),aa(A,B,F,X4)))
                 => pp(aa(A,bool,P,Y5)) )
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct
tff(fact_226_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,bool),A3: A] :
          ( ! [X4: A] :
              ( ! [Y5: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Y5)),aa(A,B,F,X4)))
                 => pp(aa(A,bool,P,Y5)) )
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A3)) ) ) ).

% measure_induct_rule
tff(fact_227_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_228_verit__negate__coefficient_I3_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B2) ) ) ) ).

% verit_negate_coefficient(3)
tff(fact_229_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_230_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] :
          ( ( X != Y3 )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_231_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_232_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_233_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F22: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option) = aa(A,B,F22,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_234_option_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,B),Option: option(A)] : aa(B,C,H,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)) = aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),case_option(C,A),aa(B,C,H,F1)),aa(fun(A,B),fun(A,C),aTP_Lamp_an(fun(B,C),fun(fun(A,B),fun(A,C)),H),F22)),Option) ).

% option.case_distrib
tff(fact_235_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_236_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_237_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_238_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K: A,A3: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,uminus_uminus(A),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_239_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A3)) ) ).

% is_num_normalize(8)
tff(fact_240_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% one_neq_neg_one
tff(fact_241_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_242_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_243_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_244_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_245_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_246_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_247_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_248_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_249_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_250_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_251_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_252_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = 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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_253_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_254_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_255_curry__K,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: C,X2: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,aTP_Lamp_ao(C,fun(product_prod(A,B),C),C2)),X2),Xa3) = C2 ).

% curry_K
tff(fact_256_infinite__descent0__measure,axiom,
    ! [A: $tType,V3: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X4: A] :
          ( ( aa(A,nat,V3,X4) = zero_zero(nat) )
         => pp(aa(A,bool,P,X4)) )
     => ( ! [X4: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V3,X4)))
           => ( ~ pp(aa(A,bool,P,X4))
             => ? [Y5: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V3,Y5)),aa(A,nat,V3,X4)))
                  & ~ pp(aa(A,bool,P,Y5)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_257_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( ~ pp(aa(nat,bool,P,N2))
             => ? [M3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
                  & ~ pp(aa(nat,bool,P,M3)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_258_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_259_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_260_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_261_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_262_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_263_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_264_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: 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),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% less_add_eq_less
tff(fact_265_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_266_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_267_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_less_mono1
tff(fact_268_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I)) ).

% not_add_less2
tff(fact_269_not__add__less1,axiom,
    ! [I: 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),I),J)),I)) ).

% not_add_less1
tff(fact_270_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_less_mono
tff(fact_271_add__lessD1,axiom,
    ! [I: 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),I),J)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K)) ) ).

% add_lessD1
tff(fact_272_uminus__unit__def,axiom,
    ! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).

% uminus_unit_def
tff(fact_273_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,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,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_274_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,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,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_275_Ints__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,member(A,aa(A,A,uminus_uminus(A),A3)),ring_1_Ints(A))) ) ) ).

% Ints_minus
tff(fact_276_minus__in__Ints__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,aa(A,A,uminus_uminus(A),X)),ring_1_Ints(A)))
        <=> pp(aa(set(A),bool,member(A,X),ring_1_Ints(A))) ) ) ).

% minus_in_Ints_iff
tff(fact_277_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_278_option_Osel,axiom,
    ! [A: $tType,X22: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X22)) = X22 ).

% option.sel
tff(fact_279_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F22: fun(A,B)] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),none(A)) = F1 ).

% option.simps(4)
tff(fact_280_option_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F22: fun(A,B),X22: A] : aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F1),F22),aa(A,option(A),some(A),X22)) = aa(A,B,F22,X22) ).

% option.simps(5)
tff(fact_281_curryE,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A3),B2))
     => pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryE
tff(fact_282_curryD,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A3),B2))
     => pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% curryD
tff(fact_283_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fTrue),aTP_Lamp_ap(A,bool)),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_284_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),fFalse),aTP_Lamp_aq(A,bool)),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_285_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_286_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_287_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_288_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_289_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_290_zero__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( zero_zero(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% zero_neq_neg_one
tff(fact_291_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: 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),Y3)),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),Y3),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_292_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_293_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_294_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_295_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_296_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_297_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_298_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_299_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_300_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_301_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),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),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_302_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_303_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_304_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa3: A,Xb: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X2),Xa3),Xb) = aa(product_prod(A,B),C,X2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa3),Xb)) ).

% curry_def
tff(fact_305_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_306_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,bool),X: option(A),R4: fun(B,bool),F: B,G: fun(A,B)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),P),Q),X))
     => ( ( pp(P)
         => pp(aa(B,bool,R4,F)) )
       => ( ! [Q3: A] :
              ( pp(aa(A,bool,Q,Q3))
             => pp(aa(B,bool,R4,aa(A,B,G,Q3))) )
         => pp(aa(B,bool,R4,aa(option(A),B,aa(fun(A,B),fun(option(A),B),aa(B,fun(fun(A,B),fun(option(A),B)),case_option(B,A),F),G),X))) ) ) ) ).

% disjE_realizer2
tff(fact_307_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_308_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_ar(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_309_add_Ogroup__axioms,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => group(A,plus_plus(A),zero_zero(A),uminus_uminus(A)) ) ).

% add.group_axioms
tff(fact_310_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)))))
     => ( ! [F3: fun(nat,fun(A,A)),A4: nat,B3: 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))),F3),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc)))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B3),A4))
               => 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,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F3,A4),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,F3),A4),B3),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_311_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_312_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_313_is__num_Osimps,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
        <=> ( ( A3 = one_one(A) )
            | ? [X3: A] :
                ( ( A3 = aa(A,A,uminus_uminus(A),X3) )
                & neg_numeral_is_num(A,X3) )
            | ? [X3: A,Y: A] :
                ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y) )
                & neg_numeral_is_num(A,X3)
                & neg_numeral_is_num(A,Y) ) ) ) ) ).

% is_num.simps
tff(fact_314_is__num_Ocases,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A] :
          ( neg_numeral_is_num(A,A3)
         => ( ( A3 != one_one(A) )
           => ( ! [X4: A] :
                  ( ( A3 = aa(A,A,uminus_uminus(A),X4) )
                 => ~ neg_numeral_is_num(A,X4) )
             => ~ ! [X4: A,Y4: A] :
                    ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),Y4) )
                   => ( neg_numeral_is_num(A,X4)
                     => ~ neg_numeral_is_num(A,Y4) ) ) ) ) ) ) ).

% is_num.cases
tff(fact_315_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_316_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_317_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_318_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_abs
tff(fact_319_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_idempotent
tff(fact_320_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% sgn_sgn
tff(fact_321_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_322_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_323_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_324_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_325_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_326_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_327_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus
tff(fact_328_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus_cancel
tff(fact_329_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_330_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_331_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_332_sgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% sgn_minus
tff(fact_333_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_334_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_335_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_336_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_337_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: 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),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_iff
tff(fact_338_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_339_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = 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),N)) ) ).

% of_nat_add
tff(fact_340_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_341_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_342_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_343_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_344_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_345_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_346_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_347_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_348_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% nat_neq_iff
tff(fact_349_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_350_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_351_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
     => ( S != T2 ) ) ).

% less_not_refl3
tff(fact_352_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_353_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
             => pp(aa(nat,bool,P,M3)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_354_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
              & ~ pp(aa(nat,bool,P,M3)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_355_linorder__neqE__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( ( X != Y3 )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y3),X)) ) ) ).

% linorder_neqE_nat
tff(fact_356_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V3: fun(A,nat),X: A] :
      ( ! [X4: A] :
          ( ~ pp(aa(A,bool,P,X4))
         => ? [Y5: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V3,Y5)),aa(A,nat,V3,X4)))
              & ~ pp(aa(A,bool,P,Y5)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_357_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_358_group_Oinverse__distrib__swap,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,B2)),aa(A,A,Inverse,A3)) ) ) ).

% group.inverse_distrib_swap
tff(fact_359_group_Ogroup__left__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,Z),A3) = A3 ) ) ).

% group.group_left_neutral
tff(fact_360_group_Oinverse__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,Inverse,Z) = Z ) ) ).

% group.inverse_neutral
tff(fact_361_group_Oinverse__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,Inverse,aa(A,A,Inverse,A3)) = A3 ) ) ).

% group.inverse_inverse
tff(fact_362_group_Oinverse__unique,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A] :
      ( group(A,F,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = Z )
       => ( aa(A,A,Inverse,A3) = B2 ) ) ) ).

% group.inverse_unique
tff(fact_363_group_Oright__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Inverse,A3)) = Z ) ) ).

% group.right_inverse
tff(fact_364_group_Oright__cancel,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),B2: A,A3: A,C2: A] :
      ( group(A,F,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,B2),A3) = aa(A,A,aa(A,fun(A,A),F,C2),A3) )
      <=> ( B2 = C2 ) ) ) ).

% group.right_cancel
tff(fact_365_group_Oleft__inverse,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A] :
      ( group(A,F,Z,Inverse)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A3)),A3) = Z ) ) ).

% group.left_inverse
tff(fact_366_group_Oleft__cancel,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A),A3: A,B2: A,C2: A] :
      ( group(A,F,Z,Inverse)
     => ( ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,A3),C2) )
      <=> ( B2 = C2 ) ) ) ).

% group.left_cancel
tff(fact_367_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_368_nat__less__as__int,axiom,
    ! [X2: nat,Xa3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Xa3))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ) ).

% nat_less_as_int
tff(fact_369_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( ( A3 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_370_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_371_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_372_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y3: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y3) )
        <=> ( ( X = Y3 )
            | ( X = aa(A,A,uminus_uminus(A),Y3) ) ) ) ) ).

% abs_eq_iff
tff(fact_373_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: 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),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_374_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => 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),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_375_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_376_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% int_cases3
tff(fact_377_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_378_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% neg_int_cases
tff(fact_379_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% pos_int_cases
tff(fact_380_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ? [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_381_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z) ).

% zadd_int_left
tff(fact_382_int__plus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_383_int__ops_I5_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_384_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_385_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_386_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_387_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_388_Ints__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,member(A,aa(A,A,abs_abs(A),A3)),ring_1_Ints(A))) ) ) ).

% Ints_abs
tff(fact_389_Ints__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,member(A,aa(nat,A,semiring_1_of_nat(A),N)),ring_1_Ints(A))) ) ).

% Ints_of_nat
tff(fact_390_is__num__normalize_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y3: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y3)
           => neg_numeral_is_num(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) ) ) ) ).

% is_num_normalize(6)
tff(fact_391_is__num__add__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y3: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y3),X) ) ) ) ) ).

% is_num_add_commute
tff(fact_392_is__num__add__left__commute,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A,Y3: A,Z: A] :
          ( neg_numeral_is_num(A,X)
         => ( neg_numeral_is_num(A,Y3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y3),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Y3),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ) ) ) ).

% is_num_add_left_commute
tff(fact_393_is__num__normalize_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => neg_numeral_is_num(A,one_one(A)) ) ).

% is_num_normalize(4)
tff(fact_394_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_395_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_396_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_397_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_398_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_399_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_400_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = aa(A,A,uminus_uminus(A),X2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = X2 ) ) ) ) ).

% abs_if_raw
tff(fact_401_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_402_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_403_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_404_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_405_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F: 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))),F),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A3),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2)))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
         => ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_406_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb2: nat,Xc: A,Y3: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb2,Xc) = Y3 )
     => ( 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)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc)))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa))
               => ( Y3 = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa))
               => ( Y3 = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa),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)),Xa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb2),Xc))))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_407_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb2: nat,Xc: A,Y3: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb2,Xc) = Y3 )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa))
         => ( Y3 = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb2),Xa))
         => ( Y3 = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb2,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_408_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A3: nat,F: 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,F,A3,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A3))
       => ( set_fo6178422350223883121st_nat(A,F,A3,B2,Acc2) = set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A3),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_409_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_410_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_411_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: 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)),N))
       => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(nat,num,num_of_nat,M)),aa(nat,num,num_of_nat,N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_412_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,member(A,Y3),ring_1_Ints(A)))
           => ( ( X = Y3 )
            <=> 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),Y3))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_413_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_414_in__measure,axiom,
    ! [A: $tType,X: A,Y3: A,F: 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),Y3)),measure(A,F)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y3))) ) ).

% in_measure
tff(fact_415_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_416_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% diff_is_0_eq
tff(fact_417_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_418_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_419_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_420_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_421_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_422_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: 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),I),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),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_423_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: 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)),I) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_424_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: 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),I),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),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_425_diff__diff__left,axiom,
    ! [I: 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),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_426_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_427_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_428_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_429_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_430_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_431_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_432_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_433_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_434_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_435_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_436_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel_right'
tff(fact_437_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_right
tff(fact_438_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),A3) = B2 ) ).

% add_diff_cancel_left'
tff(fact_439_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ).

% add_diff_cancel_left
tff(fact_440_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),B2) = A3 ) ).

% diff_add_cancel
tff(fact_441_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = A3 ) ).

% add_diff_cancel
tff(fact_442_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: 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),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% of_nat_le_iff
tff(fact_443_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% minus_diff_eq
tff(fact_444_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_445_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_446_zero__less__diff,axiom,
    ! [N: 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),N),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% zero_less_diff
tff(fact_447_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_448_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
        <=> ~ pp(P) ) ) ).

% of_bool_eq_0_iff
tff(fact_449_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_450_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_451_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).

% power_one_right
tff(fact_452_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% zle_diff1_eq
tff(fact_453_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
        <=> pp(P) ) ) ).

% of_bool_eq_1_iff
tff(fact_454_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_455_negative__zle,axiom,
    ! [N: 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),N))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zle
tff(fact_456_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_457_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% abs_bool_eq
tff(fact_458_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_459_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_460_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_461_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_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_473_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_474_verit__minus__simplify_I3_J,axiom,
    ! [B: $tType] :
      ( group_add(B)
     => ! [B2: B] : aa(B,B,aa(B,fun(B,B),minus_minus(B),zero_zero(B)),B2) = aa(B,B,uminus_uminus(B),B2) ) ).

% verit_minus_simplify(3)
tff(fact_475_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_476_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) ) ).

% diff_minus_eq_add
tff(fact_477_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3) ) ).

% uminus_add_conv_diff
tff(fact_478_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_479_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_480_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_481_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: 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),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_482_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_483_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_484_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).

% of_bool_not_iff
tff(fact_485_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_486_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ).

% zle_add1_eq_le
tff(fact_487_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y3: 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),Y3)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y3)) ) ) ) ).

% power_increasing_iff
tff(fact_488_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_489_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_490_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y3: 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),Y3)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y3)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_491_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_492_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_493_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A3)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_494_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_495_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_496_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_497_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N)))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_498_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_499_negative__zle__0,axiom,
    ! [N: 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),N))),zero_zero(int))) ).

% negative_zle_0
tff(fact_500_zabs__def,axiom,
    ! [I: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = aa(int,int,uminus_uminus(int),I) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I) = I ) ) ) ).

% zabs_def
tff(fact_501_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_502_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_503_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_504_int__cases2,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% int_cases2
tff(fact_505_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_506_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_507_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_508_zsgn__def,axiom,
    ! [I: int] :
      ( ( ( I = zero_zero(int) )
       => ( aa(int,int,sgn_sgn(int),I) = zero_zero(int) ) )
      & ( ( I != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I))
           => ( aa(int,int,sgn_sgn(int),I) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_509_add1__zle__eq,axiom,
    ! [W: int,Z: 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),W),one_one(int))),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% add1_zle_eq
tff(fact_510_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_gr_induct
tff(fact_511_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => 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)),Z))) ) ).

% le_imp_0_less
tff(fact_512_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_513_odd__less__0__iff,axiom,
    ! [Z: 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)),Z)),Z)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_514_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),K))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_less_induct
tff(fact_515_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)) ) ).

% zless_imp_add1_zle
tff(fact_516_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_517_nat__leq__as__int,axiom,
    ! [X2: nat,Xa3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Xa3))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ) ).

% nat_leq_as_int
tff(fact_518_int__if,axiom,
    ! [P: bool,A3: nat,B2: nat] :
      ( ( pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),aa(bool,fun(nat,fun(nat,nat)),if(nat),P),A3),B2)) = aa(nat,int,semiring_1_of_nat(int),A3) ) )
      & ( ~ pp(P)
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),aa(bool,fun(nat,fun(nat,nat)),if(nat),P),A3),B2)) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ) ).

% int_if
tff(fact_519_nat__int__comparison_I1_J,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = B2 )
    <=> ( aa(nat,int,semiring_1_of_nat(int),A3) = aa(nat,int,semiring_1_of_nat(int),B2) ) ) ).

% nat_int_comparison(1)
tff(fact_520_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_521_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M4: nat,N2: nat] : Z != aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),M4)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_522_int__int__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(nat,int,semiring_1_of_nat(int),N) )
    <=> ( M = N ) ) ).

% int_int_eq
tff(fact_523_zle__int,axiom,
    ! [M: nat,N: 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),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% zle_int
tff(fact_524_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = 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),N)) ) ) ) ).

% of_nat_diff
tff(fact_525_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z))
    <=> ? [N6: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N6)) ) ).

% zle_iff_zadd
tff(fact_526_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_527_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))
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_528_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_power
tff(fact_529_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ).

% power_mono
tff(fact_530_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_531_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),minus_minus(int),K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_532_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),D3) )
         => ( ( A3 = B2 )
          <=> ( C2 = D3 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_533_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_534_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_535_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_536_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_537_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) ) ).

% diff_right_commute
tff(fact_538_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5))) ) ) ) ).

% power_increasing
tff(fact_539_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% one_le_power
tff(fact_540_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_541_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool,Q2: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = aa(bool,A,zero_neq_one_of_bool(A),Q2) )
        <=> ( pp(P2)
          <=> pp(Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_542_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_543_int__induct,axiom,
    ! [P: fun(int,bool),K: int,I: int] :
      ( pp(aa(int,bool,P,K))
     => ( ! [I2: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
           => ( pp(aa(int,bool,P,I2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_induct
tff(fact_544_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_ge_induct
tff(fact_545_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),K))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I2: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
             => ( pp(aa(int,bool,P,I2))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int)))) ) )
         => pp(aa(int,bool,P,I)) ) ) ) ).

% int_le_induct
tff(fact_546_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N5))
         => ( 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),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_decreasing
tff(fact_547_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_548_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_549_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_550_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: 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),I),K)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),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),I),K)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),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),N),K)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_551_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_552_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_553_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_554_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),A3) = B2 ) ) ) ).

% diff_add
tff(fact_555_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_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: 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),N))
       => ( 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),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% less_diff_iff
tff(fact_566_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_567_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_568_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_569_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: 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)),I))
    <=> 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),I),K))) ) ).

% le_diff_conv
tff(fact_570_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: 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),I),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),I),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_571_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: 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),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_572_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: 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),I)),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)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_573_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_574_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).

% of_nat_mono
tff(fact_575_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_576_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_577_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_578_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),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))),R4),S2)) ) ).

% pred_subset_eq2
tff(fact_579_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A3)),N))) ) ).

% zero_le_power_abs
tff(fact_580_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_581_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_582_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_le_iff
tff(fact_583_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: 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)),I))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K))) ) ) ).

% less_diff_conv2
tff(fact_584_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
              <=> ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_585_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A3 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_586_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% self_le_power
tff(fact_587_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] :
          ( ( A3 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_588_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_589_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_590_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_591_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_592_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_593_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A3 )
         => ( C2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) ) ) ) ).

% add_implies_diff
tff(fact_594_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_595_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_add_eq
tff(fact_596_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_597_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) ) ).

% add_diff_eq
tff(fact_598_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_599_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = C2 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_600_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K: A,A3: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A5),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_601_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),B2) ) ).

% minus_diff_commute
tff(fact_602_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_603_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_604_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_605_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_606_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B5: B,A6: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B5),A6))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A6),B5)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_607_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)) ) ).

% abs_minus_commute
tff(fact_608_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: 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),N)),K)) ) ).

% less_imp_diff_less
tff(fact_609_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( 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),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ) ).

% diff_less_mono2
tff(fact_610_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_611_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_612_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).

% le_iff_add
tff(fact_613_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_614_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_615_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_616_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_617_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_618_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = 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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_619_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_620_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_621_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_622_diff__add__inverse,axiom,
    ! [N: 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),N),M)),N) = M ).

% diff_add_inverse
tff(fact_623_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_cancel2
tff(fact_624_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% Nat.diff_cancel
tff(fact_625_subrelI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X4: A,Y4: 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),Y4)),R))
         => 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),Y4)),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))),R),S)) ) ).

% subrelI
tff(fact_626_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_627_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_628_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_629_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_630_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_631_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_632_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_633_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_634_less__mono__imp__le__mono,axiom,
    ! [F: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,I2)),aa(nat,nat,F,J2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F,I)),aa(nat,nat,F,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_635_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% le_neq_implies_less
tff(fact_636_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_or_eq_imp_le
tff(fact_637_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_638_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_imp_le_nat
tff(fact_639_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_640_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_641_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_le_add2
tff(fact_642_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_le_add1
tff(fact_643_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_le_mono1
tff(fact_644_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_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),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_le_mono
tff(fact_645_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_646_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% add_leD2
tff(fact_647_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% add_leD1
tff(fact_648_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

% le_add2
tff(fact_649_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% le_add1
tff(fact_650_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% add_leE
tff(fact_651_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_652_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_653_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_less_power
tff(fact_654_Ints__diff,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),minus_minus(A),A3),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_diff
tff(fact_655_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_656_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(set(A),bool,member(A,A3),ring_1_Ints(A)))
         => pp(aa(set(A),bool,member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),ring_1_Ints(A))) ) ) ).

% Ints_power
tff(fact_657_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_658_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_659_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_660_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ~ ( ( pp(P2)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P2)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_661_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ( ( pp(P2)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P2)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_662_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) ) )
          & ( ~ pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_663_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_664_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_665_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_666_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_667_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_668_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N5))) ) ) ) ).

% power_strict_increasing
tff(fact_669_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_670_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_671_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B4: A,K: A,B2: A,A3: A] :
          ( ( B4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_672_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_673_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_674_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y3: 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),Y3),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y3 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_675_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y3: 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)),Y3))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y3 = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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),N)),M)) ) ) ).

% diff_less
tff(fact_686_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_687_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I),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),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_688_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_689_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_690_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_691_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),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),I),K)),J)) ) ).

% less_diff_conv
tff(fact_692_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_693_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_694_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_695_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_696_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_697_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_698_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K2))
               => ~ pp(aa(nat,bool,P,I3)) )
            & pp(aa(nat,bool,P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_699_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_700_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_701_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_702_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_703_mono__nat__linear__lb,axiom,
    ! [F: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,M4)),aa(nat,nat,F,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,F,M)),K)),aa(nat,nat,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_704_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N5: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N5))
         => ( 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),N5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_705_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% one_less_power
tff(fact_706_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A3: A,R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A3))),R))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),R)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R))) ) ) ) ).

% abs_diff_less_iff
tff(fact_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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))) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ pp(aa(nat,bool,P,D4)) ) ) ) ).

% nat_diff_split_asm
tff(fact_717_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))) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => pp(aa(nat,bool,P,D4)) ) ) ) ).

% nat_diff_split
tff(fact_718_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E3)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_719_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_720_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_721_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_722_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_723_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_724_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A] :
          ( ! [E3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E3))
             => 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),Y3),E3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ).

% field_le_epsilon
tff(fact_725_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_726_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ? [Y5: A] :
                ( pp(aa(A,bool,P,Y5))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,Y5)),aa(A,nat,F,X4))) ) )
       => ? [Y4: A] :
            ( pp(aa(A,bool,P,Y4))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,Y4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_727_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_728_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_729_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_at(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_730_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A3: A,Y3: A,U: A,V: 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),Y3),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)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = 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),V),Y3))),A3)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_731_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% zero_less_nat_eq
tff(fact_732_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ) ).

% of_int_1_less_iff
tff(fact_733_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_734_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_735_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_736_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_737_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: 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),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_738_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_739_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_740_of__int__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W: int,Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),W) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( W = Z ) ) ) ).

% of_int_eq_iff
tff(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_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_748_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_749_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_750_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_751_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_752_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( 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)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_753_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),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),N)) ) ) ).

% mult_less_cancel2
tff(fact_754_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_755_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_756_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_757_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B2: int,W: nat] :
          ( ( 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)),W) )
        <=> ( X = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_758_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,X: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,ring_1_of_int(A),X) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_759_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),N) ) ).

% of_int_power
tff(fact_760_nat__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(nat,int,semiring_1_of_nat(int),N)) = N ).

% nat_int
tff(fact_761_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_762_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_763_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_764_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_765_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1
tff(fact_766_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1_right
tff(fact_767_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),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),N)) ) ) ).

% mult_le_cancel2
tff(fact_768_of__int__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).

% of_int_0
tff(fact_769_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_770_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = zero_zero(A) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_771_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% of_int_le_iff
tff(fact_772_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% of_int_less_iff
tff(fact_773_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_774_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_775_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_776_of__int__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),Z)) = aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_minus
tff(fact_777_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),minus_minus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_778_of__int__of__nat__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% of_int_of_nat_eq
tff(fact_779_of__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int] : aa(int,A,ring_1_of_int(A),aa(int,int,abs_abs(int),X)) = aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),X)) ) ).

% of_int_abs
tff(fact_780_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_781_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,member(A,X),ring_1_Ints(A))) ) ) ).

% frac_eq_0_iff
tff(fact_782_of__int__of__bool,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(int,A,ring_1_of_int(A),aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_int_of_bool
tff(fact_783_nat__of__bool,axiom,
    ! [P: bool] : aa(int,nat,nat2,aa(bool,int,zero_neq_one_of_bool(int),P)) = aa(bool,nat,zero_neq_one_of_bool(nat),P) ).

% nat_of_bool
tff(fact_784_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_785_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_786_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: 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)),W)))
        <=> 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),W))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_787_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: 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)),W)),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),W)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_788_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: 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)),W)))
        <=> 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),W))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_789_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: 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)),W)),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),W)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_790_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_791_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_792_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% zless_nat_conj
tff(fact_793_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_794_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_795_nat__zminus__int,axiom,
    ! [N: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_796_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_797_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: 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),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_le_iff
tff(fact_798_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_799_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_less_iff
tff(fact_800_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_801_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_802_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: 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),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z)) ) ) ).

% of_int_1_le_iff
tff(fact_803_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_804_nat__times__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X2),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_times_as_int
tff(fact_805_nat__mult__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ).

% nat_mult_distrib
tff(fact_806_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_807_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_808_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_809_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_810_nat__mono,axiom,
    ! [X: int,Y3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y3))) ) ).

% nat_mono
tff(fact_811_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K)) ) ) ).

% le_trans
tff(fact_812_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% eq_imp_le
tff(fact_813_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_814_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_815_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: 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),N))
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_816_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: 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),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% le_diff_iff
tff(fact_817_diff__commute,axiom,
    ! [I: 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),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),K)),J) ).

% diff_commute
tff(fact_818_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: 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),N))
       => ( 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),N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_819_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),L)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),L))) ) ).

% diff_le_mono
tff(fact_820_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ).

% diff_le_self
tff(fact_821_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_822_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_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),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).

% mult_le_mono
tff(fact_823_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M))) ) ).

% diff_le_mono2
tff(fact_824_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).

% mult_le_mono1
tff(fact_825_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).

% mult_le_mono2
tff(fact_826_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% nat_le_linear
tff(fact_827_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),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),N),K)) ).

% diff_mult_distrib
tff(fact_828_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% diff_mult_distrib2
tff(fact_829_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => ? [X4: nat] :
            ( pp(aa(nat,bool,P,X4))
            & ! [Y5: nat] :
                ( pp(aa(nat,bool,P,Y5))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y5),X4)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_830_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_831_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),R4: bool,X: A,Y3: 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(R4) )
     => ( pp(R4)
        & ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y3))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y3)) ) ) ) ).

% predicate2D_conj
tff(fact_832_pred__subset__eq,axiom,
    ! [A: $tType,R4: set(A),S2: 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_aa(set(A),fun(A,bool)),R4)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),S2)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R4),S2)) ) ).

% pred_subset_eq
tff(fact_833_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_au(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P))) ).

% eq_subset
tff(fact_834_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_av(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_835_prop__restrict,axiom,
    ! [A: $tType,X: A,Z3: set(A),X6: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,X),Z3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z3),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_aw(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_836_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_aw(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))),X6)) ).

% Collect_restrict
tff(fact_837_mult_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)) ) ).

% mult.left_commute
tff(fact_838_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3) ) ).

% mult.commute
tff(fact_839_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_840_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: int,Y3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),X)),Y3) = aa(A,A,aa(A,fun(A,A),times_times(A),Y3),aa(int,A,ring_1_of_int(A),X)) ) ).

% mult_of_int_commute
tff(fact_841_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_842_ab__semigroup__mult__class_Omult__ac_I1_J,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)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_843_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),W))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_844_int__ops_I7_J,axiom,
    ! [A3: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_845_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z))),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Z2))) ) ) ).

% nat_mult_distrib_neg
tff(fact_846_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_1: nat] : pp(aa(nat,bool,P,X_1))
    <=> ? [X3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X3))
          & pp(aa(nat,bool,P,aa(int,nat,nat2,X3))) ) ) ).

% ex_nat
tff(fact_847_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_1: nat] : pp(aa(nat,bool,P,X_1))
    <=> ! [X3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X3))
         => pp(aa(nat,bool,P,aa(int,nat,nat2,X3))) ) ) ).

% all_nat
tff(fact_848_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),N) ) ) ).

% nat_power_eq
tff(fact_849_eq__nat__nat__iff,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z2))
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z2) )
        <=> ( Z = Z2 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_850_nat__diff__distrib,axiom,
    ! [Z2: int,Z: 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),Z2),Z))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ) ).

% nat_diff_distrib
tff(fact_851_nat__diff__distrib_H,axiom,
    ! [X: int,Y3: 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)),Y3))
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,X)),aa(int,nat,nat2,Y3)) ) ) ) ).

% nat_diff_distrib'
tff(fact_852_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_853_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_854_int__minus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_855_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_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_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_865_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),E2)),C2) ) ).

% combine_common_factor
tff(fact_866_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y3: A,X: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3)) )
        <=> ( ( W = X )
            | ( Y3 = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_877_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y3) = aa(A,A,aa(A,fun(A,A),times_times(A),Y3),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_878_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_879_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_880_add__mult__distrib,axiom,
    ! [M: nat,N: 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),N)),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),N),K)) ).

% add_mult_distrib
tff(fact_881_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: 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),N)) = 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),N)) ).

% add_mult_distrib2
tff(fact_882_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_883_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_884_Ints__of__int,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] : pp(aa(set(A),bool,member(A,aa(int,A,ring_1_of_int(A),Z)),ring_1_Ints(A))) ) ).

% Ints_of_int
tff(fact_885_Ints__induct,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,member(A,Q2),ring_1_Ints(A)))
         => ( ! [Z4: int] : pp(aa(A,bool,P,aa(int,A,ring_1_of_int(A),Z4)))
           => pp(aa(A,bool,P,Q2)) ) ) ) ).

% Ints_induct
tff(fact_886_Ints__cases,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Q2: A] :
          ( pp(aa(set(A),bool,member(A,Q2),ring_1_Ints(A)))
         => ~ ! [Z4: int] : Q2 != aa(int,A,ring_1_of_int(A),Z4) ) ) ).

% Ints_cases
tff(fact_887_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_888_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_889_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L3: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L3)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_890_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_891_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_892_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_893_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_894_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_895_nat__minus__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_minus_as_int
tff(fact_896_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_ax(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_897_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_ay(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_898_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_899_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_900_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_901_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,member(A,zero_zero(A)),semiring_1_Nats(A))) ) ).

% Nats_0
tff(fact_902_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_903_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,member(A,one_one(A)),semiring_1_Nats(A))) ) ).

% Nats_1
tff(fact_904_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,member(A,aa(nat,A,semiring_1_of_nat(A),N)),semiring_1_Nats(A))) ) ).

% of_nat_in_Nats
tff(fact_905_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,member(A,X),semiring_1_Nats(A)))
         => ( ! [N2: nat] : pp(aa(A,bool,P,aa(nat,A,semiring_1_of_nat(A),N2)))
           => pp(aa(A,bool,P,X)) ) ) ) ).

% Nats_induct
tff(fact_906_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,member(A,X),semiring_1_Nats(A)))
         => ~ ! [N2: nat] : X != aa(nat,A,semiring_1_of_nat(A),N2) ) ) ).

% Nats_cases
tff(fact_907_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A))) ) ).

% Nats_subset_Ints
tff(fact_908_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_909_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_910_le__nat__iff,axiom,
    ! [K: int,N: 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),N),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),N)),K)) ) ) ).

% le_nat_iff
tff(fact_911_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = aa(int,nat,nat2,Z) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% int_eq_iff
tff(fact_912_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = Z ) ) ).

% nat_0_le
tff(fact_913_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_914_nat__one__as__int,axiom,
    one_one(nat) = aa(int,nat,nat2,one_one(int)) ).

% nat_one_as_int
tff(fact_915_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_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_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R: A,A3: A,B2: A,C2: A,D3: A] :
          ( ( R != 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),R),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R),D3)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_953_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y3: 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),Y3),Y3)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_954_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: 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)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).

% less_1_mult
tff(fact_955_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),E2)),C2) = D3 ) ) ) ).

% eq_add_iff1
tff(fact_956_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A3)),E2)),D3) ) ) ) ).

% eq_add_iff2
tff(fact_957_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y3: 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),Y3),Y3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y3)) ) ).

% square_diff_square_factored
tff(fact_958_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_959_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_960_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_961_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).

% mult_less_mono2
tff(fact_962_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),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),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).

% mult_less_mono1
tff(fact_963_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_add
tff(fact_964_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),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),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_965_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_966_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_967_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_968_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_969_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_970_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_971_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_972_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_973_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A] :
          ( ! [Z4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z4),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),Z4),X)),Y3)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ).

% field_le_mult_one_interval
tff(fact_974_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_nonneg
tff(fact_975_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_mono_iff
tff(fact_976_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),aa(int,nat,nat2,Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_977_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( aa(int,nat,nat2,W) = M )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = 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)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_978_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = aa(int,nat,nat2,W) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = 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)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_979_nat__add__distrib,axiom,
    ! [Z: int,Z2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( 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),plus_plus(int),Z),Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) ) ) ) ).

% nat_add_distrib
tff(fact_980_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_981_nat__int__add,axiom,
    ! [A3: nat,B2: nat] : aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2) ).

% nat_int_add
tff(fact_982_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(set(A),bool,member(A,A3),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,member(A,B2),semiring_1_Nats(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A3))
             => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_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_993_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_994_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_995_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_996_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_997_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y3: 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),Y3),Y3)))) ) ).

% sum_squares_ge_zero
tff(fact_998_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y3: 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),Y3),Y3))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_999_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: 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)),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),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),Y3),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1000_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: 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)),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),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),Y3)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1001_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_1002_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_1003_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y3: 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),Y3),Y3))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1004_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y3: 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),Y3),Y3))))
        <=> ( ( X != zero_zero(A) )
            | ( Y3 != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1005_le__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
        <=> 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)),E2)),D3))) ) ) ).

% le_add_iff2
tff(fact_1006_le__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
        <=> 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)),E2)),C2)),D3)) ) ) ).

% le_add_iff1
tff(fact_1007_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
        <=> 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)),E2)),D3))) ) ) ).

% less_add_iff2
tff(fact_1008_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: 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),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E2)),D3)))
        <=> 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)),E2)),C2)),D3)) ) ) ).

% less_add_iff1
tff(fact_1009_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_1010_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_1011_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: 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),Y3)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),X)) ) ) ) ).

% abs_mult_pos
tff(fact_1012_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1013_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1014_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_minus
tff(fact_1015_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1016_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: 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),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1017_diff__nat__eq__if,axiom,
    ! [Z2: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) = aa(int,nat,nat2,Z) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z2),zero_zero(int)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z2)) = aa(nat,nat,aa(nat,fun(nat,nat),aa(bool,fun(nat,fun(nat,nat)),if(nat),aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2)),zero_zero(int))),zero_zero(nat)),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z2))) ) ) ) ).

% diff_nat_eq_if
tff(fact_1018_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_1019_nat__plus__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_plus_as_int
tff(fact_1020_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: 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,Y3))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y3)) ) )
          & ( ~ 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,Y3))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = 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,Y3))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_1021_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))) ) ) ).

% of_int_pos
tff(fact_1022_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_1023_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: 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),N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_1024_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),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),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_1025_split__nat,axiom,
    ! [P: fun(nat,bool),I: int] :
      ( pp(aa(nat,bool,P,aa(int,nat,nat2,I)))
    <=> ( ! [N6: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N6) )
           => pp(aa(nat,bool,P,N6)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_1026_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(int,nat,nat2,W)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_1027_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_1028_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_1029_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_1030_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_1031_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_1032_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_1033_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_1034_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_1035_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A3: A,Y3: A,U: A,V: 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),Y3),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)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = 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),V),Y3))),A3)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1036_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ) ).

% power_Suc_less
tff(fact_1037_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P2: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1038_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% power_minus_mult
tff(fact_1039_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I),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)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J))) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1040_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_1041_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( 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),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1042_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( 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),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1043_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( 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),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1044_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> 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),I)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1045_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1046_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y3)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1047_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1048_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( 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),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1049_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( 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),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1050_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> 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),I)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1051_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: 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),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1052_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1053_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1054_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1055_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y3)) ) ) ) ).

% mult_less_iff1
tff(fact_1056_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1057_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),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),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = 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),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1058_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I))
     => ( 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),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1059_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) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_1060_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) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R2: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2) )
               => ( ( aa(int,int,sgn_sgn(int),R2) = 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),R2)),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),Q3),A22)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_1061_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A3,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_az(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_1062_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_aa(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),B4))) ) ).

% less_eq_set_def
tff(fact_1063_eucl__rel__int__remainderI,axiom,
    ! [R: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_1064_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q2 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1065_zminus1__lemma,axiom,
    ! [A3: int,B2: int,Q2: int,R: int] :
      ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),aa(bool,fun(int,fun(int,int)),if(int),aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))),aa(int,int,uminus_uminus(int),Q2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),aa(int,int,aa(int,fun(int,int),aa(bool,fun(int,fun(int,int)),if(int),aa(int,bool,aa(int,fun(int,bool),fequal(int),R),zero_zero(int))),zero_zero(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R)))) ) ) ).

% zminus1_lemma
tff(fact_1066_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_1067_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_1068_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_1069_unique__quotient,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q5: 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),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_1070_unique__remainder,axiom,
    ! [A3: int,B2: int,Q2: int,R: int,Q5: 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),Q2),R))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R3))
       => ( R = R3 ) ) ) ).

% unique_remainder
tff(fact_1071_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_1072_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_1073_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_1074_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_1075_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_1076_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_1077_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_1078_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_1079_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_1080_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_1081_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).

% pochhammer_product'
tff(fact_1082_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1083_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_aw(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))),A5)) ).

% Collect_subset
tff(fact_1084_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% pochhammer_product
tff(fact_1085_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R: 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),R),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_1086_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_1087_nat__ivt__aux,axiom,
    ! [N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F,N)))
         => ? [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
              & ( aa(nat,int,F,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1088_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),X)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) ) ) ) ).

% power_minus'
tff(fact_1089_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ).

% one_less_nat_eq
tff(fact_1090_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F,aa(nat,nat,suc,I2))),aa(nat,int,F,I2)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F,M)),K))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F,N)))
           => ? [I2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I2))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
                & ( aa(nat,int,F,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1091_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_1092_power_Opower__eq__if,axiom,
    ! [A: $tType,M: nat,One: A,Times: fun(A,fun(A,A)),P2: A] :
      ( ( ( M = zero_zero(nat) )
       => ( power2(A,One,Times,P2,M) = One ) )
      & ( ( M != zero_zero(nat) )
       => ( power2(A,One,Times,P2,M) = aa(A,A,aa(A,fun(A,A),Times,P2),power2(A,One,Times,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_1093_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_1094_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_1095_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_1096_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_1097_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_1098_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_eq
tff(fact_1099_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_1100_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_1101_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% Suc_le_mono
tff(fact_1102_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_1103_Suc__diff__diff,axiom,
    ! [M: nat,N: 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)),N)),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),N)),K) ).

% Suc_diff_diff
tff(fact_1104_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ).

% diff_Suc_Suc
tff(fact_1105_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_1106_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_1107_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_1108_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_1109_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1110_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1111_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_1112_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_1113_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_1114_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1115_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1116_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_1117_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_1118_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1119_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_1120_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_1121_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_1122_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( 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))),N)) ) ) ).

% one_le_mult_iff
tff(fact_1123_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: 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))),I) = 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),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1124_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: 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),I),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),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1125_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_1126_fact__Suc__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% fact_Suc_0
tff(fact_1127_negative__zless,axiom,
    ! [N: 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,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_1128_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_1129_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_1137_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1138_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_1139_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_1140_Suc__inject,axiom,
    ! [X: nat,Y3: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y3) )
     => ( X = Y3 ) ) ).

% Suc_inject
tff(fact_1141_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_1142_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_1143_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_1144_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M4: nat] : N = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_1145_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_1146_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_1147_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_1148_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_1149_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X4),zero_zero(nat)))
     => ( ! [Y4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y4)))
       => ( ! [X4: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X4),Y4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X4)),aa(nat,nat,suc,Y4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).

% diff_induct
tff(fact_1150_nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_induct
tff(fact_1151_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y3 != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_1152_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_1153_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_1154_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_1155_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_1156_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_1157_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_1158_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => pp(aa(nat,bool,P,I2)) )
       => ( ! [I2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I2)))
               => pp(aa(nat,bool,P,I2)) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% strict_inc_induct
tff(fact_1159_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( ! [I2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),aa(nat,nat,suc,I2)))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I),J)) ) ) ) ).

% less_Suc_induct
tff(fact_1160_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K)) ) ) ).

% less_trans_Suc
tff(fact_1161_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_SucD
tff(fact_1162_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_1163_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M5)) ) ) ).

% Suc_less_eq2
tff(fact_1164_All__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N))
        & ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
           => pp(aa(nat,bool,P,I4)) ) ) ) ).

% All_less_Suc
tff(fact_1165_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_1166_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_1167_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I4)) )
    <=> ( pp(aa(nat,bool,P,N))
        | ? [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
            & pp(aa(nat,bool,P,I4)) ) ) ) ).

% Ex_less_Suc
tff(fact_1168_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_1169_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_1170_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_1171_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I)),K))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_1172_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_lessD
tff(fact_1173_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K))
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_1174_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R4: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ! [X4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,X4),X4))
       => ( ! [X4: nat,Y4: nat,Z4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,X4),Y4))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,Y4),Z4))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,X4),Z4)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R4,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_1175_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_1176_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M3)),N2))
             => pp(aa(nat,bool,P,M3)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_1177_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_1178_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_1179_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_1180_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M6))
     => ? [M4: nat] : M6 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_1181_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_1182_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_1183_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_leD
tff(fact_1184_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_1185_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_1186_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_1187_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I))) ) ) ).

% zero_induct_lemma
tff(fact_1188_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: 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)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_1189_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_1190_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_1191_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_1192_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_1193_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_1194_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_1195_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N)),aa(nat,A,F,N3))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_1196_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N2))),aa(nat,A,F,N2)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N3)),aa(nat,A,F,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_1197_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),N: nat,N3: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,N3))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_1198_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),N: nat,M: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,N)),aa(nat,A,F,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_1199_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_1200_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
          & 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),N))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% Ex_less_Suc2
tff(fact_1201_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M7: nat] : N = aa(nat,nat,suc,M7) ) ).

% gr0_conv_Suc
tff(fact_1202_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,suc,N)))
         => 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),N))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I4))) ) ) ) ).

% All_less_Suc2
tff(fact_1203_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M4: nat] : N = aa(nat,nat,suc,M4) ) ).

% gr0_implies_Suc
tff(fact_1204_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( ( 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),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_1205_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1206_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1207_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_1208_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_1209_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_Suc_eq_le
tff(fact_1210_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_1211_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_lessD
tff(fact_1212_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I)) ) ) ) ).

% inc_induct
tff(fact_1213_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( pp(aa(nat,bool,P,I))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_1214_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_eq
tff(fact_1215_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_1216_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_1217_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)))) ).

% less_add_Suc1
tff(fact_1218_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)))) ).

% less_add_Suc2
tff(fact_1219_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1220_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1221_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_1222_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1223_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1224_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) ) ).

% Suc_diff_le
tff(fact_1225_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_1226_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_1227_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1228_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_1229_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1230_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1231_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1232_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_1233_int__of__nat__induct,axiom,
    ! [P: fun(int,bool),Z: int] :
      ( ! [N2: nat] : pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),N2)))
     => ( ! [N2: nat] : pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))))
       => pp(aa(int,bool,P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_1234_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_1235_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : power2(A,One,Times,A3,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_1236_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_1237_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1238_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_1239_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: 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),Y3))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y3)))) ) ).

% ceiling_add_le
tff(fact_1240_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_1241_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N))
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),K2))
               => ~ pp(aa(nat,bool,P,I3)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_1242_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,I))),N)) ) ).

% diff_Suc_less
tff(fact_1243_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1244_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1245_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),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),N))) ) ) ).

% one_less_mult
tff(fact_1246_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_1247_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K))) ) ).

% power_gt_expt
tff(fact_1248_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I))
     => 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),I),N))) ) ).

% nat_one_le_power
tff(fact_1249_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
    <=> ? [N6: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_1250_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1251_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_1252_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3)) ) ) ) ).

% power_Suc_le_self
tff(fact_1253_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_1254_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1255_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1256_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_1257_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1258_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N))),comm_s3205402744901411588hammer(A,Z,N)) ) ).

% pochhammer_rec'
tff(fact_1259_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A3,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_1260_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))) ).

% not_zle_0_negative
tff(fact_1261_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_1262_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_1263_Suc__as__int,axiom,
    ! [X2: nat] : aa(nat,nat,suc,X2) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X2)),one_one(int))) ).

% Suc_as_int
tff(fact_1264_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_1265_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: 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),Z)),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),Z)))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_1266_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_1267_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
        <=> ! [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))),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I4))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% ceiling_split
tff(fact_1268_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_1269_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z))
        <=> 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),Z)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_1270_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),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),Z)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_1271_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_1272_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X22: A] : aa(option(A),nat,size_option(A,X),aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1273_Heap_Osize__gen,axiom,
    ! [A: $tType,Xa: fun(A,nat),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,heap_Time_size_Heap(A,Xa),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size_gen
tff(fact_1274_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_1275_Heap_Osize_I2_J,axiom,
    ! [A: $tType,X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size(2)
tff(fact_1276_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_1277_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)),P2)) ) ) ).

% ceiling_divide_lower
tff(fact_1278_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_1279_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_1280_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_1281_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1282_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B2: 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) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_1283_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C2: A,B2: 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) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_1284_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_1285_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% div_0
tff(fact_1286_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_1287_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% div_by_1
tff(fact_1288_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_1289_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_1290_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_1291_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_1292_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_1293_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_1294_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_1295_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_1296_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1297_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1298_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1299_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( ( A3 = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = zero_zero(A) ) )
          & ( ( A3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_1300_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1301_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1302_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1303_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1304_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_1305_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_1306_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_1307_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_1308_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_1309_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_1310_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_1311_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_1312_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_1313_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_1314_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_1315_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_1316_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_1317_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_1318_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_1319_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_1320_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_1321_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_1322_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y3: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y3) )
         => ( X != Y3 ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1323_add__divide__distrib,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),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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_1324_nat__div__as__int,axiom,
    ! [X2: nat,Xa3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X2),Xa3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_div_as_int
tff(fact_1325_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_1326_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_1327_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_1328_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => 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),Y3))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_1329_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_1330_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => 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),Y3)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_1331_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_1332_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_1333_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3))) ) ) ) ).

% divide_neg_neg
tff(fact_1334_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_1335_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_1336_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => 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),Y3))) ) ) ) ).

% divide_pos_pos
tff(fact_1337_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_1338_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_1339_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_1340_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_1341_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_1342_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_1343_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_1344_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_1345_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_1346_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_1347_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_1348_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y3: A,Z: A,X: A,W: A] :
          ( ( Y3 != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1349_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = one_one(A) )
          <=> ( A3 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1350_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1351_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1352_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_one_over
tff(fact_1353_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_1354_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_1355_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_1356_Heap_Osize__neq,axiom,
    ! [A: $tType,X: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),X) != zero_zero(nat) ).

% Heap.size_neq
tff(fact_1357_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,X: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),W))) ) ) ) ) ) ).

% frac_le
tff(fact_1358_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A,W: A,Z: 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),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),W))) ) ) ) ) ) ).

% frac_less
tff(fact_1359_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A,W: A,Z: 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),Y3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),W))) ) ) ) ) ) ).

% frac_less2
tff(fact_1360_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_1361_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_1362_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => 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),Y3))) ) ) ) ).

% divide_nonneg_pos
tff(fact_1363_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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),Y3),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),Y3))) ) ) ) ).

% divide_nonpos_neg
tff(fact_1364_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: 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)),Y3))
           => 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),Y3)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_1365_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_1366_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_1367_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_1368_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_1369_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_1370_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_1371_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_1372_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_1373_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_1374_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1375_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y3)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1376_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_1377_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_1378_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_1379_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),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_1380_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z)),B2) = B2 ) )
          & ( ( Z != 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),Z)),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),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1381_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z)) = A3 ) )
          & ( ( Z != 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),Z)) = 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),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1382_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y3: A,Z: A,X: A,W: A] :
          ( ( Y3 != zero_zero(A) )
         => ( ( Z != 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),Y3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = 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),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3))),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1383_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y3: A,X: A,Z: A] :
          ( ( Y3 != 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),Y3)),Z) = 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),Z),Y3))),Y3) ) ) ) ).

% add_frac_num
tff(fact_1384_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y3: A,Z: A,X: A] :
          ( ( Y3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3)) = 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),Z),Y3))),Y3) ) ) ) ).

% add_num_frac
tff(fact_1385_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Y3),Z)) = 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),Z)),Y3)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1386_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Z)),Y3) = 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),Y3),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1387_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z)) = A3 ) )
          & ( ( Z != 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),Z)) = 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),Z)),B2)),Z) ) ) ) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1388_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y3: A,Z: A,X: A,W: A] :
          ( ( Y3 != zero_zero(A) )
         => ( ( Z != 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),Y3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = 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),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3))),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1389_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Y3),Z)) = 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),Z)),Y3)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1390_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Z)),Y3) = 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),Y3),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1398_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),X)),Y3) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3)) ) ) ) ).

% abs_div_pos
tff(fact_1399_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1400_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y3),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_1401_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y3),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_1402_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y3),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_1403_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,aa(B,B,aa(B,fun(B,B),divide_divide(B),X),Y3),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_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y3))
         => ( 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),Z),Y3)))
           => 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),Y3)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1412_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y3)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1413_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_1414_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_1415_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_1416_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,Z: A,X: A,W: A] :
          ( ( Y3 != zero_zero(A) )
         => ( ( Z != 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),Y3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> 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),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3))),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1417_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y3: A,Z: A,X: A,W: A] :
          ( ( Y3 != zero_zero(A) )
         => ( ( Z != 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),Y3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)))
            <=> 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),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y3))),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Z))),Y3) = 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),Y3),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1425_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z))),B2) = B2 ) )
          & ( ( Z != 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),Z))),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),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1426_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != 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),Z))),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),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1427_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A3: A,B2: A] :
          ( ( ( Z = 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),Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != 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),Z)),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),Z))),Z) ) ) ) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1428_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y3: A] :
          ( ( Z != 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),Z))),Y3) = 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),Y3),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1429_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_diff
tff(fact_1430_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = 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),N)),N)) ) ) ) ).

% div_geq
tff(fact_1431_verit__less__mono__div__int2,axiom,
    ! [A5: int,B4: int,N: 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),N)))
       => 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),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_1432_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_1433_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_1434_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_1435_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_1436_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_1437_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_1438_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R: A,S: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R),S))
             => 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),R),aa(A,A,aa(A,fun(A,A),minus_minus(A),V),U))),S))),V)) ) ) ) ) ).

% scaling_mono
tff(fact_1439_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
         => ~ ! [N2: 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,N2)))),E2)) ) ) ).

% nat_approx_posE
tff(fact_1440_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),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),P2),Q2)))),Q2))) ) ) ).

% ceiling_divide_upper
tff(fact_1441_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_1442_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A3: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_1443_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_1444_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_1445_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_1446_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_1447_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1448_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_1449_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_1450_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_1451_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_1452_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).

% div_minus1_right
tff(fact_1453_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_1454_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_1455_tuple__isomorphism_Osize__neq,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism(A,B,C)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),X) != zero_zero(nat) ).

% tuple_isomorphism.size_neq
tff(fact_1456_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_1457_div__le__mono2,axiom,
    ! [M: nat,N: 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),N))
       => 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),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M))) ) ) ).

% div_le_mono2
tff(fact_1458_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_1459_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q2)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2))) ) ) ).

% div_less_iff_less_mult
tff(fact_1460_div__eq__dividend__iff,axiom,
    ! [M: nat,N: 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),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_1461_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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),N)),M)) ) ) ).

% div_less_dividend
tff(fact_1462_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1463_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1464_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = 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),N)),N)) ) ) ) ).

% div_if
tff(fact_1465_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q2)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1466_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))))) ) ).

% dividend_less_times_div
tff(fact_1467_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1468_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => pp(aa(nat,bool,P,I4)) ) ) ) ) ) ).

% split_div
tff(fact_1469_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = 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),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_1470_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),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),N),aa(nat,nat,suc,Q4))))
            & pp(aa(nat,bool,P,Q4)) ) ) ) ).

% split_div'
tff(fact_1471_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),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),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),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),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1472_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% bits_div_by_1
tff(fact_1473_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_1474_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_1475_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_1476_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_1477_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q2),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_1478_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_ba(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_1479_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_12))
     => ( ! [X4: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X4))
           => ? [Y5: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y5))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X4),Y5)) ) )
       => ? [F3: fun(nat,A)] :
          ! [N7: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N7),aa(nat,A,F3,N7)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N7),aa(nat,A,F3,N7)),aa(nat,A,F3,aa(nat,nat,suc,N7)))) ) ) ) ).

% dependent_nat_choice
tff(fact_1480_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),zero_zero(A))) ) ).

% dvd_0_right
tff(fact_1481_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_1482_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_1483_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_1484_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,uminus_uminus(A),X)),Y3))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y3)) ) ) ).

% minus_dvd_iff
tff(fact_1485_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,uminus_uminus(A),Y3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y3)) ) ) ).

% dvd_minus_iff
tff(fact_1486_div__dvd__div,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),B2))
         => ( 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),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ) ) ).

% div_dvd_div
tff(fact_1487_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),aa(A,A,abs_abs(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).

% dvd_abs_iff
tff(fact_1488_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,abs_abs(A),M)),K))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),K)) ) ) ).

% abs_dvd_iff
tff(fact_1489_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_1490_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).

% gbinomial_1
tff(fact_1491_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_1492_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_1493_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_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_div__add,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),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(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)) ) ) ) ) ).

% div_add
tff(fact_1501_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)),one_one(A))) ) ) ) ).

% unit_div
tff(fact_1502_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),one_one(A))) ) ) ).

% unit_div_1_unit
tff(fact_1503_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_1504_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_1505_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_1506_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_1507_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( ( K = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1508_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_1509_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_1510_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_1511_int__dvd__int__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ).

% int_dvd_int_iff
tff(fact_1512_zdvd1__eq,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),one_one(int)))
    <=> ( aa(int,int,abs_abs(int),X) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_1513_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_1514_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_1515_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_1516_sgn__mult__dvd__iff,axiom,
    ! [R: int,L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),L)),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_1517_mult__sgn__dvd__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R))),K))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        & ( ( R = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_1518_dvd__sgn__mult__iff,axiom,
    ! [L: int,R: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R)),K)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_1519_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),L),K))
        | ( R = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_1520_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_1521_nat__abs__dvd__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_abs_dvd_iff
tff(fact_1522_dvd__nat__abs__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ).

% dvd_nat_abs_iff
tff(fact_1523_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_bb(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(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_1524_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != B2 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
          & ( A3 != B2 ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
tff(fact_1525_dvd__trans,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),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_trans
tff(fact_1526_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),A3)) ) ).

% dvd_refl
tff(fact_1527_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => ( A3 != B2 ) ) ).

% gcd_nat.strict_implies_not_eq
tff(fact_1528_gcd__nat_Ostrict__implies__order,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2)) ) ).

% gcd_nat.strict_implies_order
tff(fact_1529_gcd__nat_Ostrict__iff__order,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) ) ) ).

% gcd_nat.strict_iff_order
tff(fact_1530_gcd__nat_Oorder__iff__strict,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
          & ( A3 != B2 ) )
        | ( A3 = B2 ) ) ) ).

% gcd_nat.order_iff_strict
tff(fact_1531_gcd__nat_Ostrict__iff__not,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3)) ) ) ).

% gcd_nat.strict_iff_not
tff(fact_1532_gcd__nat_Ostrict__trans2,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
          & ( A3 != C2 ) ) ) ) ).

% gcd_nat.strict_trans2
tff(fact_1533_gcd__nat_Ostrict__trans1,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
          & ( A3 != C2 ) ) ) ) ).

% gcd_nat.strict_trans1
tff(fact_1534_gcd__nat_Ostrict__trans,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
          & ( B2 != C2 ) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
          & ( A3 != C2 ) ) ) ) ).

% gcd_nat.strict_trans
tff(fact_1535_gcd__nat_Oantisym,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3))
       => ( A3 = B2 ) ) ) ).

% gcd_nat.antisym
tff(fact_1536_gcd__nat_Oirrefl,axiom,
    ! [A3: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),A3))
        & ( A3 != A3 ) ) ).

% gcd_nat.irrefl
tff(fact_1537_gcd__nat_Oeq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = B2 )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3)) ) ) ).

% gcd_nat.eq_iff
tff(fact_1538_gcd__nat_Otrans,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2)) ) ) ).

% gcd_nat.trans
tff(fact_1539_gcd__nat_Orefl,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),A3)) ).

% gcd_nat.refl
tff(fact_1540_gcd__nat_Oasym,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3))
          & ( B2 != A3 ) ) ) ).

% gcd_nat.asym
tff(fact_1541_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_1542_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( ( A3 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1543_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),A3))
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1544_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_1545_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_1546_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_1547_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_1548_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_1549_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_1550_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)))
         => ? [B6: A,C5: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),C5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B6),B2))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C5),C2)) ) ) ) ).

% division_decomp
tff(fact_1551_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_1552_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))
         => ~ ! [X4: A,Y4: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X4),Y4) )
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X4),A3))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y4),B2)) ) ) ) ) ).

% dvd_productE
tff(fact_1553_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_1554_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_1555_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_1556_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(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),C2))
           => 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),C2))) ) ) ) ).

% dvd_add
tff(fact_1557_dvd__add__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),C2))
         => ( 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),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_1558_dvd__add__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),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_add_right_iff
tff(fact_1559_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),one_one(A)),A3)) ) ).

% one_dvd
tff(fact_1560_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% unit_imp_dvd
tff(fact_1561_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1562_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y3: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Z))) ) ) ) ).

% dvd_diff
tff(fact_1563_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D3: A,B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(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)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1564_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C2: A,B2: 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),dvd_dvd(A),C2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
             => ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1565_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( ( 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) )
            <=> ( A3 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1566_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat))) ).

% gcd_nat.extremum
tff(fact_1567_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_1568_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_1569_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),zero_zero(nat)))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_1570_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),zero_zero(nat)),A3))
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_1571_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),L),K)) ) ) ).

% dvd_if_abs_eq
tff(fact_1572_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: 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),N))
       => 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),N))) ) ) ).

% dvd_diff_nat
tff(fact_1573_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),M)) ) ) ).

% zdvd_zdiffD
tff(fact_1574_zdvd__antisym__abs,axiom,
    ! [A3: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
       => ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) ) ) ) ).

% zdvd_antisym_abs
tff(fact_1575_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_bb(A,fun(A,bool),A3))),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_bb(A,fun(A,bool),B2))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% subset_divisors_dvd
tff(fact_1576_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_1577_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),zero_zero(A)),one_one(A))) ) ).

% not_is_unit_0
tff(fact_1578_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_1579_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_1580_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_1581_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_1582_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_1583_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_1584_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_1585_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1586_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_1587_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_1588_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_1589_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_1590_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_1591_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_1592_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(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),divide_divide(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),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_1593_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),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)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(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)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_1594_unit__div__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),divide_divide(A),B2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1595_div__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),divide_divide(A),A3),B2)),C2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1596_dvd__div__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),divide_divide(A),C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1597_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_1598_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_1599_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ) ).

% nat_dvd_not_less
tff(fact_1600_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_1601_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: 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)),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_1602_zdvd__not__zless,axiom,
    ! [M: int,N: 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),N))
       => ~ pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ) ).

% zdvd_not_zless
tff(fact_1603_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,N))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_1604_dvd__minus__self,axiom,
    ! [M: nat,N: 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),N),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% dvd_minus_self
tff(fact_1605_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)))
     => ( ( K != zero_zero(int) )
       => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N)) ) ) ).

% zdvd_mult_cancel
tff(fact_1606_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: 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),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),M)) ) ) ) ).

% dvd_diffD
tff(fact_1607_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: 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),N)))
     => ( 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),N),M))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N)) ) ) ) ).

% dvd_diffD1
tff(fact_1608_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_1609_bezout__add__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D2: nat,X4: nat,Y4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D2) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_1610_bezout__lemma__nat,axiom,
    ! [D3: nat,A3: nat,B2: nat,X: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),D3) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y3)),D3) ) )
         => ? [X4: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),Y4)),D3) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)),D3) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1611_bezout1__nat,axiom,
    ! [A3: nat,B2: nat] :
    ? [D2: nat,X4: nat,Y4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
      & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = D2 )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_1612_zdvd__period,axiom,
    ! [A3: int,D3: int,X: int,T2: int,C2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),D3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),T2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D3))),T2))) ) ) ).

% zdvd_period
tff(fact_1613_zdvd__reduce,axiom,
    ! [K: int,N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),aa(int,int,aa(int,fun(int,int),times_times(int),K),M))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),K),N)) ) ).

% zdvd_reduce
tff(fact_1614_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(int,nat,nat2,Z)),M))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),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)),Z))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_1615_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1625_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_1626_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_1627_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_1628_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_1629_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_1630_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_1631_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1632_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% dvd_imp_le
tff(fact_1633_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
     => ( 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),N)) ) ) ).

% dvd_mult_cancel
tff(fact_1634_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: 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),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1635_bezout__add__strong__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D2: nat,X4: nat,Y4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D2),B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1636_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Z),N))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_1637_dvd__imp__le__int,axiom,
    ! [I: int,D3: int] :
      ( ( I != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),I))
       => 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),I))) ) ) ).

% dvd_imp_le_int
tff(fact_1638_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A3: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),A3))
         => pp(aa(set(A),bool,member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B2))),ring_1_Ints(A))) ) ) ).

% of_int_divide_in_Ints
tff(fact_1639_dvd__fact,axiom,
    ! [M: nat,N: 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),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_1640_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_1641_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_1642_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) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),one_one(A)))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = 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),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1643_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_1644_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_1645_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_1646_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))) ) ) ).

% dvd_power
tff(fact_1647_dvd__mult__cancel1,axiom,
    ! [M: nat,N: 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),N)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1648_dvd__mult__cancel2,axiom,
    ! [M: nat,N: 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),N),M)),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1649_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),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),N),Q2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R),M)),Q2)))) ) ) ) ).

% dvd_minus_add
tff(fact_1650_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),N)),M))
      <=> ( aa(int,int,abs_abs(int),N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_1651_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_1652_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_1653_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_1654_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_1655_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_1656_normalize__denom__pos,axiom,
    ! [R: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2)) ) ).

% normalize_denom_pos
tff(fact_1657_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_1658_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_1659_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_1660_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_1661_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_1662_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_1663_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X2: 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),X2),T2)))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T2))) ) ) ) ).

% inf_period(4)
tff(fact_1664_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D3: A,D5: A,T2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),D5))
         => ! [X2: 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),X2),T2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X2),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D5))),T2))) ) ) ) ).

% inf_period(3)
tff(fact_1665_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X3: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X3)))
        <=> ? [X3: 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),X3),zero_zero(A))))
              & pp(aa(A,bool,P,X3)) ) ) ) ).

% unity_coeff_ex
tff(fact_1666_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S: B] :
        ? [Z4: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z4))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(10)
tff(fact_1667_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S: B] :
        ? [Z4: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z4))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(9)
tff(fact_1668_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S: B] :
        ? [Z4: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z4),X2))
         => ( ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(10)
tff(fact_1669_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D3: B,S: B] :
        ? [Z4: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z4),X2))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(aa(B,bool,aa(B,fun(B,bool),dvd_dvd(B),D3),aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(9)
tff(fact_1670_tuple__isomorphism_Osize_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,X1: fun(A,product_prod(B,C)),X22: fun(product_prod(B,C),A)] : aa(tuple_isomorphism(A,B,C),nat,size_size(tuple_isomorphism(A,B,C)),tuple_1188178415141063261rphism(A,B,C,X1,X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% tuple_isomorphism.size(2)
tff(fact_1671_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_aa(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),B4))) ) ).

% less_set_def
tff(fact_1672_tuple__isomorphism_Osize__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType,Xb2: fun(A,nat),Xa: fun(B,nat),X: fun(C,nat),X1: fun(A,product_prod(B,C)),X22: fun(product_prod(B,C),A)] : tuple_9181185373184732606rphism(A,B,C,Xb2,Xa,X,tuple_1188178415141063261rphism(A,B,C,X1,X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% tuple_isomorphism.size_gen
tff(fact_1673_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_1674_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2))) ) ) ).

% floor_divide_upper
tff(fact_1675_verit__le__mono__div__int,axiom,
    ! [A5: int,B4: int,N: 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)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N)),aa(int,int,aa(int,fun(int,int),aa(bool,fun(int,fun(int,int)),if(int),aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B4,N)),zero_zero(int))),one_one(int)),zero_zero(int)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ).

% verit_le_mono_div_int
tff(fact_1676_verit__le__mono__div,axiom,
    ! [A5: nat,B4: nat,N: 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)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),N)),aa(nat,nat,aa(nat,fun(nat,nat),aa(bool,fun(nat,fun(nat,nat)),if(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B4,N)),zero_zero(nat))),one_one(nat)),zero_zero(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ).

% verit_le_mono_div
tff(fact_1677_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: 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),W))))
        <=> ( ( 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),W))),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),W))),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),W)))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_1678_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: 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),W))),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),W))),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),W))),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),W))),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_1679_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: 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,Y3))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y3)) ) )
          & ( ~ 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,Y3))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = 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,Y3))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_1680_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_1681_nat__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_1682_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_1683_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_1684_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_1685_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_1686_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,A3) = zero_zero(A) ) ).

% mod_self
tff(fact_1687_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self2
tff(fact_1688_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A3),B2) = modulo_modulo(A,A3,B2) ) ).

% mod_add_self1
tff(fact_1689_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_1690_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,N: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_1691_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_1692_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_1693_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_1694_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1695_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_1696_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_1697_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_1698_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_1699_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_1700_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_1701_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_1702_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_1703_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_1704_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_1705_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_1706_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_1707_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_1708_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_1709_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_1710_nat__neg__numeral,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_1711_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_1712_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_1713_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X: num,N: nat] :
      ( ( aa(int,nat,nat2,Y3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y3 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_1714_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y3: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N) = aa(int,nat,nat2,Y3) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_1715_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A3 )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1716_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) )
        <=> ( ( ( aa(num,A,numeral_numeral(A),W) != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) = B2 ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1717_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_1718_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_1719_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_le_iff
tff(fact_1720_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_less_iff
tff(fact_1721_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_1722_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y3) ) ).

% semiring_norm(168)
tff(fact_1723_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self1
tff(fact_1724_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self2
tff(fact_1725_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self3
tff(fact_1726_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self4
tff(fact_1727_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_1728_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y3: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) = aa(int,A,ring_1_of_int(A),Y3) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) = Y3 ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_1729_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y3: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y3 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_1730_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_1731_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: A] :
          ( ( pp(aa(set(A),bool,member(A,X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,member(A,Y3),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y3)) ) ) ) ).

% floor_add2
tff(fact_1732_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_1733_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A,W: 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),W))) )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != 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),W))) = B2 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1734_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: 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),W))) = A3 )
        <=> ( ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != 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),W))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A3 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),minus_minus(int),archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_1742_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_1743_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: 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)),N)),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)),N)),A3)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_1744_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: 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)),N)))
    <=> 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)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_1745_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(int,nat,nat2,A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_1746_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A3: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),X)),N)),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)),N)),A3)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_1747_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),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),V)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_1748_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> 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),V)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_1749_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> 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),V)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_1750_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),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),V)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_1751_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_1752_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_1753_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_1754_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),X)),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),X)),N)),A3)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_1755_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y3: int,X: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) )
        <=> ( Y3 = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_1756_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y3: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) = aa(int,A,ring_1_of_int(A),Y3) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y3 ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_1757_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_1758_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_1759_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: 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),V))))
        <=> 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),V))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_1760_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: 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),V))),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),V))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_1761_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_1762_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_1763_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_1764_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),aa(int,A,ring_1_of_int(A),A3)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A3)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_1765_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_1766_nat__numeral__as__int,axiom,
    ! [X2: num] : aa(num,nat,numeral_numeral(nat),X2) = aa(int,nat,nat2,aa(num,int,numeral_numeral(int),X2)) ).

% nat_numeral_as_int
tff(fact_1767_mod__add__right__eq,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),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_right_eq
tff(fact_1768_mod__add__left__eq,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),modulo_modulo(A,A3,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_left_eq
tff(fact_1769_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A6: A,B2: A,B5: A] :
          ( ( modulo_modulo(A,A3,C2) = modulo_modulo(A,A6,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B5,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B5),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_1770_mod__add__eq,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),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C2) ) ).

% mod_add_eq
tff(fact_1771_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),modulo_modulo(A,A3,B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1772_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(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),C2),modulo_modulo(A,A3,B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3)) ) ) ) ).

% dvd_mod_iff
tff(fact_1773_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_1774_nat__mod__as__int,axiom,
    ! [X2: nat,Xa3: nat] : modulo_modulo(nat,X2,Xa3) = aa(int,nat,nat2,modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X2),aa(nat,int,semiring_1_of_nat(int),Xa3))) ).

% nat_mod_as_int
tff(fact_1775_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: num] : pp(aa(set(A),bool,member(A,aa(num,A,numeral_numeral(A),N)),ring_1_Ints(A))) ) ).

% Ints_numeral
tff(fact_1776_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W: num] : pp(aa(set(A),bool,member(A,aa(num,A,numeral_numeral(A),W)),semiring_1_Nats(A))) ) ).

% Nats_numeral
tff(fact_1777_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_1778_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_1779_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_1780_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_1781_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1782_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
        <=> ( modulo_modulo(A,B2,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1783_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% mod_0_imp_dvd
tff(fact_1784_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( modulo_modulo(A,A3,B2) = A3 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_1785_div__add1__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),plus_plus(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),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))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,C2)),modulo_modulo(A,B2,C2))),C2)) ) ).

% div_add1_eq
tff(fact_1786_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),modulo_modulo(A,A3,B2)))) ) ).

% dvd_minus_mod
tff(fact_1787_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = aa(nat,nat,suc,modulo_modulo(nat,M,N)) ) ) ) ).

% mod_Suc
tff(fact_1788_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M4),zero_zero(nat)))
     => ( ! [M4: nat,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),P,N2),modulo_modulo(nat,M4,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M4),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_1789_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_1790_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_1791_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_1792_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_1793_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_1794_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_1795_mod__eq__0D,axiom,
    ! [M: nat,D3: nat] :
      ( ( modulo_modulo(nat,M,D3) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D3),Q3) ) ).

% mod_eq_0D
tff(fact_1796_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_1797_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_1798_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_1799_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y3: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y3,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1800_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_1801_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_1802_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
     => ( modulo_modulo(int,K,L) = R ) ) ).

% mod_int_unique
tff(fact_1803_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_1804_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_1805_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( modulo_modulo(A,A3,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: 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,Y3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)))) ) ).

% le_floor_add
tff(fact_1819_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_1820_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),X)) ) ).

% int_add_floor
tff(fact_1821_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_1822_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_1823_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_1824_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_1825_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_1826_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_1827_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1828_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_1829_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_1830_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_1831_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_1832_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_1833_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1834_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = 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),W)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1835_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_1836_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_1837_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_1838_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_1839_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_1840_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y3: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y3,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),X))
       => ? [Q3: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1841_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ~ ! [S4: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_1842_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q2) = modulo_modulo(nat,N,Q2) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ~ ! [S4: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_1843_div__less__mono,axiom,
    ! [A5: nat,B4: nat,N: 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)),N))
       => ( ( modulo_modulo(nat,A5,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B4,N) = 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),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B4),N))) ) ) ) ) ).

% div_less_mono
tff(fact_1844_zdiv__mono__strict,axiom,
    ! [A5: int,B4: int,N: 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)),N))
       => ( ( modulo_modulo(int,A5,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B4,N) = 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),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B4),N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1845_div__mod__decomp,axiom,
    ! [A5: nat,N: nat] : A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),N)),N)),modulo_modulo(nat,A5,N)) ).

% div_mod_decomp
tff(fact_1846_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N),Q2))),modulo_modulo(nat,M,N)) ).

% mod_mult2_eq
tff(fact_1847_div__mod__decomp__int,axiom,
    ! [A5: int,N: int] : A5 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),N)),N)),modulo_modulo(int,A5,N)) ).

% div_mod_decomp_int
tff(fact_1848_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_1849_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),aa(nat,num,num_of_nat,N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_1850_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R)))),R)) ) ) ).

% of_nat_floor
tff(fact_1851_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N: nat] : modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A3,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_1852_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_1853_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1854_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),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),W)),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),W)),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),W)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1855_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: 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),W)))
        <=> ( ( 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),W)),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),W)),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),W))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1856_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = 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),W))),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_1857_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: 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),W)) )
        <=> ( ( ( 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),W))),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_1858_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),aa(nat,num,num_of_nat,N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_1859_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
          | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)) ) )
      & ( ~ ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
            | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N))) ) )
          & ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_1860_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_1861_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_1862_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
        <=> ! [I4: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A)))) )
             => pp(aa(int,bool,P,I4)) ) ) ) ).

% floor_split
tff(fact_1863_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_1864_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_1865_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),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),Z)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_1866_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z))
        <=> 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),Z)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_1867_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),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),W)),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),W)),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),W)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1868_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: 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),W)))
        <=> ( ( 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),W)),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),W)),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),W))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1869_Suc__times__mod__eq,axiom,
    ! [M: nat,N: 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),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1870_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: 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),W))),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),W))),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),W))),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),W))),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_1871_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: 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),W))))
        <=> ( ( 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),W))),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),W))),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),W)))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_1872_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)),P2)) ) ) ).

% floor_divide_lower
tff(fact_1873_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_1874_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1875_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_1876_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_1877_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_1878_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_1879_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_1880_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( aa(nat,product_prod(nat,nat),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))
       => ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M) = aa(nat,product_prod(nat,nat),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_1881_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_1882_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_1883_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_1884_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_1885_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_1886_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_1887_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_1888_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_1889_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_1890_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_1891_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod(nat,nat)] :
      ~ ! [K2: nat,M4: nat] : X != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M4) ).

% prod_decode_aux.cases
tff(fact_1892_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_1893_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_1894_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_1895_numeral__1__eq__Suc__0,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% numeral_1_eq_Suc_0
tff(fact_1896_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(nat,num,num_of_nat,aa(nat,nat,suc,N)) = inc(aa(nat,num,num_of_nat,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(nat,num,num_of_nat,aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_1897_num__of__nat_Osimps_I1_J,axiom,
    aa(nat,num,num_of_nat,zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_1898_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_1899_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_1900_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_1901_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( aa(nat,num,num_of_nat,N) = one2 ) ) ).

% num_of_nat_One
tff(fact_1902_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_1903_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y3: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa) = Y3 )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y3 = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y3 = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_1904_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),comm_s3205402744901411588hammer(A,Z,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ).

% pochhammer_double
tff(fact_1905_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_1906_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( 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),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_1907_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),zero_zero(int)))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1908_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_1909_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1910_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),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),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_1911_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( bit0(X22) = bit0(Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1912_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_1913_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% odd_add
tff(fact_1914_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) ) ) ) ).

% even_add
tff(fact_1915_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_1916_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_1917_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_1918_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_1919_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_1920_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_1921_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_1922_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_1923_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_1924_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_1925_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_1926_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_1927_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_1928_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).

% even_plus_one_iff
tff(fact_1929_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ).

% even_diff
tff(fact_1930_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1931_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1932_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_1933_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_1934_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y3: 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)),Y3))
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( X = Y3 ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1935_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_1936_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_1937_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_1938_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(bool,A,zero_neq_one_of_bool(A),B2)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_1939_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_1940_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_1941_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_1942_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_1943_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_1944_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_1945_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_1946_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: 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),W))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_1947_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: 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),W))),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_1948_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_1949_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_1950_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_1951_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_1952_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_1953_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).

% even_diff_nat
tff(fact_1954_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),bit0(one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_1955_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_1956_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1957_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: 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),W))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_1958_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_1959_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_1960_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1961_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: 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),W))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W)))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W)))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_1962_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_1963_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_1964_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_1965_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_1966_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(N)),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_1967_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_1968_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),bit0(W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_even
tff(fact_1969_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),B2) ) ).

% left_add_twice
tff(fact_1970_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_1971_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_1972_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A))) ) ).

% even_zero
tff(fact_1973_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2))) ) ) ) ).

% odd_even_add
tff(fact_1974_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A))) ) ).

% odd_one
tff(fact_1975_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_1976_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_1977_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_1978_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_1979_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = bit0(aa(nat,num,num_of_nat,N)) ) ) ).

% num_of_nat_double
tff(fact_1980_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),bit0(one2))))) ) ) ).

% half_gt_zero
tff(fact_1981_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),bit0(one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ).

% half_gt_zero_iff
tff(fact_1982_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% zero_le_power2
tff(fact_1983_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y3))
             => ( X = Y3 ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1984_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3)) ) ) ) ).

% power2_le_imp_le
tff(fact_1985_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1986_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_1987_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_1988_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1989_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1990_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1991_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1992_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] : 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,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_1993_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A3 = one_one(A) )
            | ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_1994_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_1995_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_1996_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_1997_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_1998_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_1999_nat__induct2,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct2
tff(fact_2000_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = modulo_modulo(A,A3,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2001_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ~ ! [B3: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_2002_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y3)) ) ) ) ).

% power2_less_imp_less
tff(fact_2003_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_2004_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y3 = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2005_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2006_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_2007_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2008_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_2009_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y3 != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2010_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y3)) ) ).

% power2_sum
tff(fact_2011_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2012_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ) ).

% zero_le_even_power'
tff(fact_2013_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y3: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y3)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2014_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% zero_le_even_power
tff(fact_2015_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% zero_le_odd_power
tff(fact_2016_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2017_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2018_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),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_2019_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A3: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => pp(aa(A,bool,P,A4)) )
         => ( ! [A4: A,B3: bool] :
                ( pp(aa(A,bool,P,A4))
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => 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),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4)))) ) )
           => pp(aa(A,bool,P,A3)) ) ) ) ).

% bits_induct
tff(fact_2020_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat,M: nat] : modulo_modulo(A,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),bit0(one2))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),modulo_modulo(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2021_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => 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),bit0(one2))),N2)))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_2022_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% minus_one_power_iff
tff(fact_2023_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% div_2_gt_zero
tff(fact_2024_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2025_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2026_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),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),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2027_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),X)),Y3)) ) ).

% power2_diff
tff(fact_2028_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2029_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_2030_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & ( A3 != zero_zero(A) ) )
            | ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2031_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2032_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),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ? [N2: 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),N2)),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),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_2033_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),bit0(one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
       => ? [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),B2),N2)),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),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_2034_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),bit0(one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2035_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))))
             => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2) = modulo_modulo(A,A3,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2036_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_2037_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),zero_zero(A))) )
              | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2038_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% even_mod_4_div_2
tff(fact_2039_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_2040_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N))) ) ).

% exp_div_exp_eq
tff(fact_2041_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A3: int,Q2: int,R: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A3,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2042_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R: A,Q2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R) ) ) ) ) ).

% divmod_step_eq
tff(fact_2043_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_2044_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_2045_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_2046_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2047_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_2048_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2049_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2050_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A3)))
        <=> ~ ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2051_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M,A3)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2052_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_2053_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y3: 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),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))))
           => ( archimedean_round(A,X) = Y3 ) ) ) ) ).

% round_unique
tff(fact_2054_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),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),bit0(one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_2055_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
           => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)))
           => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_2056_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_2057_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),bit0(one2))))) ) ).

% of_int_round_abs_le
tff(fact_2058_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),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_2059_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2060_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_2061_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_2062_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2063_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2064_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_2065_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ).

% binomial_eq_0_iff
tff(fact_2066_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2067_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_2068_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2069_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2070_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2071_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% zero_less_binomial_iff
tff(fact_2072_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : bit_ri4674362597316999326ke_bit(A,zero_zero(nat),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2073_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_2074_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2075_choose__mult__lemma,axiom,
    ! [M: nat,R: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R)),M)) ).

% choose_mult_lemma
tff(fact_2076_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K))) ) ).

% zero_less_binomial
tff(fact_2077_Suc__times__binomial__add,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A3)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),aa(nat,nat,suc,A3))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2))),A3)) ).

% Suc_times_binomial_add
tff(fact_2078_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_2079_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_2080_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2081_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: 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(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2082_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2083_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_2084_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_2085_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_ri4674362597316999326ke_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_2086_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),bit0(one2)))))) ) ).

% of_int_round_le
tff(fact_2087_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),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_2088_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X)) ) ).

% set_decode_0
tff(fact_2089_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_bc(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_2090_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_bd(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2091_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( bit_se2584673776208193580ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_2092_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_2093_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,bit0(N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_2094_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% Suc_0_xor_eq
tff(fact_2095_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_2096_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_2097_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_2098_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_2099_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_2100_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).

% case_prod_conv
tff(fact_2101_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),product_curry(A,B,C,F)) = F ).

% case_prod_curry
tff(fact_2102_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: 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),F)) = F ).

% curry_case_prod
tff(fact_2103_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2584673776208193580ke_bit(A,zero_zero(nat),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_2104_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_2105_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_2106_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( bit_se2584673776208193580ke_bit(A,N,one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2107_take__bit__of__Suc__0,axiom,
    ! [N: nat] : bit_se2584673776208193580ke_bit(nat,N,aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_2108_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_2109_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_2110_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2584673776208193580ke_bit(A,N,A3)))
        <=> ( ( N = zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% even_take_bit_eq
tff(fact_2111_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,bit0(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2112_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat)),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_2113_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,bit0(M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_be(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_2114_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),X: product_prod(B,C),Y3: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F),X),Y3) = 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_bf(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y3)),X) ).

% case_prod_app
tff(fact_2115_prod_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F: 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),F),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_bg(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F)),Prod) ).

% prod.case_distrib
tff(fact_2116_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A,B2: A] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se2584673776208193580ke_bit(A,N,A3)),bit_se2584673776208193580ke_bit(A,N,B2))) = bit_se2584673776208193580ke_bit(A,N,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) ) ).

% take_bit_add
tff(fact_2117_old_Oprod_Ocase,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(B,C)),X1: A,X22: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = aa(B,C,aa(A,fun(B,C),F,X1),X22) ).

% old.prod.case
tff(fact_2118_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q2: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P2: product_prod(A,B)] :
      ( ! [X4: A,Y4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) = Q2 )
         => ( aa(B,C,aa(A,fun(B,C),F,X4),Y4) = aa(B,C,aa(A,fun(B,C),G,X4),Y4) ) )
     => ( ( P2 = Q2 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q2) ) ) ) ).

% split_cong
tff(fact_2119_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_2120_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,bool),P: fun(B,fun(C,A)),Z: 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),Z)))
     => ~ ! [X4: B,Y4: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y4) )
           => ~ pp(aa(A,bool,Q,aa(C,A,aa(B,fun(C,A),P,X4),Y4))) ) ) ).

% case_prodE2
tff(fact_2121_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_bh(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).

% case_prod_eta
tff(fact_2122_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X4: A,Y4: B] : aa(B,C,aa(A,fun(B,C),F,X4),Y4) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).

% cond_case_prod_eta
tff(fact_2123_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_bi(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_2124_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_2125_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_2126_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_2127_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A3)) ) ) ).

% take_bit_eq_0_iff
tff(fact_2128_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_2129_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_2130_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_2131_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_2132_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_2133_take__bit__int__less__eq,axiom,
    ! [N: 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),bit0(one2))),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),bit_se2584673776208193580ke_bit(int,N,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),bit0(one2))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_2134_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_2135_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_2136_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))) ) ).

% xor_one_eq
tff(fact_2137_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)))) ) ).

% one_xor_eq
tff(fact_2138_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( bit_se2584673776208193580ke_bit(A,N,A3) = zero_zero(A) ) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
             => ( bit_se2584673776208193580ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2139_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_bj(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2140_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_bk(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_2141_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% xor_Suc_0_eq
tff(fact_2142_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),aTP_Lamp_bl(num,fun(num,int),M)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_2143_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_bm(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_2144_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_bn(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_2145_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,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_2146_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_2147_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2148_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% odd_mod_4_div_2
tff(fact_2149_verit__eq__simplify_I9_J,axiom,
    ! [X32: num,Y32: num] :
      ( ( aa(num,num,bit1,X32) = aa(num,num,bit1,Y32) )
    <=> ( X32 = Y32 ) ) ).

% verit_eq_simplify(9)
tff(fact_2150_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,bool))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => pp(aa(B,bool,aa(A,fun(B,bool),C2,A4),B3)) )
     => 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),P2)) ) ).

% case_prodI2
tff(fact_2151_case__prodI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),F,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),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).

% case_prodI
tff(fact_2152_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => pp(aa(set(C),bool,member(C,Z),aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3))) )
     => pp(aa(set(C),bool,member(C,Z),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),P2))) ) ).

% mem_case_prodI2
tff(fact_2153_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A3: B,B2: C] :
      ( pp(aa(set(A),bool,member(A,Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B2)))
     => pp(aa(set(A),bool,member(A,Z),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_2154_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool))),X: C] :
      ( ! [A4: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P2 )
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A4),B3),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),P2),X)) ) ).

% case_prodI2'
tff(fact_2155_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_2156_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),bit0(M)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_bo(num,option(num))),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_2157_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_2158_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(X)) ) ).

% xor_numerals(8)
tff(fact_2159_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_2160_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(num,A,numeral_numeral(A),bit0(Y3)) ) ).

% xor_numerals(2)
tff(fact_2161_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y3))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3)) ) ).

% xor_numerals(1)
tff(fact_2162_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_2163_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2164_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2165_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2166_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_2167_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_2168_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(X)) ).

% xor_nat_numerals(4)
tff(fact_2169_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_2170_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3))) = aa(num,nat,numeral_numeral(nat),bit0(Y3)) ).

% xor_nat_numerals(2)
tff(fact_2171_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y3))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3)) ).

% xor_nat_numerals(1)
tff(fact_2172_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2173_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_2174_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% xor_numerals(4)
tff(fact_2175_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% xor_numerals(6)
tff(fact_2176_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_2177_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : bit0(X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_2178_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_2179_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
      ( pp(aa(set(A),bool,member(A,Z),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),P2)))
     => ~ ! [X4: B,Y4: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X4),Y4) )
           => ~ pp(aa(set(A),bool,member(A,Z),aa(C,set(A),aa(B,fun(C,set(A)),C2,X4),Y4))) ) ) ).

% mem_case_prodE
tff(fact_2180_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,bool)),P2: 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),P2))
     => ~ ! [X4: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X4),Y4)) ) ) ).

% case_prodE
tff(fact_2181_case__prodD,axiom,
    ! [A: $tType,B: $tType,F: 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),F),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),F,A3),B2)) ) ).

% case_prodD
tff(fact_2182_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,bool))),P2: product_prod(A,B),Z: 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),P2),Z))
     => ~ ! [X4: A,Y4: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
           => ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X4),Y4),Z)) ) ) ).

% case_prodE'
tff(fact_2183_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R4: 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)),R4),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)),R4,A3),B2),C2)) ) ).

% case_prodD'
tff(fact_2184_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) )
     => ( ! [N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N2))
       => ( ! [N2: 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,N2))
         => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),one2)
           => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),bit0(N2))
             => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M4)),aa(num,num,bit1,N2))
               => ( ! [M4: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),one2)
                 => ( ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),bit0(N2))
                   => ~ ! [M4: num,N2: num] : X != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M4)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_2185_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_2186_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: 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,F,G)),H) = aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),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_bq(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))))),G))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,F),H)) ).

% execute_bind_case
tff(fact_2187_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2188_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_2189_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_odd
tff(fact_2190_Heap__Time__Monad_Obind__def,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(A),G: fun(A,heap_Time_Heap(B))] : heap_Time_bind(A,B,F,G) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),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_bt(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))))),F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_2191_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2192_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_2193_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_2194_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( bit_se2584673776208193580ke_bit(A,M,aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_2195_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_2196_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_num(aa(num,num,bit1,X32)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X32)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_2197_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2198_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q2))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2199_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = none(num) ) )
      & ( ( bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = aa(num,option(num),some(num),aa(nat,num,num_of_nat,bit_se2584673776208193580ke_bit(nat,N,aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_2200_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),bit0(N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_2201_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2202_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_2203_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_bu(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_2204_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,N,aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_2205_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_2206_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),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),bit0(one2))),K))) ).

% bit_cut_integer_def
tff(fact_2207_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) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),aa(bool,fun(code_integer,fun(code_integer,code_integer)),if(code_integer),aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),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),bit0(one2)))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),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),bit0(one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_2208_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_bv(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than_def
tff(fact_2209_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_bw(int,fun(int,fun(int,bool)),D3))) ).

% int_ge_less_than2_def
tff(fact_2210_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),bit_se2584673776208193580ke_bit(A,pred_numeral(L),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2211_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),aa(fun(code_integer,code_integer),fun(fun(num,code_integer),fun(num,code_integer)),comp(code_integer,code_integer,num),uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_2212_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_2213_split__part,axiom,
    ! [B: $tType,A: $tType,P: bool,Q: fun(A,fun(B,bool)),X2: 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_bx(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P),Q)),X2))
    <=> ( 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),X2)) ) ) ).

% split_part
tff(fact_2214_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_2215_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_2216_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).

% push_bit_push_bit
tff(fact_2217_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2218_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% push_bit_of_1
tff(fact_2219_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A3)))
        <=> ( ( N != zero_zero(nat) )
            | pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% even_push_bit_iff
tff(fact_2220_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% push_bit_of_Suc_0
tff(fact_2221_take__bit__num__simps_I7_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),aa(num,num,bit1,M)) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(pred_numeral(R),M))) ).

% take_bit_num_simps(7)
tff(fact_2222_take__bit__num__simps_I6_J,axiom,
    ! [R: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R),bit0(M)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_bo(num,option(num))),bit_take_bit_num(pred_numeral(R),M)) ).

% take_bit_num_simps(6)
tff(fact_2223_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_by(A,fun(B,bool))),Prod)) ).

% prod.disc_eq_case
tff(fact_2224_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),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,bit_se4730199178511100633sh_bit(A,N),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_add
tff(fact_2225_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),bit_se2584673776208193580ke_bit(A,N,A3)) = bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(A,A,bit_se4730199178511100633sh_bit(A,M),A3)) ) ).

% push_bit_take_bit
tff(fact_2226_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_2227_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2228_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_2229_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_bz(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),bit0(one2)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_2230_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_2231_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_2232_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(N)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_ca(num,option(num))),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_2233_take__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_ce(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% take_bit_num_code
tff(fact_2234_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)))
       => ( aa(code_integer,num,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)))
       => ( aa(code_integer,num,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_cf(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).

% num_of_integer_code
tff(fact_2235_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_2236_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2237_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_2238_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F: fun(nat,A),V: num] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),A3),F),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_2239_mask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% mask_Suc_0
tff(fact_2240_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : bit_se2584673776208193580ke_bit(A,N,aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_2241_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F: fun(nat,A),V: num,N: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),A3),F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_2242_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F22: fun(nat,A),Nat: nat] : aa(A,B,H,aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),F1),F22),Nat)) = aa(nat,B,aa(fun(nat,B),fun(nat,B),aa(B,fun(fun(nat,B),fun(nat,B)),case_nat(B),aa(A,B,H,F1)),aa(fun(nat,A),fun(nat,B),aTP_Lamp_cg(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F22)),Nat) ).

% nat.case_distrib
tff(fact_2243_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_ch(fun(A,B),fun(fun(num,A),fun(num,B)),H),F22),aa(fun(num,A),fun(num,B),aTP_Lamp_ch(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32),Num) ).

% num.case_distrib
tff(fact_2244_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X22: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),F1),F22),aa(nat,nat,suc,X22)) = aa(nat,A,F22,X22) ).

% old.nat.simps(5)
tff(fact_2245_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),F1),F22),zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_2246_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X22: num] : case_num(A,F1,F22,F32,bit0(X22)) = aa(num,A,F22,X22) ).

% verit_eq_simplify(17)
tff(fact_2247_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F22,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_2248_verit__eq__simplify_I18_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X32: num] : case_num(A,F1,F22,F32,aa(num,num,bit1,X32)) = aa(num,A,F32,X32) ).

% verit_eq_simplify(18)
tff(fact_2249_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_2250_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_2251_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_2252_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),X),F),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),X),F),N) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_2253_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_2254_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_2255_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,N)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ).

% mask_half_int
tff(fact_2256_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_2257_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_2258_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_2259_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_ci(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ).

% nat_of_integer_code
tff(fact_2260_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_cj(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)),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,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_ck(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_2261_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))),aa(fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(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))),aa(fun(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(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),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),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_cl(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_2262_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_cm(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_2263_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_2264_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_2265_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(8)
tff(fact_2266_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_2267_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_2268_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_2269_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_2270_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X: A,Y3: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y3)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),aa(C,B,F,Y3)) ).

% apsnd_conv
tff(fact_2271_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_2272_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).

% and.right_neutral
tff(fact_2273_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).

% and.left_neutral
tff(fact_2274_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_2275_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_2276_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_2277_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y3))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2278_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% and_numerals(7)
tff(fact_2279_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(4)
tff(fact_2280_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(C,B),G: 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),F),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),G),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),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),G)),X) ).

% apsnd_compose
tff(fact_2281_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_2282_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(fun(nat,bool),fun(nat,bool),aa(bool,fun(fun(nat,bool),fun(nat,bool)),case_nat(bool),fFalse),aTP_Lamp_cn(nat,bool)),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_2283_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(fun(nat,bool),fun(nat,bool),aa(bool,fun(fun(nat,bool),fun(nat,bool)),case_nat(bool),fTrue),aTP_Lamp_co(nat,bool)),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_2284_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(fun(nat,bool),fun(nat,bool),aa(bool,fun(fun(nat,bool),fun(nat,bool)),case_nat(bool),fFalse),aa(nat,fun(nat,bool),ord_less_eq(nat),M)),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_2285_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_2286_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_2287_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_cp(nat,nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)) ).

% diff_Suc
tff(fact_2288_integer__of__num_I3_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit1,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N))),one_one(code_integer)) ).

% integer_of_num(3)
tff(fact_2289_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,aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),F1),F22),Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred2(Nat)) )
         => pp(aa(A,bool,P,aa(nat,A,F22,pred2(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_2290_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,aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),F1),F22),Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred2(Nat)) )
            & ~ pp(aa(A,bool,P,aa(nat,A,F22,pred2(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_2291_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_2292_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num,N: nat] : aa(nat,A,rec_nat(A,A3,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(A,A,aa(nat,fun(A,A),F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)),aa(nat,A,rec_nat(A,A3,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N))) ).

% rec_nat_add_eq_if
tff(fact_2293_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_2294_Ints__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(B)
     => ! [A5: set(A),F: fun(A,B)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(set(B),bool,member(B,aa(A,B,F,X4)),ring_1_Ints(B))) )
         => pp(aa(set(B),bool,member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A5)),ring_1_Ints(B))) ) ) ).

% Ints_sum
tff(fact_2295_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),aa(fun(product_prod(A,C),A),fun(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),F)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_2296_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: 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),F),X)) = aa(product_prod(A,C),A,product_fst(A,C),X) ).

% fst_apsnd
tff(fact_2297_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_2298_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_2299_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cr(fun(B,int),fun(B,A),F)),A5) ) ).

% of_int_sum
tff(fact_2300_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cs(fun(B,nat),fun(B,A),F)),A5) ) ).

% of_nat_sum
tff(fact_2301_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F: fun(nat,fun(A,A)),V: num] : aa(nat,A,rec_nat(A,A3,F),aa(num,nat,numeral_numeral(nat),V)) = aa(A,A,aa(nat,fun(A,A),F,pred_numeral(V)),aa(nat,A,rec_nat(A,A3,F),pred_numeral(V))) ).

% rec_nat_numeral
tff(fact_2302_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y3))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2303_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2304_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2305_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_2306_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_2307_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_2308_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_div_numeral
tff(fact_2309_fstI,axiom,
    ! [B: $tType,A: $tType,X: product_prod(A,B),Y3: A,Z: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),X) = Y3 ) ) ).

% fstI
tff(fact_2310_fst__eqD,axiom,
    ! [B: $tType,A: $tType,X: A,Y3: 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),Y3)) = A3 )
     => ( X = A3 ) ) ).

% fst_eqD
tff(fact_2311_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).

% fst_conv
tff(fact_2312_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: 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_ct(fun(B,A),fun(A,fun(B,A)),F),A3)),A5),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5),A3) ) ).

% mod_sum_eq
tff(fact_2313_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_cu(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_2314_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_cv(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_2315_pred__def,axiom,
    ! [Nat: nat] : pred2(Nat) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_cp(nat,nat)),Nat) ).

% pred_def
tff(fact_2316_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(bit0(N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N)) ).

% integer_of_num(2)
tff(fact_2317_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_2318_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2319_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_2320_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: 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_cw(fun(A,B),fun(A,B),F)),A5))) ) ).

% sum_abs_ge_zero
tff(fact_2321_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: 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),F),A5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_cw(fun(A,B),fun(A,B),F)),A5))) ) ).

% sum_abs
tff(fact_2322_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] :
          ( ! [I2: A] :
              ( pp(aa(set(A),bool,member(A,I2),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I2))) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),X),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( pp(aa(set(A),bool,member(A,I2),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,I2)),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_cx(fun(A,B),fun(fun(A,B),fun(A,B)),X),A3)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_2323_abs__sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F: 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_cw(fun(A,B),fun(A,B),F)),A5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_cw(fun(A,B),fun(A,B),F)),A5) ) ).

% abs_sum_abs
tff(fact_2324_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_cy(B,A)),A5) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2325_sum__SucD,axiom,
    ! [A: $tType,F: fun(A,nat),A5: set(A),N: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A5) = aa(nat,nat,suc,N) )
     => ? [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,F,X4))) ) ) ).

% sum_SucD
tff(fact_2326_int__sum,axiom,
    ! [B: $tType,F: 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),F),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7311177749621191930dd_sum(B,int),aTP_Lamp_cz(fun(B,nat),fun(B,int),F)),A5) ).

% int_sum
tff(fact_2327_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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_da(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),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_dc(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A5)),B4) ) ).

% sum.swap
tff(fact_2328_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A5: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,member(B,A4),A5))
               => ( aa(B,A,G,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_2329_sum_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,G,X4) = zero_zero(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = zero_zero(A) ) ) ) ).

% sum.neutral
tff(fact_2330_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),K5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I2)),aa(B,A,G,I2))) )
         => 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),F),K5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),K5))) ) ) ).

% sum_mono
tff(fact_2331_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R: A,F: fun(B,A),A5: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),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_dd(A,fun(fun(B,A),fun(B,A)),R),F)),A5) ) ).

% sum_distrib_left
tff(fact_2332_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A5: set(B),R: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),R) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_de(fun(B,A),fun(A,fun(B,A)),F),R)),A5) ) ).

% sum_distrib_right
tff(fact_2333_sum__product,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_0(B)
     => ! [F: fun(A,B),A5: set(A),G: 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),F),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),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_dg(fun(A,B),fun(fun(C,B),fun(set(C),fun(A,B))),F),G),B4)),A5) ) ).

% sum_product
tff(fact_2334_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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_dh(fun(B,A),fun(fun(B,A),fun(B,A)),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),A5)) ) ).

% sum.distrib
tff(fact_2335_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: 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_di(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5)) ) ).

% sum_subtractf
tff(fact_2336_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_dj(fun(B,A),fun(B,A),F)),A5) = aa(A,A,uminus_uminus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)) ) ).

% sum_negf
tff(fact_2337_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),A5: set(B),R: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)),R) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(A,fun(B,A),aTP_Lamp_dk(fun(B,A),fun(A,fun(B,A)),F),R)),A5) ) ).

% sum_divide_distrib
tff(fact_2338_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [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,F,X4)),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),F),A5)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_2339_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X4))) )
         => 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),F),A5))) ) ) ).

% sum_nonneg
tff(fact_2340_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(nat),F: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5))
         => ( ! [X4: nat] :
                ( pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,X4)),A5))
               => ( aa(nat,A,F,aa(nat,nat,suc,X4)) = aa(nat,A,G,aa(nat,nat,suc,X4)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),A5) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A5) ) ) ) ) ).

% sum_cong_Suc
tff(fact_2341_sum__subtractf__nat,axiom,
    ! [A: $tType,A5: set(A),G: fun(A,nat),F: fun(A,nat)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X4)),aa(A,nat,F,X4))) )
     => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_dl(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F)),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),F),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A5)) ) ) ).

% sum_subtractf_nat
tff(fact_2342_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A5: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X4: B,Y4: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X4),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X4)),aa(B,A,H,Y4))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),G)),A5) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A5)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_2343_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(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)),aa(fun(A,product_prod(A,A)),fun(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_dm(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_2344_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dn(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_2345_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_do(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_2346_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_dp(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),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_2347_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),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,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_2348_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_2349_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_2350_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_2351_atMost__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_2352_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_dq(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_2353_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_2354_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_2355_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2356_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_2357_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_2358_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_2359_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_du(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_2360_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2361_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_2362_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2363_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2364_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2365_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_2366_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dv(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2367_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2368_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dw(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,aa(nat,nat,suc,N))),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_2369_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_2370_sum__choose__lower,axiom,
    ! [R: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dy(nat,fun(nat,nat),R)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R),N))),N) ).

% sum_choose_lower
tff(fact_2371_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dz(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_2372_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dz(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_2373_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ea(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2374_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))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B4))
           => ( pp(aa(A,bool,Q,X4))
             => pp(aa(A,bool,P,X4)) ) )
       => 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_aw(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_aw(set(A),fun(fun(A,bool),fun(A,bool)),A5),P)))) ) ) ).

% subset_CollectI
tff(fact_2375_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_aw(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),B4))
           => pp(aa(A,bool,P,X3)) ) ) ) ).

% subset_Collect_iff
tff(fact_2376_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_2377_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: 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),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2378_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eb(A,fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_2379_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ec(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ee(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_2380_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ef(nat,fun(nat,fun(nat,nat)),M),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_2381_vandermonde,axiom,
    ! [M: nat,N: nat,R: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_eg(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R)),aa(nat,set(nat),set_ord_atMost(nat),R)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R) ).

% vandermonde
tff(fact_2382_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_2383_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eh(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_2384_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ei(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).

% sum_telescope''
tff(fact_2385_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% choose_row_sum
tff(fact_2386_binomial,axiom,
    ! [A3: nat,B2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B2)),N) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_ej(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_2387_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_2388_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_el(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_2389_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_em(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_2390_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_en(A,fun(A,fun(nat,fun(nat,A))),A3),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% pochhammer_binomial_sum
tff(fact_2391_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2392_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ek(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_2393_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_eo(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N) ).

% choose_square_sum
tff(fact_2394_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ep(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_eq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2395_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y3: 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_er(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y3)),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_es(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y3)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_2396_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_et(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_2397_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_2398_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_2399_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,X: A,Y3: 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_er(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y3)),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_eu(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),X),Y3)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_2400_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ev(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2401_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_ew(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),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_2402_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ex(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3))) ) ).

% double_arith_series
tff(fact_2403_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_2404_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).

% binomial_r_part_sum
tff(fact_2405_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_ey(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_2406_arith__series__nat,axiom,
    ! [A3: nat,D3: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_ez(nat,fun(nat,fun(nat,nat)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D3)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_2407_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_2408_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N: nat] :
      ( ( F = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F22,N),aa(nat,A,F,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_2409_rec__nat__0__imp,axiom,
    ! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
      ( ( F = rec_nat(A,F1,F22) )
     => ( aa(nat,A,F,zero_zero(nat)) = F1 ) ) ).

% rec_nat_0_imp
tff(fact_2410_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fa(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2411_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_2412_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D3: A,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fb(A,fun(A,fun(nat,A)),A3),D3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D3)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_2413_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_2414_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_2415_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_2416_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dp(A,fun(nat,A),R)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_2417_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fc(A,fun(nat,A),Z)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_2418_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N))
          & pp(aa(nat,bool,P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X3)) ) ) ).

% ex_nat_less
tff(fact_2419_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N))
         => pp(aa(nat,bool,P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X3)) ) ) ).

% all_nat_less
tff(fact_2420_bezw_Osimps,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ( Y3 = zero_zero(nat) )
       => ( bezw(X,Y3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
      & ( ( Y3 != zero_zero(nat) )
       => ( bezw(X,Y3) = 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(Y3,modulo_modulo(nat,X,Y3)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y3,modulo_modulo(nat,X,Y3)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y3,modulo_modulo(nat,X,Y3)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y3))))) ) ) ) ).

% bezw.simps
tff(fact_2421_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y3: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y3 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y3 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y3 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa))))) ) ) ) ) ).

% bezw.elims
tff(fact_2422_bezw__non__0,axiom,
    ! [Y3: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y3))
     => ( bezw(X,Y3) = 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(Y3,modulo_modulo(nat,X,Y3)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y3,modulo_modulo(nat,X,Y3)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y3,modulo_modulo(nat,X,Y3)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Y3))))) ) ) ).

% bezw_non_0
tff(fact_2423_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_2424_Ints__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(B)
        & ring_1(B) )
     => ! [A5: set(A),F: fun(A,B)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => pp(aa(set(B),bool,member(B,aa(A,B,F,X4)),ring_1_Ints(B))) )
         => pp(aa(set(B),bool,member(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),A5)),ring_1_Ints(B))) ) ) ).

% Ints_prod
tff(fact_2425_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F),X)) = aa(C,A,F,aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% snd_apsnd
tff(fact_2426_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,B),X: product_prod(A,C),G: 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),F),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),G),X) )
    <=> ( aa(C,B,F,aa(product_prod(A,C),C,product_snd(A,C),X)) = aa(C,B,G,aa(product_prod(A,C),C,product_snd(A,C),X)) ) ) ).

% apsnd_eq_conv
tff(fact_2427_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_fd(B,A)),A5) = one_one(A) ) ).

% prod.neutral_const
tff(fact_2428_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fe(fun(B,nat),fun(B,A),F)),A5) ) ).

% of_nat_prod
tff(fact_2429_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_ff(fun(B,int),fun(B,A),F)),A5) ) ).

% of_int_prod
tff(fact_2430_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_2431_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),aa(fun(product_prod(A,C),C),fun(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),F)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),F),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_2432_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_2433_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_mod_numeral
tff(fact_2434_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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_fg(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),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_fi(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A5)),B4) ) ).

% prod.swap
tff(fact_2435_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A5: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) != one_one(A) )
         => ~ ! [A4: B] :
                ( pp(aa(set(B),bool,member(B,A4),A5))
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_2436_prod_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( aa(B,A,G,X4) = one_one(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = one_one(A) ) ) ) ).

% prod.neutral
tff(fact_2437_snd__conv,axiom,
    ! [Aa: $tType,A: $tType,X1: Aa,X22: 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),X22)) = X22 ).

% snd_conv
tff(fact_2438_snd__eqD,axiom,
    ! [B: $tType,A: $tType,X: B,Y3: 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),Y3)) = A3 )
     => ( Y3 = A3 ) ) ).

% snd_eqD
tff(fact_2439_sndI,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B),Y3: A,Z: B] :
      ( ( X = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),X) = Z ) ) ).

% sndI
tff(fact_2440_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_2441_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),Q2: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P2) = aa(product_prod(A,B),A,product_fst(A,B),Q2) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P2) = aa(product_prod(A,B),B,product_snd(A,B),Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_2442_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S: product_prod(A,B),T2: product_prod(A,B)] :
      ( ( S = T2 )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S) = aa(product_prod(A,B),A,product_fst(A,B),T2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S) = aa(product_prod(A,B),B,product_snd(A,B),T2) ) ) ) ).

% prod_eq_iff
tff(fact_2443_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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_fj(fun(B,A),fun(fun(B,A),fun(B,A)),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),A5)) ) ).

% prod.distrib
tff(fact_2444_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),G: 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_fk(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5)) ) ).

% prod_dividef
tff(fact_2445_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [F: fun(A,B),A5: set(A),N: 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),F),A5)),N) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(nat,fun(A,B),aTP_Lamp_fl(fun(A,B),fun(nat,fun(A,B)),F),N)),A5) ) ).

% prod_power_distrib
tff(fact_2446_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: 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_ct(fun(B,A),fun(A,fun(B,A)),F),A3)),A5),A3) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5),A3) ) ).

% mod_prod_eq
tff(fact_2447_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_fm(fun(B,A),fun(B,A),F)),A5) ) ).

% abs_prod
tff(fact_2448_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_fn(A,fun(B,B))),Prod) ).

% snd_def
tff(fact_2449_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X4))) )
         => 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),F),A5))) ) ) ).

% prod_nonneg
tff(fact_2450_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I2)),aa(B,A,G,I2))) ) )
         => 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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5))) ) ) ).

% prod_mono
tff(fact_2451_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,X4))) )
         => 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),F),A5))) ) ) ).

% prod_pos
tff(fact_2452_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F,X4))) )
         => 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),F),A5))) ) ) ).

% prod_ge_1
tff(fact_2453_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_2454_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).

% surjective_pairing
tff(fact_2455_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),P2: A,Q: fun(B,bool),Q2: B] :
      ( pp(aa(A,bool,P,P2))
     => ( pp(aa(B,bool,Q,Q2))
       => ( 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),P2),Q2))))
          & pp(aa(B,bool,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2)))) ) ) ) ).

% conjI_realizer
tff(fact_2456_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Y3: A,X: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,Y3),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),Y3))),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),Y3)))) ) ).

% exI_realizer
tff(fact_2457_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X: A,Y3: B,A3: product_prod(A,B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y3))
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
       => 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_2458_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_2459_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_2460_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,A)),P2: 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),F),P2) = aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),P2)),aa(product_prod(B,C),C,product_snd(B,C),P2)) ).

% case_prod_beta
tff(fact_2461_split__beta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: 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),F),Prod) = aa(B,C,aa(A,fun(B,C),F,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_2462_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_2463_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2464_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_2465_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F: 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),F),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_fq(A,fun(fun(B,nat),fun(B,A)),C2),F)),A5) ) ).

% power_sum
tff(fact_2466_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,D: $tType,F: fun(A,fun(B,C)),G: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_fr(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F),G) = 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_fs(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F),G)) ).

% split_comp_eq
tff(fact_2467_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),X2: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),X2) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),X2)),aa(product_prod(A,B),B,product_snd(A,B),X2)) ).

% case_prod_beta'
tff(fact_2468_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X2: 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),X2),Xa3) = aa(B,C,aa(A,fun(B,C),X2,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_2469_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),P2: product_prod(B,A),Q: fun(C,bool),F: 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),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2)))
     => ( ! [X4: B,Y4: A] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),P,Y4),X4))
           => pp(aa(C,bool,Q,aa(A,C,aa(B,fun(A,C),F,X4),Y4))) )
       => 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),F),P2))) ) ) ).

% exE_realizer
tff(fact_2470_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,X4)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),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),F),A5)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_2471_divides__aux__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Qr: product_prod(A,A)] :
          ( unique5940410009612947441es_aux(A,Qr)
        <=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).

% divides_aux_def
tff(fact_2472_prod_Osplit__sel__asm,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F: 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),F),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),F,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_2473_prod_Osplit__sel,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(C,bool),F: 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),F),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),F,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_2474_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ft(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_2475_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(D,fun(C,A)),G: fun(B,D),X: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),aa(fun(D,fun(C,A)),fun(fun(B,D),fun(B,fun(C,A))),comp(D,fun(C,A),B),F),G)),X) = aa(C,A,aa(D,fun(C,A),F,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X))),aa(product_prod(B,C),C,product_snd(B,C),X)) ).

% case_prod_comp
tff(fact_2476_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2477_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_2478_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(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)),aa(fun(B,product_prod(B,B)),fun(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_fu(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_2479_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2480_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_2481_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_2482_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_2483_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F: fun(nat,A),A3: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F),set_or1337092689740270186AtMost(nat,A3,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fv(fun(nat,A),fun(nat,fun(A,A)),F),A3,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_2484_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),aa(fun(product_prod(B,A),A),fun(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_fw(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_2485_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),aa(fun(product_prod(A,B),A),fun(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_fx(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_2486_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_2487_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: 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),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2488_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(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)),aa(fun(B,product_prod(B,B)),fun(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_fu(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_2489_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(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)),aa(fun(A,product_prod(A,A)),fun(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_dm(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_2490_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_2491_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fy(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_2492_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fz(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_2493_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_cp(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_2494_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_2495_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ga(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_2496_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fz(A,fun(nat,fun(nat,A)),A3),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_2497_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_gc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2498_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_gd(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_2499_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : aa(product_prod(A,B),nat,basic_BNF_size_prod(A,B,F,G),P2) = 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,F,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_2500_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y3: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y3 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y3 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X),Xa))))) ) ) )
           => ~ 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),Xa))) ) ) ) ).

% bezw.pelims
tff(fact_2501_normalize__def,axiom,
    ! [P2: 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),P2)))
       => ( normalize(P2) = 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),P2)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))) ) )
      & ( ~ 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),P2)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int) )
           => ( normalize(P2) = 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),P2) != zero_zero(int) )
           => ( normalize(P2) = 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),P2)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))) ) ) ) ) ) ).

% normalize_def
tff(fact_2502_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ge(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ee(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_2503_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N))
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_gf(int,int)),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,aa(int,fun(int,int),minus_minus(int),M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_2504_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_2505_gcd__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_right_idem
tff(fact_2506_gcd__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_left_idem
tff(fact_2507_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_2508_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add1
tff(fact_2509_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add2
tff(fact_2510_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),one_one(A)),A3) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_2511_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_2512_gcd__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg2
tff(fact_2513_gcd__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd_neg1
tff(fact_2514_gcd__greatest__iff,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),gcd_gcd(A),B2),C2)))
        <=> ( 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),C2)) ) ) ) ).

% gcd_greatest_iff
tff(fact_2515_gcd__dvd2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2)),B2)) ) ).

% gcd_dvd2
tff(fact_2516_gcd__dvd1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2)),A3)) ) ).

% gcd_dvd1
tff(fact_2517_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,N: nat,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),N) ) ).

% gcd_exp
tff(fact_2518_gcd__1__int,axiom,
    ! [M: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_2519_gcd__neg2__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% gcd_neg2_int
tff(fact_2520_gcd__neg1__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y3) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% gcd_neg1_int
tff(fact_2521_abs__gcd__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% abs_gcd_int
tff(fact_2522_gcd__abs1__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,abs_abs(int),X)),Y3) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% gcd_abs1_int
tff(fact_2523_gcd__abs2__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,abs_abs(int),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% gcd_abs2_int
tff(fact_2524_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% gcd_neg_numeral_1
tff(fact_2525_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_2526_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2)),one_one(A)))
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_2527_gcd__pos__int,axiom,
    ! [M: int,N: 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),N)))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_2528_gcd__neg__numeral__1__int,axiom,
    ! [N: num,X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),X) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(num,int,numeral_numeral(int),N)),X) ).

% gcd_neg_numeral_1_int
tff(fact_2529_gcd__neg__numeral__2__int,axiom,
    ! [X: int,N: num] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(num,int,numeral_numeral(int),N)) ).

% gcd_neg_numeral_2_int
tff(fact_2530_gcd__0__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),zero_zero(int)) = aa(int,int,abs_abs(int),X) ).

% gcd_0_int
tff(fact_2531_gcd__0__left__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),zero_zero(int)),X) = aa(int,int,abs_abs(int),X) ).

% gcd_0_left_int
tff(fact_2532_gcd__proj2__if__dvd__int,axiom,
    ! [Y3: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Y3),X))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) = aa(int,int,abs_abs(int),Y3) ) ) ).

% gcd_proj2_if_dvd_int
tff(fact_2533_gcd__proj1__if__dvd__int,axiom,
    ! [X: int,Y3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),Y3))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) = aa(int,int,abs_abs(int),X) ) ) ).

% gcd_proj1_if_dvd_int
tff(fact_2534_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).

% sum.lessThan_Suc
tff(fact_2535_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_2536_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_2537_Suc__0__mod__numeral,axiom,
    ! [K: num] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_2538_int__prod,axiom,
    ! [B: $tType,F: 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),F),A5)) = aa(set(B),int,aa(fun(B,int),fun(set(B),int),groups7121269368397514597t_prod(B,int),aTP_Lamp_cz(fun(B,nat),fun(B,int),F)),A5) ).

% int_prod
tff(fact_2539_lessThan__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_2540_gcd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)) ) ).

% gcd.left_commute
tff(fact_2541_gcd_Ocommute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A3) ) ).

% gcd.commute
tff(fact_2542_gcd_Oassoc,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2)) ) ).

% gcd.assoc
tff(fact_2543_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: A,M: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),N),M)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_diff2
tff(fact_2544_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_diff1
tff(fact_2545_gcd__dvd__antisym,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A,D3: 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),gcd_gcd(A),C2),D3)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)))
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),C2),D3) ) ) ) ) ).

% gcd_dvd_antisym
tff(fact_2546_gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ) ) ).

% gcd_greatest
tff(fact_2547_gcd__dvdI2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,C2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2))
         => 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)),C2)) ) ) ).

% gcd_dvdI2
tff(fact_2548_gcd__dvdI1,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),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C2)) ) ) ).

% gcd_dvdI1
tff(fact_2549_dvd__gcdD2,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),gcd_gcd(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ).

% dvd_gcdD2
tff(fact_2550_dvd__gcdD1,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),gcd_gcd(A),B2),C2)))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_gcdD1
tff(fact_2551_gcd__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A,D3: 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),D3))
           => 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),gcd_gcd(A),C2),D3))) ) ) ) ).

% gcd_mono
tff(fact_2552_prod__int__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_gf(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_2553_gcd__idem__int,axiom,
    ! [X: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),X) = aa(int,int,abs_abs(int),X) ).

% gcd_idem_int
tff(fact_2554_gcd__red__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y3),modulo_modulo(int,X,Y3)) ).

% gcd_red_int
tff(fact_2555_effect__heapE,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_heap(A,F),H,H2,R,N)
     => ~ ( ( H2 = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( N = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
           => ( R != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ) ) ).

% effect_heapE
tff(fact_2556_effect__heapI,axiom,
    ! [A: $tType,N: nat,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A] :
      ( ( N = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
     => ( ( H2 = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
       => ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
         => heap_Time_effect(A,heap_Time_heap(A,F),H,H2,R,N) ) ) ) ).

% effect_heapI
tff(fact_2557_effect__guardE,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_guard(A,P,F),H,H2,R,N)
     => ~ ( ( H2 = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( N = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
           => ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
             => ~ pp(aa(heap_ext(product_unit),bool,P,H)) ) ) ) ) ).

% effect_guardE
tff(fact_2558_effect__guardI,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),H2: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),N: nat,R: A] :
      ( pp(aa(heap_ext(product_unit),bool,P,H))
     => ( ( H2 = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
       => ( ( N = 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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H))) )
         => ( ( R = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) )
           => heap_Time_effect(A,heap_Time_guard(A,P,F),H,H2,R,N) ) ) ) ) ).

% effect_guardI
tff(fact_2559_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_2560_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : aa(set(nat),int,aa(fun(nat,int),fun(set(nat),int),groups7121269368397514597t_prod(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_gf(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_2561_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gg(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_2562_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gh(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_2563_effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R,N)
     => ~ ( ( R = 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))) )
         => ( ( H2 = 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)))) )
           => ( ( N = 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_2564_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_gi(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_2565_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add_mult
tff(fact_2566_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_2567_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N))
          & pp(aa(nat,bool,P,M7)) )
    <=> ? [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X3)) ) ) ).

% ex_nat_less_eq
tff(fact_2568_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N))
         => pp(aa(nat,bool,P,M7)) )
    <=> ! [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X3)) ) ) ).

% all_nat_less_eq
tff(fact_2569_gcd__ge__0__int,axiom,
    ! [X: int,Y3: 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),Y3))) ).

% gcd_ge_0_int
tff(fact_2570_bezout__int,axiom,
    ! [X: int,Y3: int] :
    ? [U2: int,V5: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U2),X)),aa(int,int,aa(int,fun(int,int),times_times(int),V5),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) ).

% bezout_int
tff(fact_2571_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N)) ).

% gcd_mult_distrib_int
tff(fact_2572_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_success(A,F,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,F,G)),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,F),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),G,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,F),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,F),H)))))) ) ) ).

% execute_bind_success
tff(fact_2573_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2574_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_2575_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_2576_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_2577_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_2578_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_2579_gcd__div__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),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit2
tff(fact_2580_gcd__div__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),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),C2) ) ) ) ).

% gcd_div_unit1
tff(fact_2581_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_2582_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_2583_gcd__cases__int,axiom,
    ! [X: int,Y3: 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)),Y3))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3))) ) )
     => ( ( 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),Y3),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),Y3)))) ) )
       => ( ( 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)),Y3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y3))) ) )
         => ( ( 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),Y3),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),Y3)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3))) ) ) ) ) ).

% gcd_cases_int
tff(fact_2584_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_2585_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))
        & ! [E4: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),A3))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),B2)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),E4),D3)) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2) ) ) ).

% gcd_unique_int
tff(fact_2586_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_2587_gcd__non__0__int,axiom,
    ! [Y3: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y3))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y3),modulo_modulo(int,X,Y3)) ) ) ).

% gcd_non_0_int
tff(fact_2588_gcd__code__int,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),L) = aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),aa(bool,fun(int,fun(int,int)),if(int),aa(int,bool,aa(int,fun(int,bool),fequal(int),L),zero_zero(int))),K),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),L),modulo_modulo(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L))))) ).

% gcd_code_int
tff(fact_2589_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_2590_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_2591_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_2592_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_2593_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_2594_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gj(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_2595_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_2596_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gk(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_2597_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ge(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_2598_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_2599_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_2600_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_2601_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B2: nat,G: 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),G),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),G),set_or7035219750837199246ssThan(nat,A3,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_2602_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_2603_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_2604_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_2605_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2606_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dw(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,M)),aa(nat,A,F,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2607_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dv(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_2608_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2609_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X4: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X4)),aa(A,nat,P,X4)))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),N))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),N))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_gn(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_2610_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2611_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2612_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),F)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_2613_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_go(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nested_swap'
tff(fact_2614_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gr(fun(nat,fun(nat,A)),fun(nat,A),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nested_swap'
tff(fact_2615_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dw(fun(nat,A),fun(nat,A),F)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F,N)),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2616_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gu(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2617_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_gv(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_2618_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gw(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_2619_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gx(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_2620_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aa(nat,fun(nat,nat),minus_minus(nat),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_2621_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_2622_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_2623_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_1_eq
tff(fact_2624_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq
tff(fact_2625_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_2626_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_2627_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_2628_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_2629_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fo(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_2630_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_2631_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_2632_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2633_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ft(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2634_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_fy(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_2635_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_2636_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_2637_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_2638_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y3)),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_gy(A,fun(nat,fun(A,fun(nat,A))),X),N),Y3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_sumr2
tff(fact_2639_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y3)),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_gz(A,fun(nat,fun(A,fun(nat,A))),X),N),Y3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_2640_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),X)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ha(A,fun(nat,fun(nat,A)),X),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_2641_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ec(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_2642_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_hb(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_2643_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hc(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_2644_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_hd(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_2645_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_hd(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_2646_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_gd(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_2647_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),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),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_2648_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y3: product_prod(nat,nat)] :
      ( ( aa(nat,product_prod(nat,nat),nat_prod_decode_aux(X),Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y3 = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X),Xa)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y3 = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(aa(nat,nat,suc,X)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),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),Xa))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_2649_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_he(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_he(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D3)),A5) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_2650_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_un7362597486090784418nd_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),aa(num,option(num),some(num),one2)),aTP_Lamp_ca(num,option(num))),bit_un7362597486090784418nd_num(M,N)) ).

% and_num.simps(9)
tff(fact_2651_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% or_numerals(4)
tff(fact_2652_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),bit0(Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% or_numerals(6)
tff(fact_2653_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y3)))) ) ).

% or_numerals(7)
tff(fact_2654_gcd__0__left__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),X) = X ).

% gcd_0_left_nat
tff(fact_2655_gcd__0__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),zero_zero(nat)) = X ).

% gcd_0_nat
tff(fact_2656_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),zero_zero(nat)) = A3 ).

% gcd_nat.right_neutral
tff(fact_2657_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_2658_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A3) = A3 ).

% gcd_nat.left_neutral
tff(fact_2659_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_2660_gcd__nat_Oabsorb1,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = A3 ) ) ).

% gcd_nat.absorb1
tff(fact_2661_gcd__nat_Oabsorb2,axiom,
    ! [B2: nat,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = B2 ) ) ).

% gcd_nat.absorb2
tff(fact_2662_gcd__nat_Obounded__iff,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2)) ) ) ).

% gcd_nat.bounded_iff
tff(fact_2663_gcd__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),Y3))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = X ) ) ).

% gcd_proj1_if_dvd_nat
tff(fact_2664_gcd__proj2__if__dvd__nat,axiom,
    ! [Y3: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Y3),X))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = Y3 ) ) ).

% gcd_proj2_if_dvd_nat
tff(fact_2665_gcd__1__nat,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_2666_gcd__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_2667_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_2668_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_2669_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_2670_gcd__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)) ).

% gcd_int_int_eq
tff(fact_2671_sum__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [F5: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),F5))
         => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),F5) = zero_zero(A) )
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),F5))
               => ( aa(B,A,F,X3) = zero_zero(A) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_2672_sum_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),A5) = zero_zero(A) ) ) ) ).

% sum.infinite
tff(fact_2673_prod__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom(A)
     => ! [A5: set(B),F: 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),F),A5) = zero_zero(A) )
          <=> ? [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),A5))
                & ( aa(B,A,F,X3) = zero_zero(A) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_2674_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G: 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),G),A5) = one_one(A) ) ) ) ).

% prod.infinite
tff(fact_2675_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_2676_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_2677_sum_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hf(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hf(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = zero_zero(A) ) ) ) ) ) ).

% sum.delta'
tff(fact_2678_sum_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hg(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hg(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = zero_zero(A) ) ) ) ) ) ).

% sum.delta
tff(fact_2679_prod__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F),A5) = one_one(nat) )
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ( aa(A,nat,F,X3) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_2680_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hh(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hh(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_2681_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hi(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = aa(B,A,B2,A3) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_hi(B,fun(fun(B,A),fun(B,A)),A3),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_2682_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_2683_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3)) ).

% or_nat_numerals(2)
tff(fact_2684_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3)) ) ).

% or_numerals(2)
tff(fact_2685_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_2686_prod__pos__nat__iff,axiom,
    ! [A: $tType,A5: set(A),F: 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),F),A5)))
      <=> ! [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,F,X3))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2687_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).

% gcd_nat_abs_left_eq
tff(fact_2688_gcd__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).

% gcd_nat_abs_right_eq
tff(fact_2689_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y3))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y3)) ).

% or_nat_numerals(1)
tff(fact_2690_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_2691_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y3: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y3))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y3)) ) ).

% or_numerals(1)
tff(fact_2692_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_2693_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_hj(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_hj(fun(nat,A),fun(nat,A),C2)),A5) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_2694_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_2695_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_2696_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% gcd_mult_distrib_nat
tff(fact_2697_gcd__nat_Omono,axiom,
    ! [A3: nat,C2: nat,B2: nat,D3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),D3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),C2),D3))) ) ) ).

% gcd_nat.mono
tff(fact_2698_gcd__nat_OorderE,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ).

% gcd_nat.orderE
tff(fact_2699_gcd__nat_OorderI,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2)) ) ).

% gcd_nat.orderI
tff(fact_2700_gcd__nat_Oabsorb3,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = A3 ) ) ).

% gcd_nat.absorb3
tff(fact_2701_gcd__nat_Oabsorb4,axiom,
    ! [B2: nat,A3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3))
        & ( B2 != A3 ) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = B2 ) ) ).

% gcd_nat.absorb4
tff(fact_2702_gcd__nat_OboundedE,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2)) ) ) ).

% gcd_nat.boundedE
tff(fact_2703_gcd__nat_OboundedI,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2))) ) ) ).

% gcd_nat.boundedI
tff(fact_2704_gcd__nat_Oorder__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
    <=> ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ).

% gcd_nat.order_iff
tff(fact_2705_gcd__nat_Ocobounded1,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),A3)) ).

% gcd_nat.cobounded1
tff(fact_2706_gcd__nat_Ocobounded2,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2)) ).

% gcd_nat.cobounded2
tff(fact_2707_gcd__nat_Oabsorb__iff1,axiom,
    ! [A3: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = A3 ) ) ).

% gcd_nat.absorb_iff1
tff(fact_2708_gcd__nat_Oabsorb__iff2,axiom,
    ! [B2: nat,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),A3))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) = B2 ) ) ).

% gcd_nat.absorb_iff2
tff(fact_2709_gcd__nat_OcoboundedI1,axiom,
    ! [A3: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),C2)) ) ).

% gcd_nat.coboundedI1
tff(fact_2710_gcd__nat_OcoboundedI2,axiom,
    ! [B2: nat,C2: nat,A3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),C2)) ) ).

% gcd_nat.coboundedI2
tff(fact_2711_gcd__nat_Ostrict__boundedE,axiom,
    ! [A3: nat,B2: nat,C2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2)))
        & ( A3 != aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),B2),C2) ) )
     => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
            & ( A3 != B2 ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
              & ( A3 != C2 ) ) ) ) ).

% gcd_nat.strict_boundedE
tff(fact_2712_gcd__nat_Ostrict__order__iff,axiom,
    ! [A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B2))
        & ( A3 != B2 ) )
    <=> ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) )
        & ( A3 != B2 ) ) ) ).

% gcd_nat.strict_order_iff
tff(fact_2713_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A3: nat,C2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),C2))
        & ( A3 != C2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),C2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI1
tff(fact_2714_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B2: nat,C2: nat,A3: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),C2))
        & ( B2 != C2 ) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),C2))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) != C2 ) ) ) ).

% gcd_nat.strict_coboundedI2
tff(fact_2715_gcd__unique__nat,axiom,
    ! [D3: nat,A3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),A3))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),B2))
        & ! [E4: nat] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E4),A3))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E4),B2)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),E4),D3)) ) )
    <=> ( D3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ).

% gcd_unique_nat
tff(fact_2716_gcd__red__nat,axiom,
    ! [X: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y3),modulo_modulo(nat,X,Y3)) ).

% gcd_red_nat
tff(fact_2717_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I: nat] : pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_hk(fun(nat,bool),fun(nat,fun(nat,bool)),P),I)))) ).

% finite_M_bounded_by_nat
tff(fact_2718_finite__less__ub,axiom,
    ! [F: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,F,N2)))
     => pp(aa(set(nat),bool,finite_finite2(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_hl(fun(nat,nat),fun(nat,fun(nat,bool)),F),U)))) ) ).

% finite_less_ub
tff(fact_2719_sum_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(C),G: fun(B,fun(C,A)),R4: 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_hn(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G),R4)),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_hp(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A5),G),R4)),B4) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2720_prod_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(C),G: fun(B,fun(C,A)),R4: 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_hq(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),B4),G),R4)),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_hr(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),A5),G),R4)),B4) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2721_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_2722_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_2723_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff2_nat
tff(fact_2724_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff1_nat
tff(fact_2725_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y3 )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y3 = X ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_2726_gcd__nat_Osimps,axiom,
    ! [Y3: nat,X: nat] :
      ( ( ( Y3 = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = X ) )
      & ( ( Y3 != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y3),modulo_modulo(nat,X,Y3)) ) ) ) ).

% gcd_nat.simps
tff(fact_2727_gcd__non__0__nat,axiom,
    ! [Y3: nat,X: nat] :
      ( ( Y3 != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y3),modulo_modulo(nat,X,Y3)) ) ) ).

% gcd_non_0_nat
tff(fact_2728_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T3: set(C),S2: set(B),I: fun(C,B),J: fun(B,C),T4: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                   => pp(aa(set(C),bool,member(C,aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)))
                       => pp(aa(set(B),bool,member(B,aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,member(B,A4),S5))
                         => ( aa(B,A,G,A4) = zero_zero(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,member(C,B3),T3))
                           => ( aa(C,A,H,B3) = zero_zero(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,member(B,A4),S2))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_2729_prod__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ? [X2: B] :
                ( pp(aa(set(B),bool,member(B,X2),A5))
                & ( aa(B,A,F,X2) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5) = zero_zero(A) ) ) ) ) ).

% prod_zero
tff(fact_2730_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T3: set(C),S2: set(B),I: fun(C,B),J: fun(B,C),T4: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                 => ( aa(C,B,I,aa(B,C,J,A4)) = A4 ) )
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5)))
                   => pp(aa(set(C),bool,member(C,aa(B,C,J,A4)),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)))
                     => ( aa(B,C,J,aa(C,B,I,B3)) = B3 ) )
                 => ( ! [B3: C] :
                        ( pp(aa(set(C),bool,member(C,B3),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3)))
                       => pp(aa(set(B),bool,member(B,aa(C,B,I,B3)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5))) )
                   => ( ! [A4: B] :
                          ( pp(aa(set(B),bool,member(B,A4),S5))
                         => ( aa(B,A,G,A4) = one_one(A) ) )
                     => ( ! [B3: C] :
                            ( pp(aa(set(C),bool,member(C,B3),T3))
                           => ( aa(C,A,H,B3) = one_one(A) ) )
                       => ( ! [A4: B] :
                              ( pp(aa(set(B),bool,member(B,A4),S2))
                             => ( aa(C,A,H,aa(B,C,J,A4)) = aa(B,A,G,A4) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T4) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_2731_gcd__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa)) ).

% gcd_integer.rep_eq
tff(fact_2732_gcd__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),X)) ).

% gcd_integer.abs_eq
tff(fact_2733_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N5)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2734_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite2(nat),N5)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_2735_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se5668285175392031749et_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_2736_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y3: 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_hs(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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),Y3))))
           => 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_ht(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y3)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_2737_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y3: 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_hu(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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),Y3))))
           => 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_hv(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y3)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_2738_sum_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_hw(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_hx(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A5) ) ) ) ).

% sum.inter_filter
tff(fact_2739_sum_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),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_hy(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2740_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G: 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),G),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_hz(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2741_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G: 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),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_hw(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_ia(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A5) ) ) ) ).

% prod.inter_filter
tff(fact_2742_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_ib(A,fun(A,fun(A,bool)),A3),B2)))) ) ).

% finite_int_segment
tff(fact_2743_bezout__nat,axiom,
    ! [A3: nat,B2: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X4: nat,Y4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) ) ).

% bezout_nat
tff(fact_2744_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A3: nat] :
    ? [X4: nat,Y4: 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),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = 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),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_2745_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A5: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),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),zero_zero(A)),aa(B,A,F,X4))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5) = zero_zero(A) )
            <=> ! [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),A5))
                 => ( aa(B,A,F,X3) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_2746_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),T2: set(C),G: fun(C,A),I: fun(C,B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( pp(aa(set(C),bool,finite_finite2(C),T2))
           => ( ! [X4: C] :
                  ( pp(aa(set(C),bool,member(C,X4),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X4))) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S))
                   => ? [Xa3: C] :
                        ( pp(aa(set(C),bool,member(C,Xa3),T2))
                        & ( aa(C,B,I,Xa3) = X4 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),aa(C,A,G,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),F),S)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_2747_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R4: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R4,zero_zero(A)),zero_zero(A)))
         => ( ! [X12: A,Y12: A,X23: A,Y22: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X12),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R4,Y12),Y22)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(A,A,aa(A,fun(A,A),plus_plus(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y22))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S2))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(B,A,H,X4)),aa(B,A,G,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2))) ) ) ) ) ) ).

% sum.related
tff(fact_2748_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C6: set(B),A5: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),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),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C6) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_2749_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C6: set(B),A5: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
                   => ( aa(B,A,G,A4) = zero_zero(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4)))
                     => ( aa(B,A,H,B3) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),C6) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B4) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_2750_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T4) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_2751_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_2752_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T4: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,H,X4) = zero_zero(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T4) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_2753_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_2754_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R4: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R4,one_one(A)),one_one(A)))
         => ( ! [X12: A,Y12: A,X23: A,Y22: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X12),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R4,Y12),Y22)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(A,A,aa(A,fun(A,A),times_times(A),X12),Y12)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y22))) )
           => ( pp(aa(set(B),bool,finite_finite2(B),S2))
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(B,A,H,X4)),aa(B,A,G,X4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2))) ) ) ) ) ) ).

% prod.related
tff(fact_2755_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(B),A5: set(B),G: 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),G),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),G),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),G),B4)) ) ) ) ) ).

% sum.subset_diff
tff(fact_2756_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C6: set(B),A5: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),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),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C6) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2757_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C6: set(B),A5: set(B),B4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),C6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),C6))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),C6))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),A5)))
                   => ( aa(B,A,G,A4) = one_one(A) ) )
               => ( ! [B3: B] :
                      ( pp(aa(set(B),bool,member(B,B3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),C6),B4)))
                     => ( aa(B,A,H,B3) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C6) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B4) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2758_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T4) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2759_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2760_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T4: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,H,X4) = one_one(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T4) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2761_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T4: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( ! [X4: B] :
                    ( pp(aa(set(B),bool,member(B,X4),S2))
                   => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2762_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5668285175392031749et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_2763_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A5: set(A),F: fun(A,nat)] :
      ( 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),F),A5) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
            & ( aa(A,nat,F,X3) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),A5))
               => ( ( X3 != Xa2 )
                 => ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_2764_sum__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F: fun(A,nat)] :
      ( 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),F),A5) = one_one(nat) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
            & ( aa(A,nat,F,X3) = one_one(nat) )
            & ! [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),A5))
               => ( ( X3 != Xa2 )
                 => ( aa(A,nat,F,Xa2) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_2765_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F: fun(B,A),I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,member(B,I2),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),S) = zero_zero(A) )
             => ( pp(aa(set(B),bool,member(B,I),S))
               => ( aa(B,A,F,I) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2766_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F: fun(B,A),B4: A,I: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),S))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,member(B,I2),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2))) )
           => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),S) = B4 )
             => ( pp(aa(set(B),bool,member(B,I),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),B4)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2767_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_ic(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_2768_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),K),L) = aa(code_integer,code_integer,abs_abs(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),aa(bool,fun(code_integer,fun(code_integer,code_integer)),if(code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),L),zero_zero(code_integer))),K),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),L),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))))) ).

% gcd_code_integer
tff(fact_2769_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_id(A,fun(A,bool),A3)))) ) ).

% finite_abs_int_segment
tff(fact_2770_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,X: A,Y3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),X) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y3) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y3) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( X = Y3 ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_2771_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I: B,F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(B),bool,member(B,I),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F,I)))
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,member(B,I2),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2))) )
               => 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),F),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_2772_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B4: set(B),A5: set(B),F: 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))
           => ( ! [B3: B] :
                  ( pp(aa(set(B),bool,member(B,B3),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,F,B3))) )
             => 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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),B4))) ) ) ) ) ).

% sum_mono2
tff(fact_2773_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( pp(aa(set(A),bool,member(A,I),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F,I)))
             => ( ! [I2: A] :
                    ( pp(aa(set(A),bool,member(A,I2),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F,I2))) )
               => 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),F),I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2774_gcd__int__def,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y3) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y3)))) ).

% gcd_int_def
tff(fact_2775_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B4: set(A),A5: set(A),B2: A,F: 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,F,B2)))
               => ( ! [X4: A] :
                      ( pp(aa(set(A),bool,member(A,X4),B4))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X4))) )
                 => 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),F),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),B4))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_2776_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B4: set(A),A5: set(A),F: 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))
           => ( ! [B3: A] :
                  ( pp(aa(set(A),bool,member(A,B3),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,F,B3))) )
             => ( ! [A4: A] :
                    ( pp(aa(set(A),bool,member(A,A4),A5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A4))) )
               => 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),F),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F),B4))) ) ) ) ) ) ).

% prod_mono2
tff(fact_2777_bezw__aux,axiom,
    ! [X: nat,Y3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(X,Y3))),aa(nat,int,semiring_1_of_nat(int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(X,Y3))),aa(nat,int,semiring_1_of_nat(int),Y3))) ).

% bezw_aux
tff(fact_2778_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( bit_un7362597486090784418nd_num(M,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_2779_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% one_or_eq
tff(fact_2780_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(bool,A,zero_neq_one_of_bool(A),aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% or_one_eq
tff(fact_2781_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_2782_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un7362597486090784418nd_num(M,N)) ) ).

% numeral_and_num
tff(fact_2783_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% Suc_0_or_eq
tff(fact_2784_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% or_Suc_0_eq
tff(fact_2785_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_2786_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_ie(nat,fun(nat,bool)),K)))) ).

% finite_Collect_le_nat
tff(fact_2787_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_cq(nat,fun(nat,bool)),K)))) ).

% finite_Collect_less_nat
tff(fact_2788_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_if(set(A),fun(set(A),bool),A5)))) ) ).

% finite_Collect_subsets
tff(fact_2789_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y3 = X ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) )
           => ~ 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),Xa))) ) ) ) ).

% gcd_nat.pelims
tff(fact_2790_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_av(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)))) ) ).

% finite_Collect_conjI
tff(fact_2791_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_ig(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_2792_gcd__idem__nat,axiom,
    ! [X: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),X) = X ).

% gcd_idem_nat
tff(fact_2793_gcd__nat_Oright__idem,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)),B2) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ).

% gcd_nat.right_idem
tff(fact_2794_gcd__nat_Oleft__idem,axiom,
    ! [A3: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B2) ).

% gcd_nat.left_idem
tff(fact_2795_gcd__nat_Oidem,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),A3) = A3 ).

% gcd_nat.idem
tff(fact_2796_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_ih(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int1
tff(fact_2797_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_ii(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int4
tff(fact_2798_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_ij(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int2
tff(fact_2799_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_ik(int,fun(int,fun(int,bool)),A3),B2)))) ).

% finite_interval_int3
tff(fact_2800_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_aa(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),B4))) ).

% minus_set_def
tff(fact_2801_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_il(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% set_diff_eq
tff(fact_2802_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => pp(aa(set(int),bool,finite_finite2(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_im(int,fun(int,bool),I)))) ) ).

% finite_divisors_int
tff(fact_2803_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),aa(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),comp(nat,nat,nat),suc),suc) = aa(fun(nat,nat),fun(nat,nat),aa(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),comp(nat,nat,nat),suc),suc) ).

% card.comp_fun_commute_on
tff(fact_2804_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),R4: fun(A,fun(B,bool))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pp(aa(set(B),bool,finite_finite2(B),B4))
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),A5))
             => ? [Xa3: B] :
                  ( pp(aa(set(B),bool,member(B,Xa3),B4))
                  & pp(aa(B,bool,aa(A,fun(B,bool),R4,X4),Xa3)) ) )
         => ? [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),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_in(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),A5),R4),X4)))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_2805_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_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% not_finite_existsD
tff(fact_2806_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_2807_even__set__encode__iff,axiom,
    ! [A5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),nat_set_encode(A5)))
      <=> ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) ) ) ).

% even_set_encode_iff
tff(fact_2808_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_2809_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A5: set(B),F: 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),bit0(one2))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),A5)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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_io(set(B),fun(fun(B,A),fun(B,bool)),A5),F))))) ) ) ) ).

% even_sum_iff
tff(fact_2810_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = A3 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_2811_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_ip(set(A),fun(set(A),bool)))) ).

% finite_psubset_def
tff(fact_2812_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N))) = N ).

% card_Collect_less_nat
tff(fact_2813_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_2814_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A3)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).

% drop_bit_drop_bit
tff(fact_2815_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P,Q)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_or_iff
tff(fact_2816_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).

% max_Suc_Suc
tff(fact_2817_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_2818_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_2819_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_2820_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_2821_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_2822_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_2823_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_ie(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_2824_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_2825_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_2826_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),one_one(A)) = one_one(A) ) ) ).

% max_0_1(1)
tff(fact_2827_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),zero_zero(A)) = one_one(A) ) ) ).

% max_0_1(2)
tff(fact_2828_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_2829_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_2830_card_Oinfinite,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) ) ) ).

% card.infinite
tff(fact_2831_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_2832_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_2833_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y3: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_iq(A,fun(B,A),Y3)),A5) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y3),aa(set(B),nat,finite_card(B),A5)) ) ).

% prod_constant
tff(fact_2834_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_2835_sum__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Y3: A,A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_ir(A,fun(B,A),Y3)),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))),Y3) ) ).

% sum_constant
tff(fact_2836_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y3),Z)) ) ).

% max_add_distrib_left
tff(fact_2837_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y3),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_2838_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Z)) ) ).

% max_diff_distrib_left
tff(fact_2839_of__int__max,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y3: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_max(int),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y3)) ) ).

% of_int_max
tff(fact_2840_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y3: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y3)) = 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),Y3)) ) ).

% of_nat_max
tff(fact_2841_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ).

% nat_add_max_right
tff(fact_2842_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_2843_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_max_right
tff(fact_2844_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_2845_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa3) = Xa3 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa3) = X2 ) ) ) ) ).

% max_def_raw
tff(fact_2846_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_is(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_2847_nat__minus__add__max,axiom,
    ! [N: 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),N),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).

% nat_minus_add_max
tff(fact_2848_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( bit_se2584673776208193580ke_bit(A,N,A3) = A3 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_2849_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,M,aa(A,A,bit_se4197421643247451524op_bit(A,N),A3)) = aa(A,A,bit_se4197421643247451524op_bit(A,N),bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),A3)) ) ).

% take_bit_drop_bit
tff(fact_2850_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S: set(A),T2: set(B),R4: fun(A,fun(B,bool)),K: fun(B,nat)] :
      ( pp(aa(set(A),bool,finite_finite2(A),S))
     => ( pp(aa(set(B),bool,finite_finite2(B),T2))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),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_in(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S),R4),X4))) = aa(B,nat,K,X4) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,bool)),fun(A,nat),aTP_Lamp_iu(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T2),R4)),S) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_2851_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_iv(A,nat)),A5) ).

% card_eq_sum
tff(fact_2852_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),aa(nat,nat,suc,N)),aTP_Lamp_iw(nat,fun(nat,nat),N)),M) ).

% max_Suc2
tff(fact_2853_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),aa(nat,nat,suc,N)),aTP_Lamp_ix(nat,fun(nat,nat),N)),M) ).

% max_Suc1
tff(fact_2854_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_2855_card__less__Suc2,axiom,
    ! [M2: set(nat),I: nat] :
      ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M2))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_iy(set(nat),fun(nat,fun(nat,bool)),M2),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_iz(set(nat),fun(nat,fun(nat,bool)),M2),I))) ) ) ).

% card_less_Suc2
tff(fact_2856_card__less__Suc,axiom,
    ! [M2: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M2))
     => ( 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_iy(set(nat),fun(nat,fun(nat,bool)),M2),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_iz(set(nat),fun(nat,fun(nat,bool)),M2),I))) ) ) ).

% card_less_Suc
tff(fact_2857_card__less,axiom,
    ! [M2: set(nat),I: nat] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M2))
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_iz(set(nat),fun(nat,fun(nat,bool)),M2),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_2858_sum__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),A5: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ja(fun(A,nat),fun(A,nat),F)),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),F),A5)),aa(set(A),nat,finite_card(A),A5)) ).

% sum_Suc
tff(fact_2859_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T4: set(B),R4: fun(A,fun(B,bool)),K: nat] :
      ( pp(aa(set(A),bool,finite_finite2(A),S2))
     => ( pp(aa(set(B),bool,finite_finite2(B),T4))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),T4))
             => ( 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_in(set(A),fun(fun(A,fun(B,bool)),fun(B,fun(A,bool))),S2),R4),X4))) = 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_iu(set(B),fun(fun(A,fun(B,bool)),fun(A,nat)),T4),R4)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T4)) ) ) ) ) ).

% sum_multicount
tff(fact_2860_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_2861_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se4197421643247451524op_bit(A,N),A3))),bit_se2584673776208193580ke_bit(A,N,A3)) = A3 ) ).

% bits_ident
tff(fact_2862_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_2863_set__encode__inf,axiom,
    ! [A5: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),A5))
     => ( nat_set_encode(A5) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_2864_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))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => ! [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),A5))
               => ( X3 = Xa2 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_2865_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N5)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_2866_card__sum__le__nat__sum,axiom,
    ! [S2: 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_cp(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S2)))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),S2))) ).

% card_sum_le_nat_sum
tff(fact_2867_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F: fun(B,A),N: A,K: nat] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I2)),N)) ) )
         => ( 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)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)),aa(nat,A,aa(A,fun(nat,A),power_power(A),N),K))) ) ) ) ) ).

% prod_le_power
tff(fact_2868_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A5: set(B),F: fun(B,A),K5: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I2)),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),F),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_2869_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_jb(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S2) = 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),S2)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_jb(B,fun(fun(B,A),fun(A,fun(B,A))),A3),B2),C2)),S2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,finite_card(B),S2)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_2870_set__encode__insert,axiom,
    ! [A5: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
     => ( ~ pp(aa(set(nat),bool,member(nat,N),A5))
       => ( nat_set_encode(aa(set(nat),set(nat),insert2(nat,N),A5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),nat_set_encode(A5)) ) ) ) ).

% set_encode_insert
tff(fact_2871_card__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),N: 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_jc(set(A),fun(nat,fun(list(A),bool)),A5),N))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_2872_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_2873_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A,N: 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),bit0(one2))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_2874_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ie(nat,fun(nat,bool)),aTP_Lamp_cq(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_2875_apsnd__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),X: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X))),aa(C,B,F,aa(product_prod(D,C),C,product_snd(D,C),X))) ).

% apsnd_apfst
tff(fact_2876_apfst__conv,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(C,A),X: C,Y3: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y3)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X)),Y3) ).

% apfst_conv
tff(fact_2877_apfst__eq__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,A),X: product_prod(C,B),G: fun(C,A)] :
      ( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),X) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G),X) )
    <=> ( aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X)) = aa(C,A,G,aa(product_prod(C,B),C,product_fst(C,B),X)) ) ) ).

% apfst_eq_conv
tff(fact_2878_fst__apfst,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: 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,F),X)) = aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X)) ).

% fst_apfst
tff(fact_2879_snd__apfst,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: 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,F),X)) = aa(product_prod(C,A),A,product_snd(C,A),X) ).

% snd_apfst
tff(fact_2880_snd__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),aa(fun(product_prod(C,B),B),fun(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,F)) = product_snd(A,B) ).

% snd_comp_apfst
tff(fact_2881_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),X: B,G: 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),G),aa(set(B),set(B),insert2(B,X),A5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5)) ) ) ) ) ).

% sum.insert
tff(fact_2882_fst__comp__apfst,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),aa(fun(product_prod(C,B),C),fun(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,F)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).

% fst_comp_apfst
tff(fact_2883_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A5: set(A),B4: set(A)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( ~ pp(aa(set(A),bool,member(A,A3),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),aa(set(A),set(A),insert2(A,A3),B4))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_2884_apfst__apsnd,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X))) ).

% apfst_apsnd
tff(fact_2885_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),zero_zero(nat)))
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ).

% bit_0
tff(fact_2886_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)) ) ) ) ).

% bit_mod_2_iff
tff(fact_2887_insert__compr,axiom,
    ! [A: $tType,A3: A,B4: set(A)] : aa(set(A),set(A),insert2(A,A3),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_jd(A,fun(set(A),fun(A,bool)),A3),B4)) ).

% insert_compr
tff(fact_2888_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,bool)] : aa(set(A),set(A),insert2(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_je(A,fun(fun(A,bool),fun(A,bool)),A3),P)) ).

% insert_Collect
tff(fact_2889_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B2: A,N: nat] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_2890_apfst__compose,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,C),X: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G),X)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),aa(fun(C,A),fun(fun(D,C),fun(D,A)),comp(C,A,D),F),G)),X) ).

% apfst_compose
tff(fact_2891_apsnd__apfst__commute,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),P2: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),P2)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G),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),F),P2)) ).

% apsnd_apfst_commute
tff(fact_2892_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N))) ) ).

% not_bit_1_Suc
tff(fact_2893_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_2894_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% bit_numeral_simps(1)
tff(fact_2895_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_2896_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_2897_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B2) ) ) ) ).

% disjunctive_add
tff(fact_2898_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_2899_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,N),A3)) = aa(fun(nat,nat),fun(nat,bool),aa(fun(nat,bool),fun(fun(nat,nat),fun(nat,bool)),comp(nat,bool,nat),bit_se5641148757651400278ts_bit(A,A3)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_2900_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_2901_finite__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),N: 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_jf(set(A),fun(nat,fun(list(A),bool)),A5),N)))) ) ).

% finite_lists_length_eq
tff(fact_2902_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_2903_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_2904_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),X: B,G: 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),G),aa(set(B),set(B),insert2(B,X),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),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),G),aa(set(B),set(B),insert2(B,X),A5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5)) ) ) ) ) ) ).

% sum.insert_if
tff(fact_2905_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_2906_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_2907_finite__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),N: 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_jc(set(A),fun(nat,fun(list(A),bool)),A5),N)))) ) ).

% finite_lists_length_le
tff(fact_2908_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(W))),N))
        <=> pp(aa(nat,bool,aa(fun(nat,bool),fun(nat,bool),aa(bool,fun(fun(nat,bool),fun(nat,bool)),case_nat(bool),fFalse),bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W))),N)) ) ) ).

% bit_numeral_rec(1)
tff(fact_2909_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),N))
        <=> pp(aa(nat,bool,aa(fun(nat,bool),fun(nat,bool),aa(bool,fun(fun(nat,bool),fun(nat,bool)),case_nat(bool),fTrue),bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W))),N)) ) ) ).

% bit_numeral_rec(2)
tff(fact_2910_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_2911_card__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),N: 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_jf(set(A),fun(nat,fun(list(A),bool)),A5),N))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5)),N) ) ) ).

% card_lists_length_eq
tff(fact_2912_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_2913_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M3))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M3))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N2),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_2914_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_2915_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_jg(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_2916_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y3: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),insert2(A,X),Y3))),N)) ) ) ).

% card_insert_le_m1
tff(fact_2917_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] : bit_se2584673776208193580ke_bit(A,N,A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_jh(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_2918_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_2919_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_2920_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa(set(nat),bool,member(nat,N),nat_set_decode(Z)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Z)) = aa(set(nat),set(nat),insert2(nat,N),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2921_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_ri4674362597316999326ke_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))))),bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N),A3)) ) ).

% signed_take_bit_code
tff(fact_2922_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_ji(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_cp(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_2923_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_jj(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_cp(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_2924_apfst__convE,axiom,
    ! [C: $tType,A: $tType,B: $tType,Q2: product_prod(A,B),F: fun(C,A),P2: product_prod(C,B)] :
      ( ( Q2 = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),P2) )
     => ~ ! [X4: C,Y4: B] :
            ( ( P2 = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X4),Y4) )
           => ( Q2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X4)),Y4) ) ) ) ).

% apfst_convE
tff(fact_2925_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_jl(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% times_int.abs_eq
tff(fact_2926_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_jn(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_2927_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_2928_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_2929_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_2930_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_2931_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_2932_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),N: 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_ji(set(A),fun(nat,fun(list(A),bool)),A5),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_2933_int_Oabs__induct,axiom,
    ! [P: fun(int,bool),X: int] :
      ( ! [Y4: product_prod(nat,nat)] : pp(aa(int,bool,P,aa(product_prod(nat,nat),int,abs_Integ,Y4)))
     => pp(aa(int,bool,P,X)) ) ).

% int.abs_induct
tff(fact_2934_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),Q: fun(B,bool),G: fun(A,B),X2: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),aa(fun(B,bool),fun(B,bool),aTP_Lamp_jo(fun(B,bool),fun(fun(B,bool),fun(B,bool)),P),Q)),G),X2))
    <=> ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),P),G),X2))
        & pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),G),X2)) ) ) ).

% conj_comp_iff
tff(fact_2935_quotient__of__div,axiom,
    ! [R: rat,N: int,D3: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),N),D3) )
     => ( R = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),N)),aa(int,rat,ring_1_of_int(rat),D3)) ) ) ).

% quotient_of_div
tff(fact_2936_eq__Abs__Integ,axiom,
    ! [Z: int] :
      ~ ! [X4: nat,Y4: nat] : Z != 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),X4),Y4)) ).

% eq_Abs_Integ
tff(fact_2937_nat_Oabs__eq,axiom,
    ! [X: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),X) ).

% nat.abs_eq
tff(fact_2938_quotient__of__denom__pos,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2)) ) ).

% quotient_of_denom_pos
tff(fact_2939_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_2940_int__def,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),N) = aa(product_prod(nat,nat),int,abs_Integ,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N),zero_zero(nat))) ).

% int_def
tff(fact_2941_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = 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(P2)) ).

% rat_floor_code
tff(fact_2942_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_jp(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_2943_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_jq(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_2944_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P2),Q2))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_js(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P2))) ) ).

% rat_less_eq_code
tff(fact_2945_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P2),Q2))
    <=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_ju(rat,fun(int,fun(int,bool)),Q2)),quotient_of(P2))) ) ).

% rat_less_code
tff(fact_2946_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_jv(nat,fun(nat,product_prod(nat,nat)))),X)) ).

% uminus_int.abs_eq
tff(fact_2947_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_2948_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_2949_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_jw(nat,fun(nat,A))),X) ) ).

% of_int.abs_eq
tff(fact_2950_less__int_Oabs__eq,axiom,
    ! [Xa: 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,Xa)),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_jy(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_2951_less__eq__int_Oabs__eq,axiom,
    ! [Xa: 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,Xa)),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_ka(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_eq_int.abs_eq
tff(fact_2952_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% plus_int.abs_eq
tff(fact_2953_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ke(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_2954_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_kg(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_2955_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa)),aa(product_prod(nat,nat),int,abs_Integ,X)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_ki(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% minus_int.abs_eq
tff(fact_2956_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_kk(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_2957_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_2958_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_kl(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_2959_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(option(num),A,aa(fun(num,A),fun(option(num),A),aa(A,fun(fun(num,A),fun(option(num),A)),case_option(A,num),zero_zero(A)),numeral_numeral(A)),bit_un2480387367778600638or_num(M,N)) ) ).

% numeral_xor_num
tff(fact_2960_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).

% diff_numeral_special(8)
tff(fact_2961_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).

% diff_numeral_special(7)
tff(fact_2962_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_2963_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_2964_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_2965_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_2966_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_2967_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_2968_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_2969_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_2970_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_2971_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_2972_sub__num__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_sub(A,one2,one2) = zero_zero(A) ) ) ).

% sub_num_simps(1)
tff(fact_2973_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_2974_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_2975_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_2976_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_2977_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_2978_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_2979_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_2980_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V)),Y3) ) ).

% semiring_norm(167)
tff(fact_2981_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W)),Y3) ) ).

% semiring_norm(166)
tff(fact_2982_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_2983_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),M)),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).

% diff_numeral_special(2)
tff(fact_2984_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).

% diff_numeral_special(1)
tff(fact_2985_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(1)
tff(fact_2986_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(2)
tff(fact_2987_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_2988_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_2989_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).

% minus_sub_one_diff_one
tff(fact_2990_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_2991_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_2992_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A3 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_2993_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_2994_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_2995_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_2996_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_2997_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_2998_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_2999_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_3000_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_3001_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_3002_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_3003_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_3004_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3005_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_3006_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).

% inverse_eq_divide
tff(fact_3007_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3008_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_3009_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3010_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: int,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xa))) ) ).

% mult_inverse_of_int_commute
tff(fact_3011_prod__inversef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: 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),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),inverse_inverse(A)),F)),A5) = aa(A,A,inverse_inverse(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)) ) ).

% prod_inversef
tff(fact_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3024_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_3025_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_3026_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_3027_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_3028_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_3029_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))
         => ? [N2: 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,N2)))),X)) ) ) ).

% reals_Archimedean
tff(fact_3030_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% sub_non_positive
tff(fact_3031_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% sub_non_negative
tff(fact_3032_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% sub_positive
tff(fact_3033_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% sub_negative
tff(fact_3034_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))
         => ? [N2: nat] :
              ( 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(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3035_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3036_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub(int,bitM(N),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),neg_numeral_sub(int,N,one2)) ).

% sub_BitM_One_eq
tff(fact_3037_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( bit_un2480387367778600638or_num(M,N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_3038_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_3039_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_km(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_3040_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
    <=> 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_ka(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,Xa))) ) ).

% less_eq_int.rep_eq
tff(fact_3041_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa))
    <=> 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_jy(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,Xa))) ) ).

% less_int.rep_eq
tff(fact_3042_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_3043_nat_Orep__eq,axiom,
    ! [X: int] : aa(int,nat,nat2,X) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,X)) ).

% nat.rep_eq
tff(fact_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_jw(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X)) ) ).

% of_int.rep_eq
tff(fact_3050_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_3051_old_Orec__unit__def,axiom,
    ! [T: $tType,X2: T,Xa3: product_unit] : product_rec_unit(T,X2,Xa3) = the(T,product_rec_set_unit(T,X2,Xa3)) ).

% old.rec_unit_def
tff(fact_3052_old_Orec__prod__def,axiom,
    ! [T: $tType,B: $tType,A: $tType,X2: fun(A,fun(B,T)),Xa3: product_prod(A,B)] : product_rec_prod(A,B,T,X2,Xa3) = the(T,product_rec_set_prod(A,B,T,X2,Xa3)) ).

% old.rec_prod_def
tff(fact_3053_old_Orec__nat__def,axiom,
    ! [T: $tType,X2: T,Xa3: fun(nat,fun(T,T)),Xb: nat] : aa(nat,T,rec_nat(T,X2,Xa3),Xb) = the(T,rec_set_nat(T,X2,Xa3,Xb)) ).

% old.rec_nat_def
tff(fact_3054_the__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ( X4 = A3 ) )
       => ( the(A,P) = A3 ) ) ) ).

% the_equality
tff(fact_3055_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_kn(A,fun(A,bool),A3)) = A3 ).

% the_eq_trivial
tff(fact_3056_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_3057_The__split__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: 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_ko(A,fun(B,fun(A,fun(B,bool))),X),Y3))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) ).

% The_split_eq
tff(fact_3058_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_kp(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).

% The_case_prod
tff(fact_3059_the1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ? [X2: A] :
          ( pp(aa(A,bool,P,X2))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => ( Y4 = X2 ) ) )
     => ( pp(aa(A,bool,P,A3))
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality
tff(fact_3060_the1I2,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(A,bool,P,X2))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => ( Y4 = X2 ) ) )
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => pp(aa(A,bool,Q,X4)) )
       => pp(aa(A,bool,Q,the(A,P))) ) ) ).

% the1I2
tff(fact_3061_If__def,axiom,
    ! [A: $tType,P: bool,X: A,Y3: A] :
      ( ( pp(P)
       => ( X = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_kq(bool,fun(A,fun(A,fun(A,bool))),P),X),Y3)) ) )
      & ( ~ pp(P)
       => ( Y3 = the(A,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aTP_Lamp_kq(bool,fun(A,fun(A,fun(A,bool))),P),X),Y3)) ) ) ) ).

% If_def
tff(fact_3062_theI2,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ( X4 = A3 ) )
       => ( ! [X4: A] :
              ( pp(aa(A,bool,P,X4))
             => pp(aa(A,bool,Q,X4)) )
         => pp(aa(A,bool,Q,the(A,P))) ) ) ) ).

% theI2
tff(fact_3063_theI_H,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(A,bool,P,X2))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => ( Y4 = X2 ) ) )
     => pp(aa(A,bool,P,the(A,P))) ) ).

% theI'
tff(fact_3064_theI,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ( X4 = A3 ) )
       => pp(aa(A,bool,P,the(A,P))) ) ) ).

% theI
tff(fact_3065_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_ks(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_3066_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_jv(nat,fun(nat,product_prod(nat,nat))))) ).

% uminus_int_def
tff(fact_3067_prod__encode__def,axiom,
    nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_kt(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_3068_eq__snd__iff,axiom,
    ! [B: $tType,A: $tType,B2: A,P2: product_prod(B,A)] :
      ( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P2) )
    <=> ? [A7: B] : P2 = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B2) ) ).

% eq_snd_iff
tff(fact_3069_eq__fst__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,P2: product_prod(A,B)] :
      ( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P2) )
    <=> ? [B7: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B7) ) ).

% eq_fst_iff
tff(fact_3070_map__fun__apply,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType,F: fun(B,C),G: fun(D,A),H: fun(C,D),X: B] : aa(B,A,aa(fun(C,D),fun(B,A),map_fun(B,C,D,A,F,G),H),X) = aa(D,A,G,aa(C,D,H,aa(B,C,F,X))) ).

% map_fun_apply
tff(fact_3071_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B5: B,R: set(product_prod(A,A)),S: 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),A6),B5))),lex_prod(A,B,R,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),A3),A6)),R))
        | ( ( A3 = A6 )
          & 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),B5)),S)) ) ) ) ).

% in_lex_prod
tff(fact_3072_map__fun__def,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(B,D),H: fun(A,B)] : aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,F,G),H) = aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),G),H)),F) ).

% map_fun_def
tff(fact_3073_map__fun_Ocompositionality,axiom,
    ! [D: $tType,F2: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: fun(E,C),G: fun(D,F2),H: fun(C,A),I: fun(B,D),Fun: fun(A,B)] : aa(fun(C,D),fun(E,F2),map_fun(E,C,D,F2,F,G),aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,H,I),Fun)) = aa(fun(A,B),fun(E,F2),map_fun(E,A,B,F2,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Fun) ).

% map_fun.compositionality
tff(fact_3074_map__fun_Ocomp,axiom,
    ! [E: $tType,C: $tType,A: $tType,F2: $tType,D: $tType,B: $tType,F: fun(E,C),G: fun(D,F2),H: fun(C,A),I: fun(B,D)] : aa(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F2)),aa(fun(fun(C,D),fun(E,F2)),fun(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F2))),comp(fun(C,D),fun(E,F2),fun(A,B)),map_fun(E,C,D,F2,F,G)),map_fun(C,A,B,D,H,I)) = map_fun(E,A,B,F2,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).

% map_fun.comp
tff(fact_3075_sum_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X) != zero_zero(nat) ).

% sum.size_neq
tff(fact_3076_prod_Osize__neq,axiom,
    ! [A: $tType,B: $tType,X: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X) != zero_zero(nat) ).

% prod.size_neq
tff(fact_3077_le__prod__encode__1,axiom,
    ! [A3: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).

% le_prod_encode_1
tff(fact_3078_le__prod__encode__2,axiom,
    ! [B2: nat,A3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),B2)))) ).

% le_prod_encode_2
tff(fact_3079_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_3080_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_jl(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% times_int_def
tff(fact_3081_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_ki(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% minus_int_def
tff(fact_3082_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_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).

% plus_int_def
tff(fact_3083_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),R4: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P,R4) = 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_kv(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P),R4)))) ).

% same_fst_def
tff(fact_3084_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_kw(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_3085_same__fstI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),X: A,Y6: B,Y3: B,R4: 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),Y6),Y3)),aa(A,set(product_prod(B,B)),R4,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),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3))),same_fst(A,B,P,R4))) ) ) ).

% same_fstI
tff(fact_3086_gcd__integer__def,axiom,
    gcd_gcd(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),gcd_gcd(int)) ).

% gcd_integer_def
tff(fact_3087_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y3: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y3)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Y3))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_3088_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_3089_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: fun(B,A),I: 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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,member(B,I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P2,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_3090_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),bool),aTP_Lamp_kx(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_3091_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_ka(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_eq_int_def
tff(fact_3092_id__apply,axiom,
    ! [A: $tType,X: A] : aa(A,A,id(A),X) = X ).

% id_apply
tff(fact_3093_fun_Omap__id,axiom,
    ! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)),T2) = T2 ).

% fun.map_id
tff(fact_3094_id__comp,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B)] : aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),G) = G ).

% id_comp
tff(fact_3095_comp__id,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),id(A)) = F ).

% comp_id
tff(fact_3096_apfst__id,axiom,
    ! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).

% apfst_id
tff(fact_3097_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_3098_group__add__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% group_add_class.minus_comp_minus
tff(fact_3099_boolean__algebra__class_Ominus__comp__minus,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).

% boolean_algebra_class.minus_comp_minus
tff(fact_3100_push__bit__0__id,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4730199178511100633sh_bit(A,zero_zero(nat)) = id(A) ) ) ).

% push_bit_0_id
tff(fact_3101_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_3102_comp__the__Some,axiom,
    ! [A: $tType] : aa(fun(A,option(A)),fun(A,A),aa(fun(option(A),A),fun(fun(A,option(A)),fun(A,A)),comp(option(A),A,A),the2(A)),some(A)) = id(A) ).

% comp_the_Some
tff(fact_3103_fun_Omap__id0,axiom,
    ! [A: $tType,D: $tType] : aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)) = id(fun(D,A)) ).

% fun.map_id0
tff(fact_3104_pointfree__idE,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B),X: A] :
      ( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
     => ( aa(B,A,F,aa(A,B,G,X)) = X ) ) ).

% pointfree_idE
tff(fact_3105_comp__eq__id__dest,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C2: fun(A,B),V: A] :
      ( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),C2) )
     => ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C2,V) ) ) ).

% comp_eq_id_dest
tff(fact_3106_id__def,axiom,
    ! [A: $tType,X2: A] : aa(A,A,id(A),X2) = X2 ).

% id_def
tff(fact_3107_eq__id__iff,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( ! [X3: A] : aa(A,A,F,X3) = X3
    <=> ( F = id(A) ) ) ).

% eq_id_iff
tff(fact_3108_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_3109_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_3110_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_3111_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_ky(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5) ) ).

% prod.non_neutral'
tff(fact_3112_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y3: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Y3) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y3)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_3113_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_3114_not__iszero__neg__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_iszero_neg_1
tff(fact_3115_prod_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fj(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_3116_fst__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_fst(A,A)),aTP_Lamp_dm(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% fst_diag_id
tff(fact_3117_snd__diag__id,axiom,
    ! [A: $tType,Z: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_snd(A,A)),aTP_Lamp_dm(A,product_prod(A,A))),Z) = aa(A,A,id(A),Z) ).

% snd_diag_id
tff(fact_3118_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3119_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T4: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,H,I2) = one_one(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),T4) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3120_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3121_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = one_one(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),T4) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3122_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y3: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y3)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Y3)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_3123_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_3124_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G: 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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( 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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fj(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),H),I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_3125_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: 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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ 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_hu(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),I5) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_3126_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Y3: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Y3) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Y3)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_3127_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_3128_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_jw(nat,fun(nat,A)))) ) ) ).

% ring_1_class.of_int_def
tff(fact_3129_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_jy(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).

% less_int_def
tff(fact_3130_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_kz(int,int)) ).

% Code_Numeral.dup_def
tff(fact_3131_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_la(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_3132_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_lb(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_3133_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: nat] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_3134_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A5: set(B),F: fun(B,A),A3: B] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ( aa(B,A,F,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),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert2(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),F),A5)),aa(B,A,F,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),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_3135_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_3136_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3137_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).

% min_Suc_Suc
tff(fact_3138_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_3139_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_3140_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_3141_set__encode__empty,axiom,
    nat_set_encode(bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3142_singleton__conv,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_kn(A,fun(A,bool),A3)) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3143_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),insert2(A,A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3144_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_3145_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_3146_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_3147_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3148_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_3149_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_3150_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_3151_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_3152_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_3153_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1962203154675924110t_prod(B,A),P2),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_3154_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3155_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_3156_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_3157_sum__mult__of__bool__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A5: set(B),F: 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_lc(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P)),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),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_3158_sum__of__bool__mult__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [A5: set(B),P: fun(B,bool),F: 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_ld(fun(B,bool),fun(fun(B,A),fun(B,A)),P),F)),A5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),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_3159_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),I)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),I) ).

% min_diff
tff(fact_3160_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_3161_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_3162_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Z)) ) ).

% min_diff_distrib_left
tff(fact_3163_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y3: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y3)) = 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),Y3)) ) ).

% of_nat_min
tff(fact_3164_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y3),Z)) ) ).

% min_add_distrib_left
tff(fact_3165_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y3: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_min(A),Y3),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_3166_of__int__min,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,Y3: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),ord_min(int),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(int,A,ring_1_of_int(A),X)),aa(int,A,ring_1_of_int(A),Y3)) ) ).

% of_int_min
tff(fact_3167_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ap(A,bool)) ).

% Set.empty_def
tff(fact_3168_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_av(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_3169_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_3170_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_le(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% Int_def
tff(fact_3171_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa3) = X2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa3) = Xa3 ) ) ) ) ).

% min_def_raw
tff(fact_3172_map__fun_Oidentity,axiom,
    ! [B: $tType,A: $tType] : map_fun(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)) = id(fun(A,B)) ).

% map_fun.identity
tff(fact_3173_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_lg(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),insert2(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_lg(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if
tff(fact_3174_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_lh(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = aa(set(A),set(A),insert2(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_lh(fun(A,bool),fun(A,fun(A,bool)),P),A3)) = bot_bot(set(A)) ) ) ) ).

% Collect_conv_if2
tff(fact_3175_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y3)) ) ).

% minus_max_eq_min
tff(fact_3176_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y3)) ) ).

% minus_min_eq_max
tff(fact_3177_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_3178_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3179_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_3180_card__1__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
     => ~ ! [X4: A] : A5 != aa(set(A),set(A),insert2(A,X4),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_3181_sum_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),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_li(fun(B,A),fun(set(B),fun(B,A)),G),B4)),A5) ) ) ) ).

% sum.inter_restrict
tff(fact_3182_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),P: fun(B,bool),H: fun(B,A),G: 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_lj(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),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),G),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_3183_prod_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),P: fun(B,bool),H: fun(B,A),G: 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_lk(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),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),G),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_3184_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),G: 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),G),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_ll(fun(B,A),fun(set(B),fun(B,A)),G),B4)),A5) ) ) ) ).

% prod.inter_restrict
tff(fact_3185_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),P2)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_3186_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y3),P2)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_3187_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y3)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_3188_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y3)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y3)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_3189_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),P2)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_3190_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),X),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y3),P2)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_3191_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_lm(nat,fun(nat,nat),N)),M) ).

% min_Suc1
tff(fact_3192_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),case_nat(nat),zero_zero(nat)),aTP_Lamp_ln(nat,fun(nat,nat),N)),M) ).

% min_Suc2
tff(fact_3193_sum_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T4: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,finite_finite2(B),S2))
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,H,I2) = zero_zero(A) ) )
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),T4)))
                   => ( aa(B,A,G,I2) = zero_zero(A) ) )
               => ( ! [X4: B] :
                      ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),T4)))
                     => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),T4) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_3194_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),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),G),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),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B4))) ) ) ) ).

% sum.Int_Diff
tff(fact_3195_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T4: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),T4))
         => ( pp(aa(set(B),bool,finite_finite2(B),S2))
           => ( ! [I2: B] :
                  ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
                 => ( aa(B,A,H,I2) = one_one(A) ) )
             => ( ! [I2: B] :
                    ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),T4)))
                   => ( aa(B,A,G,I2) = one_one(A) ) )
               => ( ! [X4: B] :
                      ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),T4)))
                     => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T4) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_3196_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I2: B] :
                  ( 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,F,I2))) )
             => 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),F),I5))) ) ) ) ) ).

% sum_pos
tff(fact_3197_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( 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,F,I2))) )
             => 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),F),I5))) ) ) ) ) ).

% less_1_prod
tff(fact_3198_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_3199_card__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B8: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert2(A,B7),B8) )
          & ~ pp(aa(set(A),bool,member(A,B7),B8))
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_3200_card__eq__SucD,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
     => ? [B3: A,B9: set(A)] :
          ( ( A5 = aa(set(A),set(A),insert2(A,B3),B9) )
          & ~ pp(aa(set(A),bool,member(A,B3),B9))
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_3201_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)) )
    <=> ? [X3: A] : A5 = aa(set(A),set(A),insert2(A,X3),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_3202_num__of__integer__def,axiom,
    code_num_of_integer = aa(fun(int,num),fun(code_integer,num),map_fun(code_integer,int,num,num,code_int_of_integer,id(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)) ).

% num_of_integer_def
tff(fact_3203_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_3204_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,member(B,I2),A5))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F,I2)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I2)),aa(B,A,G,I2))) ) )
           => ( ( 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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5))) ) ) ) ) ).

% prod_mono_strict
tff(fact_3205_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A5: set(B),F: 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),F),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_lo(fun(B,A),fun(A,fun(B,A)),F),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_lp(fun(B,A),fun(A,fun(B,bool)),F),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),F),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_lq(fun(B,A),fun(A,fun(B,bool)),F),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_3206_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),G: 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),G),aa(set(B),set(B),insert2(B,X),A5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3207_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),X: B,G: 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),G),A5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert2(B,X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_3208_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),insert2(A,X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_3209_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),insert2(A,X),bot_bot(set(A))))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),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),insert2(A,X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Diff_singleton_if
tff(fact_3210_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_atMost(nat),N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3211_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),aa(nat,set(nat),set_ord_lessThan(nat),N)),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_3212_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = 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)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = 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)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_3213_prod_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = 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)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
             => ( 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_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = 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)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% prod.delta_remove
tff(fact_3214_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: C,A5: set(C),F: fun(C,B)] :
          ( pp(aa(set(C),bool,member(C,I),A5))
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,member(C,X4),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),A5),aa(set(C),set(C),insert2(C,I),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F,X4))) )
           => ( 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,F,I)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),F),A5))) ) ) ) ) ).

% member_le_sum
tff(fact_3215_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),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(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),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),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),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,member(int,K),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(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),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),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),bit0(one2))),K)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_3216_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,member(int,Xa),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y3 = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa))))) ) )
              & ( ~ ( pp(aa(set(int),bool,member(int,X),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,member(int,Xa),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),X)),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),X),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) )
           => ~ 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),Xa))) ) ) ) ).

% and_int.pelims
tff(fact_3217_the__elem__def,axiom,
    ! [A: $tType,X6: set(A)] : the_elem(A,X6) = the(A,aTP_Lamp_lt(set(A),fun(A,bool),X6)) ).

% the_elem_def
tff(fact_3218_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F5: set(A),I5: set(A),F: fun(A,B),I: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),F5))
         => ( 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_lu(set(A),fun(fun(A,B),fun(A,bool)),I5),F))),F5))
           => ( ( pp(aa(set(A),bool,member(A,I),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I5)),aa(A,B,F,I)) ) )
              & ( ~ pp(aa(set(A),bool,member(A,I),I5))
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3219_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_3220_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3221_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A),I: 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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,member(B,I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) ) )
            & ( ~ pp(aa(set(B),bool,member(B,I),I5))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P2,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_3222_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_aa(set(A),fun(A,bool)),A5))) ).

% uminus_set_def
tff(fact_3223_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_lv(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_3224_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_lw(set(A),fun(A,bool),A5)) ).

% Compl_eq
tff(fact_3225_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),aa(fun(B,bool),set(B),collect(B),aa(set(B),fun(B,bool),aTP_Lamp_lx(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5) ) ).

% sum.non_neutral'
tff(fact_3226_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_3227_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_dh(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3228_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3229_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T4: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [I2: B] :
                ( pp(aa(set(B),bool,member(B,I2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,H,I2) = zero_zero(A) ) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),S2))
                 => ( aa(B,A,G,X4) = aa(B,A,H,X4) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),T4) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3230_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3231_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T4))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T4),S2)))
               => ( aa(B,A,G,X4) = zero_zero(A) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),T4) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3232_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: 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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( 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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_dh(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),H),I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3233_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: 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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P2),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aTP_Lamp_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ 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_hs(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups1027152243600224163dd_sum(B,A),P2),I5) = zero_zero(A) ) ) ) ) ).

% sum.G_def
tff(fact_3234_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,L3: 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),L3)))
           => ( ( ~ ( pp(aa(set(int),bool,member(int,K2),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,member(int,L3),aa(set(int),set(int),insert2(int,zero_zero(int)),aa(set(int),set(int),insert2(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),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L3),aa(num,int,numeral_numeral(int),bit0(one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L3)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_3235_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F: fun(A,B),I: 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_lu(set(A),fun(fun(A,B),fun(A,bool)),I5),F))))
         => ( ( pp(aa(set(A),bool,member(A,I),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I5)),aa(A,B,F,I)) ) )
            & ( ~ pp(aa(set(A),bool,member(A,I),I5))
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),insert2(A,I),bot_bot(set(A))))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups1027152243600224163dd_sum(A,B),F),I5) ) ) ) ) ) ).

% sum_diff1'
tff(fact_3236_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)))
     => ( ! [I2: 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),I2),J2)))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J2))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I2),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_3237_Code__Numeral_Osub__def,axiom,
    code_sub = aa(fun(num,fun(num,int)),fun(num,fun(num,code_integer)),map_fun(num,num,fun(num,int),fun(num,code_integer),id(num),map_fun(num,num,int,code_integer,id(num),code_integer_of_int)),aTP_Lamp_ly(num,fun(num,int))) ).

% Code_Numeral.sub_def
tff(fact_3238_Gcd__remove0__nat,axiom,
    ! [M2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M2))
     => ( gcd_Gcd(nat,M2) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M2),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_3239_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y3: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y3))
       => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_lz(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y3)) = 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),Y3),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y3))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y3))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_lz(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y3)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y3))
           => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_lz(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y3)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_3240_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_lb(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_3241_image__ident,axiom,
    ! [A: $tType,Y7: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ak(A,A)),Y7) = Y7 ).

% image_ident
tff(fact_3242_image__id,axiom,
    ! [A: $tType] : image2(A,A,id(A)) = id(set(A)) ).

% image_id
tff(fact_3243_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_3244_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_3245_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_3246_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),aa(A,set(A),set_ord_atMost(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)) ) ).

% image_add_atMost
tff(fact_3247_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_3248_Gcd__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] : gcd_Gcd(A,aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),gcd_Gcd(A,A5)) ) ).

% Gcd_insert
tff(fact_3249_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ma(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_3250_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_mb(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_3251_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_ma(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_3252_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% Gcd_2
tff(fact_3253_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_3254_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),insert2(A,zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_3255_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_mc(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_3256_image__comp,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),R: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(C),set(B),image2(C,B,G),R)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),R) ).

% image_comp
tff(fact_3257_image__eq__imp__comp,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(B,A),A5: set(B),G: fun(C,A),B4: set(C),H: fun(A,D)] :
      ( ( aa(set(B),set(A),image2(B,A,F),A5) = aa(set(C),set(A),image2(C,A,G),B4) )
     => ( aa(set(B),set(D),image2(B,D,aa(fun(B,A),fun(B,D),aa(fun(A,D),fun(fun(B,A),fun(B,D)),comp(A,D,B),H),F)),A5) = aa(set(C),set(D),image2(C,D,aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),H),G)),B4) ) ) ).

% image_eq_imp_comp
tff(fact_3258_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F: 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_md(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F),A5),P)) = aa(set(B),set(A),image2(B,A,F),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_me(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F),A5),P))) ).

% Compr_image_eq
tff(fact_3259_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A5: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(C),set(B),image2(C,B,G),A5)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mf(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A5) ).

% image_image
tff(fact_3260_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F),A5)))
     => ~ ! [X4: B] :
            ( ( B2 = aa(B,A,F,X4) )
           => ~ pp(aa(set(B),bool,member(B,X4),A5)) ) ) ).

% imageE
tff(fact_3261_Gcd__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F,X4)),aa(B,A,G,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,F),A5))),gcd_Gcd(A,aa(set(B),set(A),image2(B,A,G),A5)))) ) ) ).

% Gcd_mono
tff(fact_3262_Gcd__greatest__nat,axiom,
    ! [A5: set(nat),A3: nat] :
      ( ! [B3: nat] :
          ( pp(aa(set(nat),bool,member(nat,B3),A5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),B3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),gcd_Gcd(nat,A5))) ) ).

% Gcd_greatest_nat
tff(fact_3263_Gcd__dvd__nat,axiom,
    ! [A3: nat,A5: set(nat)] :
      ( pp(aa(set(nat),bool,member(nat,A3),A5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),gcd_Gcd(nat,A5)),A3)) ) ).

% Gcd_dvd_nat
tff(fact_3264_Gcd__nat__eq__one,axiom,
    ! [N5: set(nat)] :
      ( pp(aa(set(nat),bool,member(nat,one_one(nat)),N5))
     => ( gcd_Gcd(nat,N5) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_3265_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),aTP_Lamp_mg(C,set(B))),F) = aa(fun(A,set(D)),fun(A,set(B)),aa(fun(set(D),set(B)),fun(fun(A,set(D)),fun(A,set(B))),comp(set(D),set(B),A),image2(D,B,G)),aTP_Lamp_mh(A,set(D))) ).

% empty_natural
tff(fact_3266_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,one_one(A)),A5))
         => ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_3267_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: 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,F),A5)))
       => ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),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_mi(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A5),F),X4)))) ) ) ) ).

% pigeonhole_infinite
tff(fact_3268_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(A,B),B4: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(set(B),bool,member(B,aa(A,B,F,X4)),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,F),aa(fun(A,bool),set(A),collect(A),P))),B4)) ) ).

% image_Collect_subsetI
tff(fact_3269_Gcd__greatest,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),A3: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,member(A,B3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),gcd_Gcd(A,A5))) ) ) ).

% Gcd_greatest
tff(fact_3270_dvd__Gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),gcd_Gcd(A,A5)))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),X3)) ) ) ) ).

% dvd_Gcd_iff
tff(fact_3271_dvd__GcdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [X: A,A5: set(A),Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),gcd_Gcd(A,A5)))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y3)) ) ) ) ).

% dvd_GcdD
tff(fact_3272_Gcd__dvd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Gcd(A,A5)),A3)) ) ) ).

% Gcd_dvd
tff(fact_3273_zero__notin__Suc__image,axiom,
    ! [A5: set(nat)] : ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),A5))) ).

% zero_notin_Suc_image
tff(fact_3274_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).

% translation_Int
tff(fact_3275_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).

% translation_diff
tff(fact_3276_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T2: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).

% translation_Compl
tff(fact_3277_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_cu(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_cu(A,fun(B,A)),C2)),A5) = aa(set(A),set(A),insert2(A,C2),bot_bot(set(A))) ) ) ) ).

% image_constant_conv
tff(fact_3278_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_mj(B,fun(A,B),C2)),A5) = aa(set(B),set(B),insert2(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_3279_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A5: set(A),F: fun(nat,A),N: nat] :
      ( ( A5 = aa(set(nat),set(A),image2(nat,A,F),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N))) )
     => pp(aa(set(A),bool,finite_finite2(A),A5)) ) ).

% nat_seg_image_imp_finite
tff(fact_3280_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
    <=> ? [N6: nat,F6: fun(nat,A)] : A5 = aa(set(nat),set(A),image2(nat,A,F6),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_3281_translation__subtract__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),S)),aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),T2)) ) ).

% translation_subtract_Int
tff(fact_3282_sum_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) = 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_mm(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S2),H),G)),aa(set(B),set(C),image2(B,C,G),S2)) ) ) ) ).

% sum.image_gen
tff(fact_3283_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),S)),aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),T2)) ) ).

% translation_subtract_diff
tff(fact_3284_prod_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),H: fun(B,A),G: fun(B,C)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) = 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_mn(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S2),H),G)),aa(set(B),set(C),image2(B,C,G),S2)) ) ) ) ).

% prod.image_gen
tff(fact_3285_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,T2: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),T2)) ) ).

% translation_subtract_Compl
tff(fact_3286_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( pp(aa(set(A),bool,member(A,A3),A5))
           => ( gcd_Gcd(A,A5) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_3287_Gcd__in,axiom,
    ! [A5: set(nat)] :
      ( ! [A4: nat,B3: nat] :
          ( pp(aa(set(nat),bool,member(nat,A4),A5))
         => ( pp(aa(set(nat),bool,member(nat,B3),A5))
           => pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B3)),A5)) ) )
     => ( ( A5 != bot_bot(set(nat)) )
       => pp(aa(set(nat),bool,member(nat,gcd_Gcd(nat,A5)),A5)) ) ) ).

% Gcd_in
tff(fact_3288_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T4: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( 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,G),S2)),T4))
             => ( 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_mo(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_3289_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T4: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( pp(aa(set(C),bool,finite_finite2(C),T4))
           => ( 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,G),S2)),T4))
             => ( 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_mp(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ).

% prod.group
tff(fact_3290_sum_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X4: B,Y4: B] :
                ( pp(aa(set(B),bool,member(B,X4),A5))
               => ( pp(aa(set(B),bool,member(B,Y4),A5))
                 => ( ( X4 != Y4 )
                   => ( ( aa(B,C,H,X4) = aa(B,C,H,Y4) )
                     => ( aa(C,A,G,aa(B,C,H,X4)) = zero_zero(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_3291_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X4: B,Y4: B] :
                ( pp(aa(set(B),bool,member(B,X4),A5))
               => ( pp(aa(set(B),bool,member(B,Y4),A5))
                 => ( ( X4 != Y4 )
                   => ( ( aa(B,C,H,X4) = aa(B,C,H,Y4) )
                     => ( aa(C,A,G,aa(B,C,H,X4)) = one_one(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_3292_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_3293_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_3294_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_3295_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_3296_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N))) ).

% atMost_Suc_eq_insert_0
tff(fact_3297_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_3298_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G: fun(A,B),F: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite2(C),I5))
         => ( ! [I2: C] :
                ( pp(aa(set(C),bool,member(C,I2),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F,I2)))) )
           => 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),G),aa(set(C),set(A),image2(C,A,F),I5))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),G),F)),I5))) ) ) ) ).

% sum_image_le
tff(fact_3299_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y3: 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,Y3)) = 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),Y3)) ) )
          & ( ~ 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),Y3))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y3),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),Y3))
               => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y3)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_3300_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3))
           => ( ( 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_mq(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y3)) = 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),Y3),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_mq(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y3),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),Y3))
           => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_mq(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y3)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_3301_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_mr(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_mr(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_mr(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_3302_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_ms(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_ms(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_ms(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_3303_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_mt(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_mt(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_mt(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_3304_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_mu(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_mu(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_mu(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_3305_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R4: set(B),G: fun(A,B),F: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( pp(aa(set(B),bool,finite_finite2(B),R4))
           => ( 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,G),S2)),R4))
             => ( 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_mv(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mx(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G),F)),R4) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_3306_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_my(nat,bool))) ) ).

% semiring_char_def
tff(fact_3307_Inf_OINF__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,Inf: fun(set(A),A),G: fun(B,A),F: fun(C,B),A5: set(C)] : aa(set(A),A,Inf,aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A5))) = aa(set(A),A,Inf,aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A5)) ).

% Inf.INF_image
tff(fact_3308_Sup_OSUP__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,Sup: fun(set(A),A),G: fun(B,A),F: fun(C,B),A5: set(C)] : aa(set(A),A,Sup,aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A5))) = aa(set(A),A,Sup,aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A5)) ).

% Sup.SUP_image
tff(fact_3309_card__UNION,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite2(set(A)),A5))
     => ( ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),X4),A5))
           => pp(aa(set(A),bool,finite_finite2(A),X4)) )
       => ( 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_mz(set(set(A)),int)),aa(fun(set(set(A)),bool),set(set(set(A))),collect(set(set(A))),aTP_Lamp_na(set(set(A)),fun(set(set(A)),bool),A5)))) ) ) ) ).

% card_UNION
tff(fact_3310_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A3: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),bit_se2584673776208193580ke_bit(A,M,aa(A,A,bit_se4197421643247451524op_bit(A,N),A3))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_3311_abs__Gcd__eq,axiom,
    ! [K5: set(int)] : aa(int,int,abs_abs(int),gcd_Gcd(int,K5)) = gcd_Gcd(int,K5) ).

% abs_Gcd_eq
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_nb(A,fun(fun(A,B),B),X)),A5)) ) ).

% Sup_apply
tff(fact_3313_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_nc(A,fun(fun(A,B),B),X)),A5)) ) ).

% Inf_apply
tff(fact_3314_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_3315_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))))
    <=> ? [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_iff
tff(fact_3316_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_3317_INT__I,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B2: B,B4: fun(A,set(B))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B4,X4))) )
     => 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_3318_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))))
    <=> ! [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))) ) ) ).

% INT_iff
tff(fact_3319_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_nd(A,A)),A5)) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ).

% SUP_identity_eq
tff(fact_3320_SUP__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [F: 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),F),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_ne(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X)),A5)) ) ).

% SUP_apply
tff(fact_3321_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A5: set(product_prod(A,B)),F: 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),F,A3),B2)),aa(set(product_prod(A,B)),set(C),image2(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),A5))) ) ).

% pair_imageI
tff(fact_3322_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_nf(A,A)),A5)) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ).

% INF_identity_eq
tff(fact_3323_INF__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [F: 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),F),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_ng(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X)),A5)) ) ).

% INF_apply
tff(fact_3324_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_nh(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_nh(set(A),fun(B,set(A)),C2)),A5)) = C2 ) ) ) ).

% UN_constant
tff(fact_3325_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))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,member(A,A4),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A4))) )
       => 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_3326_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,member(A,X2),I5))
          & pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A5,X2))) )
     => 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_3327_Gcd__int__eq,axiom,
    ! [N5: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N5)) ).

% Gcd_int_eq
tff(fact_3328_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_3329_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_3330_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_ni(B,A)),A5)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_3331_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) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => ( aa(B,A,B4,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_3332_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)) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => ( aa(B,A,B4,X3) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_3333_SUP__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_nj(A,fun(B,A),F)),A5)) = F ) ) ) ).

% SUP_const
tff(fact_3334_INF__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_nj(A,fun(B,A),F)),A5)) = F ) ) ) ).

% INF_const
tff(fact_3335_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C6: set(B),A3: A,B4: fun(B,set(A))] :
      ( ( ( C6 = 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_nk(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C6)) = bot_bot(set(A)) ) )
      & ( ( C6 != 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_nk(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C6)) = aa(set(A),set(A),insert2(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C6))) ) ) ) ).

% UN_simps(1)
tff(fact_3336_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_nl(A,set(A))),A5)) = A5 ).

% UN_singleton
tff(fact_3337_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),insert2(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_3338_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_nm(fun(B,set(A)),fun(B,set(A)),B4)),A5)) ).

% Compl_UN
tff(fact_3339_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_nm(fun(B,set(A)),fun(B,set(A)),B4)),A5)) ).

% Compl_INT
tff(fact_3340_Gcd__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_nn(int,nat)),K5)) = aa(int,nat,nat2,gcd_Gcd(int,K5)) ).

% Gcd_nat_abs_eq
tff(fact_3341_bit_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.compl_zero
tff(fact_3342_bit_Ocompl__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% bit.compl_one
tff(fact_3343_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_3344_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_3345_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_3346_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_3347_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_3348_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_3349_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5)) = bot_bot(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X3))
             => ? [Xa2: B] :
                  ( pp(aa(set(B),bool,member(B,Xa2),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,Xa2)),X3)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_3350_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_3351_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% not_one_eq
tff(fact_3352_Int__Inter__eq_I2_J,axiom,
    ! [A: $tType,B10: set(set(A)),A5: set(A)] :
      ( ( ( B10 = 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)),B10)),A5) = A5 ) )
      & ( ( B10 != 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)),B10)),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_no(set(A),fun(set(A),set(A)),A5)),B10)) ) ) ) ).

% Int_Inter_eq(2)
tff(fact_3353_Int__Inter__eq_I1_J,axiom,
    ! [A: $tType,B10: set(set(A)),A5: set(A)] :
      ( ( ( B10 = 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)),B10)) = A5 ) )
      & ( ( B10 != 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)),B10)) = 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)),B10)) ) ) ) ).

% Int_Inter_eq(1)
tff(fact_3354_INT__extend__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C6: set(D),A5: set(C),B4: fun(D,set(C))] :
      ( ( ( C6 = 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),C6))) = A5 ) )
      & ( ( C6 != 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),C6))) = 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_np(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C6)) ) ) ) ).

% INT_extend_simps(2)
tff(fact_3355_INT__extend__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C6: set(A),A5: fun(A,set(B)),B4: set(B)] :
      ( ( ( C6 = 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),C6))),B4) = B4 ) )
      & ( ( C6 != 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),C6))),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_nq(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C6)) ) ) ) ).

% INT_extend_simps(1)
tff(fact_3356_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)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
         => ( aa(B,set(A),B4,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_3357_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)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
         => ( aa(B,set(A),B4,X3) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_3358_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_3359_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_nr(B,set(A))),A5)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_3360_INT__extend__simps_I4_J,axiom,
    ! [G2: $tType,H6: $tType,C6: set(H6),A5: set(G2),B4: fun(H6,set(G2))] :
      ( ( ( C6 = bot_bot(set(H6)) )
       => ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A5),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),B4),C6))) = A5 ) )
      & ( ( C6 != bot_bot(set(H6)) )
       => ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A5),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),B4),C6))) = aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),aa(fun(H6,set(G2)),fun(H6,set(G2)),aTP_Lamp_ns(set(G2),fun(fun(H6,set(G2)),fun(H6,set(G2))),A5),B4)),C6)) ) ) ) ).

% INT_extend_simps(4)
tff(fact_3361_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C6: set(B),A3: A,B4: fun(B,set(A))] :
      ( ( ( C6 = bot_bot(set(B)) )
       => ( aa(set(A),set(A),insert2(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C6))) = aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))) ) )
      & ( ( C6 != bot_bot(set(B)) )
       => ( aa(set(A),set(A),insert2(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C6))) = 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_nk(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B4)),C6)) ) ) ) ).

% UN_extend_simps(1)
tff(fact_3362_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_nt(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B4)),A5)) = aa(set(B),set(B),insert2(B,A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))) ) ) ).

% INT_insert_distrib
tff(fact_3363_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_nt(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B4)),A5)) = aa(set(B),set(B),insert2(B,A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B4),A5))) ) ) ).

% UN_insert_distrib
tff(fact_3364_INT__extend__simps_I5_J,axiom,
    ! [I6: $tType,J4: $tType,A3: I6,B4: fun(J4,set(I6)),C6: set(J4)] : aa(set(I6),set(I6),insert2(I6,A3),aa(set(set(I6)),set(I6),complete_Inf_Inf(set(I6)),aa(set(J4),set(set(I6)),image2(J4,set(I6),B4),C6))) = 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_nu(I6,fun(fun(J4,set(I6)),fun(J4,set(I6))),A3),B4)),C6)) ).

% INT_extend_simps(5)
tff(fact_3365_UN__extend__simps_I6_J,axiom,
    ! [L5: $tType,K6: $tType,A5: fun(K6,set(L5)),C6: 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),C6))),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_nv(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),A5),B4)),C6)) ).

% UN_extend_simps(6)
tff(fact_3366_UN__extend__simps_I7_J,axiom,
    ! [M8: $tType,N8: $tType,A5: set(M8),B4: fun(N8,set(M8)),C6: set(N8)] : aa(set(M8),set(M8),aa(set(M8),fun(set(M8),set(M8)),minus_minus(set(M8)),A5),aa(set(set(M8)),set(M8),complete_Inf_Inf(set(M8)),aa(set(N8),set(set(M8)),image2(N8,set(M8),B4),C6))) = aa(set(set(M8)),set(M8),complete_Sup_Sup(set(M8)),aa(set(N8),set(set(M8)),image2(N8,set(M8),aa(fun(N8,set(M8)),fun(N8,set(M8)),aTP_Lamp_nw(set(M8),fun(fun(N8,set(M8)),fun(N8,set(M8))),A5),B4)),C6)) ).

% UN_extend_simps(7)
tff(fact_3367_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_nx(fun(B,A),fun(B,A),B4)),A5)) ) ).

% uminus_INF
tff(fact_3368_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_nx(fun(B,A),fun(B,A),B4)),A5)) ) ).

% uminus_SUP
tff(fact_3369_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))))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),aa(B,set(A),A5,X3))) ) ) ).

% INT_subset_iff
tff(fact_3370_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))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A5,X3)),B4)) ) ) ).

% UN_subset_iff
tff(fact_3371_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( 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(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,X4)),aa(A,set(B),G,X4))) )
       => 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),F),B4))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),A5)))) ) ) ).

% INT_anti_mono
tff(fact_3372_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C6: set(B),B4: fun(A,set(B))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),aa(A,set(B),B4,X4))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),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_3373_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_3374_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_3375_UN__least,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: fun(A,set(B)),C6: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X4)),C6)) )
     => 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))),C6)) ) ).

% UN_least
tff(fact_3376_UN__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
      ( 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(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,X4)),aa(A,set(B),G,X4))) )
       => 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),F),A5))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),B4)))) ) ) ).

% UN_mono
tff(fact_3377_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),G: 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),F),aa(C,set(B),G,X)))),X6))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G,X)),aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ny(fun(B,set(A)),fun(set(A),fun(B,bool)),F),X6)))) ) ).

% UN_image_subset
tff(fact_3378_UN__extend__simps_I10_J,axiom,
    ! [V6: $tType,U3: $tType,T: $tType,B4: fun(U3,set(V6)),F: fun(T,U3),A5: set(T)] : aa(set(set(V6)),set(V6),complete_Sup_Sup(set(V6)),aa(set(T),set(set(V6)),image2(T,set(V6),aa(fun(T,U3),fun(T,set(V6)),aTP_Lamp_nz(fun(U3,set(V6)),fun(fun(T,U3),fun(T,set(V6))),B4),F)),A5)) = aa(set(set(V6)),set(V6),complete_Sup_Sup(set(V6)),aa(set(U3),set(set(V6)),image2(U3,set(V6),B4),aa(set(T),set(U3),image2(T,U3,F),A5))) ).

% UN_extend_simps(10)
tff(fact_3379_UN__extend__simps_I9_J,axiom,
    ! [S6: $tType,R6: $tType,Q6: $tType,C6: fun(R6,set(S6)),B4: fun(Q6,set(R6)),A5: set(Q6)] : aa(set(set(S6)),set(S6),complete_Sup_Sup(set(S6)),aa(set(Q6),set(set(S6)),image2(Q6,set(S6),aa(fun(Q6,set(R6)),fun(Q6,set(S6)),aTP_Lamp_oa(fun(R6,set(S6)),fun(fun(Q6,set(R6)),fun(Q6,set(S6))),C6),B4)),A5)) = aa(set(set(S6)),set(S6),complete_Sup_Sup(set(S6)),aa(set(R6),set(set(S6)),image2(R6,set(S6),C6),aa(set(set(R6)),set(R6),complete_Sup_Sup(set(R6)),aa(set(Q6),set(set(R6)),image2(Q6,set(R6),B4),A5)))) ).

% UN_extend_simps(9)
tff(fact_3380_UN__extend__simps_I8_J,axiom,
    ! [P3: $tType,O: $tType,B4: fun(O,set(P3)),A5: set(set(O))] : aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(set(O)),set(set(P3)),image2(set(O),set(P3),aTP_Lamp_ob(fun(O,set(P3)),fun(set(O),set(P3)),B4)),A5)) = aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(O),set(set(P3)),image2(O,set(P3),B4),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A5))) ).

% UN_extend_simps(8)
tff(fact_3381_INT__extend__simps_I10_J,axiom,
    ! [V6: $tType,U3: $tType,T: $tType,B4: fun(U3,set(V6)),F: fun(T,U3),A5: set(T)] : aa(set(set(V6)),set(V6),complete_Inf_Inf(set(V6)),aa(set(T),set(set(V6)),image2(T,set(V6),aa(fun(T,U3),fun(T,set(V6)),aTP_Lamp_nz(fun(U3,set(V6)),fun(fun(T,U3),fun(T,set(V6))),B4),F)),A5)) = aa(set(set(V6)),set(V6),complete_Inf_Inf(set(V6)),aa(set(U3),set(set(V6)),image2(U3,set(V6),B4),aa(set(T),set(U3),image2(T,U3,F),A5))) ).

% INT_extend_simps(10)
tff(fact_3382_INT__extend__simps_I9_J,axiom,
    ! [S6: $tType,R6: $tType,Q6: $tType,C6: fun(R6,set(S6)),B4: fun(Q6,set(R6)),A5: set(Q6)] : aa(set(set(S6)),set(S6),complete_Inf_Inf(set(S6)),aa(set(Q6),set(set(S6)),image2(Q6,set(S6),aa(fun(Q6,set(R6)),fun(Q6,set(S6)),aTP_Lamp_oc(fun(R6,set(S6)),fun(fun(Q6,set(R6)),fun(Q6,set(S6))),C6),B4)),A5)) = aa(set(set(S6)),set(S6),complete_Inf_Inf(set(S6)),aa(set(R6),set(set(S6)),image2(R6,set(S6),C6),aa(set(set(R6)),set(R6),complete_Sup_Sup(set(R6)),aa(set(Q6),set(set(R6)),image2(Q6,set(R6),B4),A5)))) ).

% INT_extend_simps(9)
tff(fact_3383_INT__extend__simps_I8_J,axiom,
    ! [P3: $tType,O: $tType,B4: fun(O,set(P3)),A5: set(set(O))] : aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(set(O)),set(set(P3)),image2(set(O),set(P3),aTP_Lamp_od(fun(O,set(P3)),fun(set(O),set(P3)),B4)),A5)) = aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(O),set(set(P3)),image2(O,set(P3),B4),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A5))) ).

% INT_extend_simps(8)
tff(fact_3384_INF__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: 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_oe(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),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_og(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A5)),B4)) ) ).

% INF_commute
tff(fact_3385_SUP__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: 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_oh(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),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_oi(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A5)),B4)) ) ).

% SUP_commute
tff(fact_3386_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))))
     => ~ ! [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_E
tff(fact_3387_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_3388_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_3389_image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),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_oj(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F),B4)),A5)) ).

% image_UN
tff(fact_3390_SUP__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),G: fun(C,set(B)),A5: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),G),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_ok(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F),G)),A5)) ) ).

% SUP_UNION
tff(fact_3391_Inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [A5: set(fun(A,B)),X2: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A5),X2) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_nc(A,fun(fun(A,B),B),X2)),A5)) ) ).

% Inf_fun_def
tff(fact_3392_Sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [A5: set(fun(A,B)),X2: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A5),X2) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),image2(fun(A,B),B,aTP_Lamp_nb(A,fun(fun(A,B),B),X2)),A5)) ) ).

% Sup_fun_def
tff(fact_3393_image__Union,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),S2: set(set(B))] : aa(set(B),set(A),image2(B,A,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F)),S2)) ).

% image_Union
tff(fact_3394_UN__UN__flatten,axiom,
    ! [B: $tType,A: $tType,C: $tType,C6: 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),C6),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_ol(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C6),B4)),A5)) ).

% UN_UN_flatten
tff(fact_3395_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_on(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_3396_Int__Inter__image,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),C6: 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_oo(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),C6)) = 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),C6))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C6))) ).

% Int_Inter_image
tff(fact_3397_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_oo(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_3398_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_op(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A5)),I5)) ).

% Int_UN_distrib
tff(fact_3399_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_no(set(A),fun(set(A),set(A)),A5)),B4)) ).

% Int_Union2
tff(fact_3400_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_3401_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_3402_UN__extend__simps_I4_J,axiom,
    ! [H6: $tType,G2: $tType,A5: fun(G2,set(H6)),C6: set(G2),B4: set(H6)] : aa(set(H6),set(H6),aa(set(H6),fun(set(H6),set(H6)),inf_inf(set(H6)),aa(set(set(H6)),set(H6),complete_Sup_Sup(set(H6)),aa(set(G2),set(set(H6)),image2(G2,set(H6),A5),C6))),B4) = aa(set(set(H6)),set(H6),complete_Sup_Sup(set(H6)),aa(set(G2),set(set(H6)),image2(G2,set(H6),aa(set(H6),fun(G2,set(H6)),aTP_Lamp_oq(fun(G2,set(H6)),fun(set(H6),fun(G2,set(H6))),A5),B4)),C6)) ).

% UN_extend_simps(4)
tff(fact_3403_UN__extend__simps_I5_J,axiom,
    ! [I6: $tType,J4: $tType,A5: set(I6),B4: fun(J4,set(I6)),C6: 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),C6))) = 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_or(set(I6),fun(fun(J4,set(I6)),fun(J4,set(I6))),A5),B4)),C6)) ).

% UN_extend_simps(5)
tff(fact_3404_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)),aa(fun(B,set(A)),fun(fun(C,B),fun(C,set(A))),comp(B,set(A),C),aa(fun(B,set(set(A))),fun(B,set(A)),aa(fun(set(set(A)),set(A)),fun(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)),aa(fun(set(set(A)),set(A)),fun(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))),aa(fun(B,set(set(A))),fun(fun(C,B),fun(C,set(set(A)))),comp(B,set(set(A)),C),Gset),Gmap)),A5))) ) ).

% in_Union_o_assoc
tff(fact_3405_INF__image,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Inf(A)
     => ! [G: fun(B,A),F: fun(C,B),A5: set(C)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A5))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A5)) ) ).

% INF_image
tff(fact_3406_SUP__image,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( complete_Sup(A)
     => ! [G: fun(B,A),F: fun(C,B),A5: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),aa(set(C),set(B),image2(C,B,F),A5))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A5)) ) ).

% SUP_image
tff(fact_3407_Gcd__dvd__int,axiom,
    ! [A3: int,A5: set(int)] :
      ( pp(aa(set(int),bool,member(int,A3),A5))
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),gcd_Gcd(int,A5)),A3)) ) ).

% Gcd_dvd_int
tff(fact_3408_Gcd__greatest__int,axiom,
    ! [A5: set(int),A3: int] :
      ( ! [B3: int] :
          ( pp(aa(set(int),bool,member(int,B3),A5))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),B3)) )
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),gcd_Gcd(int,A5))) ) ).

% Gcd_greatest_int
tff(fact_3409_Union__natural,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),aa(fun(set(set(B)),set(B)),fun(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B))),comp(set(set(B)),set(B),set(set(A))),complete_Sup_Sup(set(B))),image2(set(A),set(B),image2(A,B,F))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),aa(fun(set(A),set(B)),fun(fun(set(set(A)),set(A)),fun(set(set(A)),set(B))),comp(set(A),set(B),set(set(A))),image2(A,B,F)),complete_Sup_Sup(set(A))) ).

% Union_natural
tff(fact_3410_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),X: A,F: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F,I2))) )
         => ( ! [Y4: 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),Y4),aa(B,A,F,I3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5)) = X ) ) ) ) ).

% INF_eqI
tff(fact_3411_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A5: set(C),F: fun(C,A),G: fun(B,A)] :
          ( ! [M4: B] :
              ( pp(aa(set(B),bool,member(B,M4),B4))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,member(C,X2),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X2)),aa(B,A,G,M4))) ) )
         => 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,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B4)))) ) ) ).

% INF_mono
tff(fact_3412_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A5: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I),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,F),A5))),aa(B,A,F,I))) ) ) ).

% INF_lower
tff(fact_3413_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
          ( ! [X4: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),aa(B,A,G,X4)))
         => 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,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),A5)))) ) ) ).

% INF_mono'
tff(fact_3414_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A5: set(B),F: fun(B,A),U: A] :
          ( 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,F,I)),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,F),A5))),U)) ) ) ) ).

% INF_lower2
tff(fact_3415_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F: 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,F),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),U),aa(B,A,F,X3))) ) ) ) ).

% le_INF_iff
tff(fact_3416_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),U: A,F: fun(B,A)] :
          ( ! [I2: B] :
              ( 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,F,I2))) )
         => 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,F),A5)))) ) ) ).

% INF_greatest
tff(fact_3417_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: fun(B,A),X: A] :
          ( ! [I2: B] :
              ( 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,F,I2)),X)) )
         => ( ! [Y4: 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,F,I3)),Y4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_3418_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(C),F: fun(B,A),G: fun(C,A)] :
          ( ! [N2: B] :
              ( pp(aa(set(B),bool,member(B,N2),A5))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,member(C,X2),B4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N2)),aa(C,A,G,X2))) ) )
         => 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,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),B4)))) ) ) ).

% SUP_mono
tff(fact_3419_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: fun(B,A),U: A] :
          ( ! [I2: B] :
              ( 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,F,I2)),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,F),A5))),U)) ) ) ).

% SUP_least
tff(fact_3420_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
          ( ! [X4: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),aa(B,A,G,X4)))
         => 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,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A5)))) ) ) ).

% SUP_mono'
tff(fact_3421_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A5: set(B),F: fun(B,A)] :
          ( 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,F,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5)))) ) ) ).

% SUP_upper
tff(fact_3422_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: 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,F),A5))),U))
        <=> ! [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,F,X3)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_3423_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I: B,A5: set(B),U: A,F: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,I),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,I)))
           => 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,F),A5)))) ) ) ) ).

% SUP_upper2
tff(fact_3424_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: 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,F),A5))),A3))
        <=> ? [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,F,X3)),A3)) ) ) ) ).

% INF_less_iff
tff(fact_3425_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y3: A,F: fun(B,A),A5: set(B),I: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5))))
         => ( pp(aa(set(B),bool,member(B,I),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),aa(B,A,F,I))) ) ) ) ).

% less_INF_D
tff(fact_3426_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F: 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,F),A5))))
        <=> ? [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X3))) ) ) ) ).

% less_SUP_iff
tff(fact_3427_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A5: set(B),Y3: A,I: 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,F),A5))),Y3))
         => ( pp(aa(set(B),bool,member(B,I),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y3)) ) ) ) ).

% SUP_lessD
tff(fact_3428_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_3429_INF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A5: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),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_os(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5)) ) ).

% INF_inf_distrib
tff(fact_3430_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image2(int,int,aTP_Lamp_ot(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_3431_UN__le__add__shift,axiom,
    ! [A: $tType,M2: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_ou(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M2),K)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift
tff(fact_3432_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M2: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_ou(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M2),K)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift_strict
tff(fact_3433_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(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,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).

% not_diff_distrib
tff(fact_3434_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B2) ) ).

% not_add_distrib
tff(fact_3435_sum_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),A5: fun(B,set(C)),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A5,X4))) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X4 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A5,X4)),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),G),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_ov(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A5),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_3436_prod_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),A5: fun(B,set(C)),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),I5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A5,X4))) )
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),I5))
                 => ! [Xa4: B] :
                      ( pp(aa(set(B),bool,member(B,Xa4),I5))
                     => ( ( X4 != Xa4 )
                       => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A5,X4)),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),G),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_ow(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A5),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_3437_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_3438_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_ox(fun(A,set(B)),fun(A,nat),A5)),I5))) ) ).

% card_UN_le
tff(fact_3439_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B4: set(B),A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),A5))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),B4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),aa(B,A,G,X4))) )
           => 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,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),B4)))) ) ) ) ).

% INF_superset_mono
tff(fact_3440_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: set(B),F: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4))
         => ( ! [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,F,X4)),aa(B,A,G,X4))) )
           => 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,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),B4)))) ) ) ) ).

% SUP_subset_mono
tff(fact_3441_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_nj(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_nj(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% SUP_constant
tff(fact_3442_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_3443_INF__inf__const2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F: 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_oy(fun(B,A),fun(A,fun(B,A)),F),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,F),I5))),X) ) ) ) ).

% INF_inf_const2
tff(fact_3444_INF__inf__const1,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),X: A,F: 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_oz(A,fun(fun(B,A),fun(B,A)),X),F)),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,F),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_3445_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: B,A5: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),insert2(B,A3),A5))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5))) ) ).

% INF_insert
tff(fact_3446_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F: 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_pa(fun(B,A),fun(B,set(A)),F)),A5)) = aa(set(B),set(A),image2(B,A,F),A5) ).

% UNION_singleton_eq_range
tff(fact_3447_SUP__inf__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A5: set(B),G: 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,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,G),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_pc(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B4)),A5)) ) ).

% SUP_inf_distrib2
tff(fact_3448_inf__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F: 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,F),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_pd(A,fun(fun(B,A),fun(B,A)),A3),F)),B4)) ) ).

% inf_SUP
tff(fact_3449_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_pe(A,fun(A,A),A3)),B4)) ) ).

% Sup_inf
tff(fact_3450_SUP__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: 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,F),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_pf(fun(B,A),fun(A,fun(B,A)),F),A3)),B4)) ) ).

% SUP_inf
tff(fact_3451_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_3452_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),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))),K5))) ).

% Gcd_int_def
tff(fact_3453_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))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),I5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),A5,X4))) )
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),I5))
             => ! [Xa4: A] :
                  ( pp(aa(set(A),bool,member(A,Xa4),I5))
                 => ( ( X4 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X4)),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_ox(fun(A,set(B)),fun(A,nat),A5)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_3454_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_3455_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_3456_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).

% not_eq_complement
tff(fact_3457_sum_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B4: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),B4))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [A12: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),A12),B4))
               => ! [A23: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),A23),B4))
                   => ( ( A12 != A23 )
                     => ! [X4: B] :
                          ( pp(aa(set(B),bool,member(B,X4),A12))
                         => ( pp(aa(set(B),bool,member(B,X4),A23))
                           => ( aa(B,A,G,X4) = zero_zero(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),G),B4) ) ) ) ) ).

% sum.Union_comp
tff(fact_3458_prod_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B4: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),B4))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [A12: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),A12),B4))
               => ! [A23: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),A23),B4))
                   => ( ( A12 != A23 )
                     => ! [X4: B] :
                          ( pp(aa(set(B),bool,member(B,X4),A12))
                         => ( pp(aa(set(B),bool,member(B,X4),A23))
                           => ( aa(B,A,G,X4) = one_one(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),G),B4) ) ) ) ) ).

% prod.Union_comp
tff(fact_3459_sum_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C6: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [X4: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),Xa4),C6))
                   => ( ( X4 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C6)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),G),C6) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_3460_prod_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C6: set(set(B)),G: fun(B,A)] :
          ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
             => pp(aa(set(B),bool,finite_finite2(B),X4)) )
         => ( ! [X4: set(B)] :
                ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
               => ! [Xa4: set(B)] :
                    ( pp(aa(set(set(B)),bool,member(set(B),Xa4),C6))
                   => ( ( X4 != Xa4 )
                     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),Xa4) = bot_bot(set(B)) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C6)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),G),C6) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_3461_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A5: set(product_prod(A,B)),Y7: 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) )
     => ( ( Y7 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A5) )
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),X6))
             => ! [Xa4: B] :
                  ( pp(aa(set(B),bool,member(B,Xa4),Y7))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X4),Xa4))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q,X4),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_3462_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( bit_se2584673776208193580ke_bit(A,M,aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_3463_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_3464_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : bit_se2638667681897837118et_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_3465_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_ak(A,A)),A5)) = aa(set(A),A,Sup,A5) ).

% Sup.SUP_identity_eq
tff(fact_3466_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_ak(A,A)),A5)) = aa(set(A),A,Inf,A5) ).

% Inf.INF_identity_eq
tff(fact_3467_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y3) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y3) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y3 ) ) ) ) ).

% bit.compl_unique
tff(fact_3468_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_3469_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(option(num),int,aa(fun(num,int),fun(option(num),int),aa(int,fun(fun(num,int),fun(option(num),int)),case_option(int,num),zero_zero(int)),numeral_numeral(int)),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_3470_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),N))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_3471_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_3472_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_3473_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_pg(A,fun(B,A),C2)),A5)) = C2 ) ) ) ).

% cINF_const
tff(fact_3474_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_pg(A,fun(B,A),C2)),A5)) = C2 ) ) ) ).

% cSUP_const
tff(fact_3475_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea3799213064322606851m_diff(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_3476_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))))),Y7: 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_ph(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),A5),Y7))),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_pk(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),A5),Y7)),X6)) ).

% UN_UN_split_split_eq
tff(fact_3477_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F: fun(nat,set(A)),S2: set(A)] :
      ( ! [I2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F,I2)),S2))
     => ( pp(aa(set(A),bool,finite_finite2(A),S2))
       => ( ? [N9: nat] :
              ( ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N9))
                 => ! [M4: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N9))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N2))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F,M4)),aa(nat,set(A),F,N2))) ) ) )
              & ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N2))
                 => ( aa(nat,set(A),F,N9) = aa(nat,set(A),F,N2) ) ) )
         => ( aa(nat,set(A),F,aa(set(A),nat,finite_card(A),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_3478_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_3479_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_3480_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_3481_Collect__const,axiom,
    ! [A: $tType,P: bool] :
      ( ( pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_pl(bool,fun(A,bool),P)) = top_top(set(A)) ) )
      & ( ~ pp(P)
       => ( aa(fun(A,bool),set(A),collect(A),aTP_Lamp_pl(bool,fun(A,bool),P)) = bot_bot(set(A)) ) ) ) ).

% Collect_const
tff(fact_3482_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_lv(fun(A,bool),fun(A,bool),P))))
      <=> pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_Collect_not
tff(fact_3483_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_3484_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_3485_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image2(A,A,aTP_Lamp_mk(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_3486_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_pm(B,A)),A5)) = top_top(A) ) ).

% INF_top
tff(fact_3487_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) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => ( aa(B,A,B4,X3) = top_top(A) ) ) ) ) ).

% INF_top_conv(1)
tff(fact_3488_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)) )
        <=> ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),A5))
             => ( aa(B,A,B4,X3) = top_top(A) ) ) ) ) ).

% INF_top_conv(2)
tff(fact_3489_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5)) = top_top(A) )
        <=> ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),top_top(A)))
             => ? [Xa2: B] :
                  ( pp(aa(set(B),bool,member(B,Xa2),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),aa(B,A,F,Xa2))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_3490_range__constant,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_cu(A,fun(B,A)),X)),top_top(set(B))) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).

% range_constant
tff(fact_3491_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_nh(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_nh(set(A),fun(B,set(A)),C2)),A5)) = C2 ) ) ) ).

% INT_constant
tff(fact_3492_INT__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,C6: set(A),A5: fun(A,set(B)),B4: set(B)] :
      ( ( ( C6 = 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_nq(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C6)) = top_top(set(B)) ) )
      & ( ( C6 != 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_nq(fun(A,set(B)),fun(set(B),fun(A,set(B))),A5),B4)),C6)) = 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),C6))),B4) ) ) ) ).

% INT_simps(1)
tff(fact_3493_INT__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C6: set(D),A5: set(C),B4: fun(D,set(C))] :
      ( ( ( C6 = 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_np(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C6)) = top_top(set(C)) ) )
      & ( ( C6 != 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_np(set(C),fun(fun(D,set(C)),fun(D,set(C))),A5),B4)),C6)) = 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),C6))) ) ) ) ).

% INT_simps(2)
tff(fact_3494_INT__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C6: set(E),A5: fun(E,set(F2)),B4: set(F2)] :
      ( ( ( C6 = bot_bot(set(E)) )
       => ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_pn(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A5),B4)),C6)) = top_top(set(F2)) ) )
      & ( ( C6 != bot_bot(set(E)) )
       => ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_pn(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A5),B4)),C6)) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A5),C6))),B4) ) ) ) ).

% INT_simps(3)
tff(fact_3495_INT__simps_I4_J,axiom,
    ! [G2: $tType,H6: $tType,C6: set(H6),A5: set(G2),B4: fun(H6,set(G2))] :
      ( ( ( C6 = bot_bot(set(H6)) )
       => ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),aa(fun(H6,set(G2)),fun(H6,set(G2)),aTP_Lamp_ns(set(G2),fun(fun(H6,set(G2)),fun(H6,set(G2))),A5),B4)),C6)) = top_top(set(G2)) ) )
      & ( ( C6 != bot_bot(set(H6)) )
       => ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),aa(fun(H6,set(G2)),fun(H6,set(G2)),aTP_Lamp_ns(set(G2),fun(fun(H6,set(G2)),fun(H6,set(G2))),A5),B4)),C6)) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A5),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H6),set(set(G2)),image2(H6,set(G2),B4),C6))) ) ) ) ).

% INT_simps(4)
tff(fact_3496_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_3497_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3498_surjD,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y3: A] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ? [X4: B] : Y3 = aa(B,A,F,X4) ) ).

% surjD
tff(fact_3499_surjE,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y3: A] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ~ ! [X4: B] : Y3 != aa(B,A,F,X4) ) ).

% surjE
tff(fact_3500_surjI,axiom,
    ! [B: $tType,A: $tType,G: fun(B,A),F: fun(A,B)] :
      ( ! [X4: A] : aa(B,A,G,aa(A,B,F,X4)) = X4
     => ( aa(set(B),set(A),image2(B,A,G),top_top(set(B))) = top_top(set(A)) ) ) ).

% surjI
tff(fact_3501_surj__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
    <=> ! [Y: A] :
        ? [X3: B] : Y = aa(B,A,F,X3) ) ).

% surj_def
tff(fact_3502_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aq(A,bool)) ).

% UNIV_def
tff(fact_3503_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_3504_bot__unit__def,axiom,
    bot_bot(product_unit) = product_Unity ).

% bot_unit_def
tff(fact_3505_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_3506_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_3507_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_3508_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S2: set(C),X2: 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_po(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S2)),X2),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),X2),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)),R),S2)))) ) ).

% INF_INT_eq2
tff(fact_3509_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X2),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),X2),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S2))) ) ).

% INF_Int_eq2
tff(fact_3510_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S2: set(C),X2: 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_po(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S2)),X2),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),X2),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)),R),S2)))) ) ).

% SUP_UN_eq2
tff(fact_3511_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool)))),S2)),X2),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),X2),Xa3)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S2))) ) ).

% SUP_Sup_eq2
tff(fact_3512_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B)),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X2),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),X2),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))),R4),S2))) ) ).

% inf_Int_eq2
tff(fact_3513_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X2),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),X2),Xa3)),bot_bot(set(product_prod(A,B))))) ) ).

% bot_empty_eq2
tff(fact_3514_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_aa(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),B4))) ).

% inf_set_def
tff(fact_3515_inf__Int__eq,axiom,
    ! [A: $tType,R4: set(A),S2: set(A),X2: 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_aa(set(A),fun(A,bool)),R4)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R4),S2))) ) ).

% inf_Int_eq
tff(fact_3516_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R: fun(B,set(A)),S2: set(B),X2: 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_pp(fun(B,set(A)),fun(B,fun(A,bool)),R)),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R),S2)))) ) ).

% SUP_UN_eq
tff(fact_3517_SUP__Sup__eq,axiom,
    ! [A: $tType,S2: set(set(A)),X2: 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_aa(set(A),fun(A,bool))),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S2))) ) ).

% SUP_Sup_eq
tff(fact_3518_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R: fun(B,set(A)),S2: set(B),X2: 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_pp(fun(B,set(A)),fun(B,fun(A,bool)),R)),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R),S2)))) ) ).

% INF_INT_eq
tff(fact_3519_INF__Int__eq,axiom,
    ! [A: $tType,S2: set(set(A)),X2: 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_aa(set(A),fun(A,bool))),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S2))) ) ).

% INF_Int_eq
tff(fact_3520_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_pq(set(set(A)),fun(A,bool),A5)) ).

% Sup_set_def
tff(fact_3521_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_pr(set(set(A)),fun(A,bool),A5)) ).

% Inf_set_def
tff(fact_3522_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ps(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_3523_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image2(B,A,F),top_top(set(B)))))
     => ~ ! [X4: B] : B2 != aa(B,A,F,X4) ) ).

% rangeE
tff(fact_3524_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)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
         => ( aa(B,set(A),B4,X3) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(1)
tff(fact_3525_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)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
         => ( aa(B,set(A),B4,X3) = top_top(set(A)) ) ) ) ).

% INTER_UNIV_conv(2)
tff(fact_3526_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_3527_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_3528_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_pu(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_pw(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% INF_SUP
tff(fact_3529_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_px(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_py(fun(C,fun(B,A)),fun(fun(B,C),A),P)),top_top(set(fun(B,C))))) ) ).

% SUP_INF
tff(fact_3530_inf__int__def,axiom,
    inf_inf(int) = ord_min(int) ).

% inf_int_def
tff(fact_3531_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(fun(A,B)),bool,finite_finite2(fun(A,B)),top_top(set(fun(A,B)))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => pp(aa(set(A),bool,finite_finite2(A),top_top(set(A)))) ) ) ).

% finite_fun_UNIVD1
tff(fact_3532_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(B,A),X6: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),X6) = top_top(set(A)) )
     => ( ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),X6))
           => ( aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G1),F),X4) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G22),F),X4) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_3533_comp__surj,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(A),set(C),image2(A,C,G),top_top(set(A))) = top_top(set(C)) )
       => ( aa(set(B),set(C),image2(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),F)),top_top(set(B))) = top_top(set(C)) ) ) ) ).

% comp_surj
tff(fact_3534_fun_Oset__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,F: fun(A,B),V: fun(D,A)] : aa(set(D),set(B),image2(D,B,aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)),top_top(set(D))) = aa(set(A),set(B),image2(A,B,F),aa(set(D),set(A),image2(D,A,V),top_top(set(D)))) ).

% fun.set_map
tff(fact_3535_fun_Omap__cong,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(set(D),set(A),image2(D,A,Ya),top_top(set(D)))))
           => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
       => ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).

% fun.map_cong
tff(fact_3536_fun_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),F: fun(A,B),G: fun(A,B)] :
      ( ! [Z4: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(set(D),set(A),image2(D,A,X),top_top(set(D)))))
         => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
     => ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),X) ) ) ).

% fun.map_cong0
tff(fact_3537_fun_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),Xa: fun(D,A),F: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z4: A,Za: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(set(D),set(A),image2(D,A,X),top_top(set(D)))))
         => ( pp(aa(set(A),bool,member(A,Za),aa(set(D),set(A),image2(D,A,Xa),top_top(set(D)))))
           => ( ( aa(A,B,F,Z4) = aa(A,B,Fa,Za) )
             => ( Z4 = Za ) ) ) )
     => ( ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),Fa),Xa) )
       => ( X = Xa ) ) ) ).

% fun.inj_map_strong
tff(fact_3538_surj__id,axiom,
    ! [A: $tType] : aa(set(A),set(A),image2(A,A,id(A)),top_top(set(A))) = top_top(set(A)) ).

% surj_id
tff(fact_3539_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_3540_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X2: 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))),S2),X2),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),X2),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)),S2))))) ) ).

% Inf_INT_eq2
tff(fact_3541_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,bool))),X2: 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))),S2),X2),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),X2),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)),S2))))) ) ).

% Sup_SUP_eq2
tff(fact_3542_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_3543_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F: fun(A,C)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,G),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_pz(fun(B,A),fun(fun(A,C),fun(B,C)),G),F)),top_top(set(B))))) ) ).

% finite_range_imageI
tff(fact_3544_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_3545_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_nj(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_nj(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% INF_constant
tff(fact_3546_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_3547_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),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,F),A5))),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),uminus_uminus(set(B)),A5)))) ) ).

% surj_Compl_image_subset
tff(fact_3548_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_3549_INT__extend__simps_I3_J,axiom,
    ! [F2: $tType,E: $tType,C6: set(E),A5: fun(E,set(F2)),B4: set(F2)] :
      ( ( ( C6 = bot_bot(set(E)) )
       => ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A5),C6))),B4) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),top_top(set(F2))),B4) ) )
      & ( ( C6 != bot_bot(set(E)) )
       => ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),A5),C6))),B4) = aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),image2(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_pn(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A5),B4)),C6)) ) ) ) ).

% INT_extend_simps(3)
tff(fact_3550_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_qa(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_3551_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_3552_UN__finite__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),C6: set(A)] :
      ( ! [N2: 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),N2)))),C6))
     => 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))))),C6)) ) ).

% UN_finite_subset
tff(fact_3553_UN__finite2__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B4: 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),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = 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),N2),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_3554_UN__constant__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,A5: set(A),F: fun(A,set(B)),C2: set(B)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => ( aa(A,set(B),F,X4) = C2 ) )
       => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),A5)) = C2 ) ) ) ).

% UN_constant_eq
tff(fact_3555_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),aa(set(B),set(A),image2(B,A,F),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,F),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_3556_UN__finite2__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B4: fun(nat,set(A)),K: nat] :
      ( ! [N2: 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),N2)))),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),N2),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_3557_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_qb(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_3558_INF__filter__not__bot,axiom,
    ! [I6: $tType,A: $tType,B4: set(I6),F5: fun(I6,filter(A))] :
      ( ! [X7: set(I6)] :
          ( pp(aa(set(I6),bool,aa(set(I6),fun(set(I6),bool),ord_less_eq(set(I6)),X7),B4))
         => ( pp(aa(set(I6),bool,finite_finite2(I6),X7))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image2(I6,filter(A),F5),X7)) != 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),F5),B4)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_3559_measure__function__int,axiom,
    fun_is_measure(int,aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))) ).

% measure_function_int
tff(fact_3560_bit_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea2506097494486148201lgebra(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_3561_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_qc(fun(B,list(A)),fun(B,set(A)),F)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_3562_UN__le__eq__Un0,axiom,
    ! [A: $tType,M2: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M2),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M2,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_3563_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_3564_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_qd(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_qd(bool,fun(A,fun(B,bool)),P))) = bot_bot(set(product_prod(A,B))) ) ) ) ).

% Collect_const_case_prod
tff(fact_3565_range__abs__Nats,axiom,
    aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).

% range_abs_Nats
tff(fact_3566_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_3567_INF1__I,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: fun(A,fun(B,bool)),B2: B] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(B,bool,aa(A,fun(B,bool),B4,X4),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_3568_INF2__I,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: fun(A,fun(B,fun(C,bool))),B2: B,C2: C] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B4,X4),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_3569_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_3570_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,bool),F: fun(B,A),G: fun(B,A),S2: 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_qe(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F),G)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,bool),set(B),collect(B),P)))),aa(set(B),set(A),image2(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_qf(fun(B,bool),fun(B,bool),P))))) ).

% if_image_distrib
tff(fact_3571_UN__Un,axiom,
    ! [A: $tType,B: $tType,M2: 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),M2),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),M2),A5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M2),B4))) ).

% UN_Un
tff(fact_3572_UN__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C6: set(F2),A5: set(E),B4: fun(F2,set(E))] :
      ( ( ( C6 = bot_bot(set(F2)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_qg(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A5),B4)),C6)) = bot_bot(set(E)) ) )
      & ( ( C6 != bot_bot(set(F2)) )
       => ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_qg(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A5),B4)),C6)) = 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(F2),set(set(E)),image2(F2,set(E),B4),C6))) ) ) ) ).

% UN_simps(3)
tff(fact_3573_UN__simps_I2_J,axiom,
    ! [C: $tType,D: $tType,C6: set(C),A5: fun(C,set(D)),B4: set(D)] :
      ( ( ( C6 = 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_qh(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C6)) = bot_bot(set(D)) ) )
      & ( ( C6 != 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_qh(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C6)) = 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),C6))),B4) ) ) ) ).

% UN_simps(2)
tff(fact_3574_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),insert2(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_3575_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F5: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( pp(aa(set(A),bool,member(A,I2),I5))
         => ! [J2: A] :
              ( pp(aa(set(A),bool,member(A,J2),I5))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,member(A,X2),I5))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F5,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F5,I2)),aa(A,filter(B),F5,J2)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F5),I5)) = bot_bot(filter(B)) )
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),I5))
            & ( aa(A,filter(B),F5,X3) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_3576_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M2: 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,M2),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,M2),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M2),B4))) ) ).

% SUP_union
tff(fact_3577_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_3578_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_3579_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_3580_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_3581_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_3582_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))
     => ~ ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),A5))
           => ~ pp(aa(A,bool,aa(B,fun(A,bool),B4,X4),B2)) ) ) ).

% SUP1_E
tff(fact_3583_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))
     => ~ ! [X4: C] :
            ( pp(aa(set(C),bool,member(C,X4),A5))
           => ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B4,X4),B2),C2)) ) ) ).

% SUP2_E
tff(fact_3584_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_qi(set(A),fun(set(A),fun(bool,set(A))),A5),B4)),top_top(set(bool)))) ).

% Un_eq_UN
tff(fact_3585_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_3586_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X2),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),X2),Xa3)),top_top(set(product_prod(A,B))))) ) ).

% top_empty_eq2
tff(fact_3587_top__unit__def,axiom,
    top_top(product_unit) = product_Unity ).

% top_unit_def
tff(fact_3588_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_qj(set(A),fun(set(A),fun(A,bool)),A5),B4)) ).

% Un_def
tff(fact_3589_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_ig(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_3590_UNIV__unit,axiom,
    top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),insert2(product_unit,product_Unity),bot_bot(set(product_unit))) ).

% UNIV_unit
tff(fact_3591_measure__fst,axiom,
    ! [B: $tType,A: $tType,F: fun(A,nat)] :
      ( fun_is_measure(A,F)
     => fun_is_measure(product_prod(A,B),aTP_Lamp_qk(fun(A,nat),fun(product_prod(A,B),nat),F)) ) ).

% measure_fst
tff(fact_3592_measure__snd,axiom,
    ! [B: $tType,A: $tType,F: fun(A,nat)] :
      ( fun_is_measure(A,F)
     => fun_is_measure(product_prod(B,A),aTP_Lamp_ql(fun(A,nat),fun(product_prod(B,A),nat),F)) ) ).

% measure_snd
tff(fact_3593_insert__def,axiom,
    ! [A: $tType,A3: A,B4: set(A)] : aa(set(A),set(A),insert2(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_kn(A,fun(A,bool),A3))),B4) ).

% insert_def
tff(fact_3594_infinite__UNIV__int,axiom,
    ~ pp(aa(set(int),bool,finite_finite2(int),top_top(set(int)))) ).

% infinite_UNIV_int
tff(fact_3595_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_qm(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_3596_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_3597_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A5: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),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_qn(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_3598_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: fun(B,A),A5: set(B),G: 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,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),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_qp(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B4)),A5)) ) ).

% INF_sup_distrib2
tff(fact_3599_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F: 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,F),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_qq(A,fun(fun(B,A),fun(B,A)),A3),F)),B4)) ) ).

% sup_INF
tff(fact_3600_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_qr(A,fun(A,A),A3)),B4)) ) ).

% Inf_sup
tff(fact_3601_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F: 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,F),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_qs(fun(B,A),fun(A,fun(B,A)),F),A3)),B4)) ) ).

% INF_sup
tff(fact_3602_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_3603_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_qt(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_3604_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),B4: fun(B,set(A)),C6: 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_qt(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A5),B4)),C6)) = 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),C6))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B4),C6))) ).

% Un_Union_image
tff(fact_3605_INT__extend__simps_I7_J,axiom,
    ! [M8: $tType,N8: $tType,A5: set(M8),B4: fun(N8,set(M8)),C6: set(N8)] : aa(set(M8),set(M8),aa(set(M8),fun(set(M8),set(M8)),sup_sup(set(M8)),A5),aa(set(set(M8)),set(M8),complete_Inf_Inf(set(M8)),aa(set(N8),set(set(M8)),image2(N8,set(M8),B4),C6))) = aa(set(set(M8)),set(M8),complete_Inf_Inf(set(M8)),aa(set(N8),set(set(M8)),image2(N8,set(M8),aa(fun(N8,set(M8)),fun(N8,set(M8)),aTP_Lamp_qu(set(M8),fun(fun(N8,set(M8)),fun(N8,set(M8))),A5),B4)),C6)) ).

% INT_extend_simps(7)
tff(fact_3606_INT__extend__simps_I6_J,axiom,
    ! [L5: $tType,K6: $tType,A5: fun(K6,set(L5)),C6: 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),C6))),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_qv(fun(K6,set(L5)),fun(set(L5),fun(K6,set(L5))),A5),B4)),C6)) ).

% INT_extend_simps(6)
tff(fact_3607_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_qw(set(A),fun(fun(B,set(A)),fun(B,set(A))),B4),A5)),I5)) ).

% Un_INT_distrib
tff(fact_3608_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_qy(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_3609_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_3610_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_3611_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_3612_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_3613_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_3614_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),A3: B,A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),insert2(B,A3),A5))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5))) ) ).

% SUP_insert
tff(fact_3615_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M2: 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,M2),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,M2),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M2),B4))) ) ).

% INF_union
tff(fact_3616_UN__extend__simps_I3_J,axiom,
    ! [E: $tType,F2: $tType,C6: set(F2),A5: set(E),B4: fun(F2,set(E))] :
      ( ( ( C6 = bot_bot(set(F2)) )
       => ( 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(F2),set(set(E)),image2(F2,set(E),B4),C6))) = A5 ) )
      & ( ( C6 != bot_bot(set(F2)) )
       => ( 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(F2),set(set(E)),image2(F2,set(E),B4),C6))) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),image2(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_qg(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A5),B4)),C6)) ) ) ) ).

% UN_extend_simps(3)
tff(fact_3617_UN__extend__simps_I2_J,axiom,
    ! [D: $tType,C: $tType,C6: set(C),A5: fun(C,set(D)),B4: set(D)] :
      ( ( ( C6 = 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),C6))),B4) = B4 ) )
      & ( ( C6 != 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),C6))),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_qh(fun(C,set(D)),fun(set(D),fun(C,set(D))),A5),B4)),C6)) ) ) ) ).

% UN_extend_simps(2)
tff(fact_3618_INT__Un,axiom,
    ! [A: $tType,B: $tType,M2: 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),M2),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),M2),A5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M2),B4))) ).

% INT_Un
tff(fact_3619_Ints__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_Ints(A) = aa(set(int),set(A),image2(int,A,ring_1_of_int(A)),top_top(set(int))) ) ) ).

% Ints_def
tff(fact_3620_int__in__range__abs,axiom,
    ! [N: nat] : pp(aa(set(int),bool,member(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(set(int),set(int),image2(int,int,abs_abs(int)),top_top(set(int))))) ).

% int_in_range_abs
tff(fact_3621_sum_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G: 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),G),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),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B4)) ) ) ) ) ).

% sum.union_inter
tff(fact_3622_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_3623_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,G,X4) = zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B4)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_3624_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A5: set(B),B4: set(B),F: 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),F),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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ).

% sum_Un
tff(fact_3625_sum_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G: 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),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),B4)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3626_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,G,X4) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),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),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B4)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_3627_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(B),G: 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),G),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),G),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),G),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),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ).

% sum.union_diff2
tff(fact_3628_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B4: set(A),F: 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),F),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),F),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),F),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),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ) ).

% sum_Un2
tff(fact_3629_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_3630_sum__Un__nat,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),F: 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),F),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),F),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),B4))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4))) ) ) ) ).

% sum_Un_nat
tff(fact_3631_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A5: set(B),B4: set(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(B),bool,finite_finite2(B),B4))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,member(B,X4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4)))
                 => ( aa(B,A,F,X4) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),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),F),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),B4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B4))) ) ) ) ) ) ).

% prod_Un
tff(fact_3632_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_qz(A,fun(A,bool)),aTP_Lamp_ra(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_3633_these__insert__Some,axiom,
    ! [A: $tType,X: A,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert2(option(A),aa(A,option(A),some(A),X)),A5)) = aa(set(A),set(A),insert2(A,X),these(A,A5)) ).

% these_insert_Some
tff(fact_3634_Gcd__eq__Max,axiom,
    ! [M2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M2))
     => ( ( M2 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M2))
         => ( gcd_Gcd(nat,M2) = 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_rb(nat,set(nat))),M2))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_3635_these__insert__None,axiom,
    ! [A: $tType,A5: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),insert2(option(A),none(A)),A5)) = these(A,A5) ).

% these_insert_None
tff(fact_3636_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_rc(set(option(A)),fun(option(A),bool),A5)) ).

% Some_image_these_eq
tff(fact_3637_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_3638_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_3639_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ic(nat,fun(nat,bool),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_3640_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_rd(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% Max_const
tff(fact_3641_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_3642_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_3643_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(A,B)),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R4)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),S2)),X2),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),X2),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))),R4),S2))) ) ).

% sup_Un_eq2
tff(fact_3644_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_aa(set(A),fun(A,bool)),A5)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),B4))) ).

% sup_set_def
tff(fact_3645_sup__Un__eq,axiom,
    ! [A: $tType,R4: set(A),S2: set(A),X2: 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_aa(set(A),fun(A,bool)),R4)),aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),S2)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R4),S2))) ) ).

% sup_Un_eq
tff(fact_3646_sup__int__def,axiom,
    sup_sup(int) = ord_max(int) ).

% sup_int_def
tff(fact_3647_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_3648_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_3649_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_3650_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( S2 != 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_re(fun(B,A),fun(A,fun(B,A)),F),K)),S2)) = 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,F),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_3651_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_rf(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_3652_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_rg(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_3653_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_rc(set(option(A)),fun(option(A),bool),A5))) ).

% Option.these_def
tff(fact_3654_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)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_3655_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)),insert2(option(A),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_3656_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_lb(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_3657_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_3658_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_rh(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Max.eq_fold'
tff(fact_3659_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R4: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F)
     => ( 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,G),top_top(set(C)))),S2))
       => finite673082921795544331dem_on(C,B,R4,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_3660_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aTP_Lamp_im(int,fun(int,bool),N))) = aa(int,int,abs_abs(int),N) ) ) ).

% Max_divisors_self_int
tff(fact_3661_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_3662_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B)),X: A] :
      ( finite673082921795544331dem_on(A,B,S2,F)
     => ( pp(aa(set(A),bool,member(A,X),S2))
       => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,X)) = aa(A,fun(B,B),F,X) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_3663_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N) = aa(set(int),int,lattic643756798349783984er_Max(int),aa(fun(int,bool),set(int),collect(int),aa(int,fun(int,bool),aTP_Lamp_ri(int,fun(int,fun(int,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_3664_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),plus_plus(A)),G),zero_zero(A),A5) ) ).

% sum.eq_fold
tff(fact_3665_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),times_times(A)),G),one_one(A),A5) ) ).

% prod.eq_fold
tff(fact_3666_sup__SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: A,F: 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,F),A5))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),B4,A5) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_3667_inf__INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),B4: A,F: 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,F),A5))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),B4,A5) ) ) ) ).

% inf_INF_fold_inf
tff(fact_3668_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_3669_SUP__fold__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: 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,F),A5)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),bot_bot(A),A5) ) ) ) ).

% SUP_fold_sup
tff(fact_3670_INF__fold__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(B),F: 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,F),A5)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),top_top(A),A5) ) ) ) ).

% INF_fold_inf
tff(fact_3671_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] : lattic5882676163264333800up_fin(A,A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_rj(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Sup_fin.eq_fold'
tff(fact_3672_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] : lattic7752659483105999362nf_fin(A,A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_rk(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Inf_fin.eq_fold'
tff(fact_3673_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_rl(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Min.eq_fold'
tff(fact_3674_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_rm(B,fun(A,set(product_prod(B,A))),X)),B4))),A5) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_rn(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X),A5,B4) ) ) ).

% fold_union_pair
tff(fact_3675_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_rd(A,fun(B,A),C2)),A5)) = C2 ) ) ) ) ).

% Min_const
tff(fact_3676_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_3677_card_Oeq__fold,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_ro(A,fun(nat,nat)),zero_zero(nat),A5) ).

% card.eq_fold
tff(fact_3678_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( lattic7752659483105999362nf_fin(A,A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_3679_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( lattic5882676163264333800up_fin(A,A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_3680_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( aa(set(A),set(B),image2(A,B,F),A5) = finite_fold(A,set(B),aTP_Lamp_rp(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A5) ) ) ).

% image_fold_insert
tff(fact_3681_Min__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite2(B),S2))
         => ( ( S2 != 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_re(fun(B,A),fun(A,fun(B,A)),F),K)),S2)) = 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,F),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_3682_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_rq(fun(A,bool),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A5) ) ) ).

% Set_filter_fold
tff(fact_3683_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_rr(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_3684_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_rs(A,set(product_prod(A,A)))),A5)) ).

% Id_on_def
tff(fact_3685_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_lb(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_3686_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_3687_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_rt(fun(A,bool),fun(set(A),fun(A,bool)),P),A5)) ).

% Set.filter_def
tff(fact_3688_Id__on__iff,axiom,
    ! [A: $tType,X: A,Y3: 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),Y3)),id_on(A,A5)))
    <=> ( ( X = Y3 )
        & pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% Id_on_iff
tff(fact_3689_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_3690_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)))
     => ~ ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => ( C2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ) ) ).

% Id_onE
tff(fact_3691_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_ru(fun(A,bool),fun(A,fun(A,bool)),A5))) ).

% Id_on_def'
tff(fact_3692_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_aa(set(A),fun(A,bool)),A5),B4) ) ) ).

% inter_Set_filter
tff(fact_3693_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_rv(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B4)) ) ).

% comp_fun_commute_product_fold
tff(fact_3694_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F: 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_rw(fun(A,set(B)),fun(A,filter(B)),F)),X6)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X6))) ) ) ).

% INF_principal_finite
tff(fact_3695_image__o__collect,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F5: 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)),aa(fun(set(C),set(B)),fun(fun(A,set(C)),fun(A,set(B))),comp(set(C),set(B),A),image2(C,B,G))),F5)) = aa(fun(A,set(C)),fun(A,set(B)),aa(fun(set(C),set(B)),fun(fun(A,set(C)),fun(A,set(B))),comp(set(C),set(B),A),image2(C,B,G)),bNF_collect(A,C,F5)) ).

% image_o_collect
tff(fact_3696_total__on__singleton,axiom,
    ! [A: $tType,X: A] : total_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_3697_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_rx(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_3698_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_ry(fun(B,B),fun(A,fun(B,B)),F)) ).

% comp_fun_commute_const
tff(fact_3699_total__onI,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ( pp(aa(set(A),bool,member(A,Y4),A5))
           => ( ( X4 != 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),X4),Y4)),R))
                | pp(aa(set(product_prod(A,A)),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),X4)),R)) ) ) ) )
     => total_on(A,A5,R) ) ).

% total_onI
tff(fact_3700_total__on__def,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( total_on(A,A5,R)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ! [Xa2: A] :
              ( pp(aa(set(A),bool,member(A,Xa2),A5))
             => ( ( X3 != Xa2 )
               => ( pp(aa(set(product_prod(A,A)),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),Xa2)),R))
                  | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X3)),R)) ) ) ) ) ) ).

% total_on_def
tff(fact_3701_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F)
    <=> ! [Y: A,X3: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y)) ) ).

% comp_fun_commute_def
tff(fact_3702_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Y3: A,X: A] :
      ( finite6289374366891150609ommute(A,B,F)
     => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y3)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y3)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_3703_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
      ( ! [Y4: A,X4: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4))
     => finite6289374366891150609ommute(A,B,F) ) ).

% comp_fun_commute.intro
tff(fact_3704_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(B,B)),G: fun(C,A)] :
      ( finite6289374366891150609ommute(A,B,F)
     => finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ).

% comp_fun_commute.comp_comp_fun_commute
tff(fact_3705_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,bool)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_rq(fun(A,bool),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_3706_collect__def,axiom,
    ! [A: $tType,B: $tType,F5: set(fun(B,set(A))),X: B] : aa(B,set(A),bNF_collect(B,A,F5),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_rz(B,fun(fun(B,set(A)),set(A)),X)),F5)) ).

% collect_def
tff(fact_3707_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S2))
     => 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_sb(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S2))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_3708_collect__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F5: set(fun(C,set(B))),G: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),bNF_collect(C,B,F5)),G) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),image2(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_sc(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G)),F5)) ).

% collect_comp
tff(fact_3709_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_sd(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_3710_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_se(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C2))) ) ).

% at_bot_sub
tff(fact_3711_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: 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)),S2))
     => ( 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)),insert2(product_prod(C,A),X),bot_bot(set(product_prod(C,A)))),S2)),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_sf(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),X6,S2) ) ) ).

% insert_relcomp_union_fold
tff(fact_3712_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_3713_relcomp_Ocases,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: 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,R,S)))
     => ~ ! [B3: 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),B3)),R))
           => ~ 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),B3),A22)),S)) ) ) ).

% relcomp.cases
tff(fact_3714_relcomp_Osimps,axiom,
    ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: 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,R,S)))
    <=> ? [A7: A,B7: B,C4: C] :
          ( ( A1 = A7 )
          & ( A22 = C4 )
          & 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),A7),B7)),R))
          & 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),B7),C4)),S)) ) ) ).

% relcomp.simps
tff(fact_3715_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(A,B)),C2: C,S: 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)),R))
     => ( 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)),S))
       => 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,R,S))) ) ) ).

% relcomp.relcompI
tff(fact_3716_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Xz),relcomp(A,C,B,R,S)))
     => ~ ! [X4: A,Y4: C,Z4: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Z4) )
           => ( 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),X4),Y4)),R))
             => ~ 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),Y4),Z4)),S)) ) ) ) ).

% relcompE
tff(fact_3717_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set(product_prod(A,C)),S: 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,R,S)))
     => ~ ! [B3: 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),B3)),R))
           => ~ 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),B3),C2)),S)) ) ) ).

% relcompEpair
tff(fact_3718_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C)),S: B] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X4)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X4)),H) ) )
     => ( aa(B,C,H,aa(B,B,fold(A,B,G,Xs),S)) = aa(C,C,fold(A,C,F,Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_3719_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X4)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X4)),H) ) )
     => ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),fold(A,B,G,Xs)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),fold(A,C,F,Xs)),H) ) ) ).

% fold_commute
tff(fact_3720_comp__fun__commute__Image__fold,axiom,
    ! [B: $tType,A: $tType,S2: 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_sg(set(A),fun(A,fun(B,fun(set(B),set(B)))),S2))) ).

% comp_fun_commute_Image_fold
tff(fact_3721_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R: fun(D,set(product_prod(A,C))),I5: set(D),S: 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)),R),I5)),S) = 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_sh(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R),S)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_3722_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S: set(product_prod(A,C)),R: fun(D,set(product_prod(C,B))),I5: set(D)] : relcomp(A,C,B,S,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)),R),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_si(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S),R)),I5)) ).

% relcomp_UNION_distrib
tff(fact_3723_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R4))
     => ( pp(aa(set(product_prod(B,C)),bool,finite_finite2(product_prod(B,C)),S2))
       => ( relcomp(A,B,C,R4,S2) = 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_sk(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S2)),bot_bot(set(product_prod(A,C))),R4) ) ) ) ).

% relcomp_fold
tff(fact_3724_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_gcd(nat),Xs),zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_3725_Gcd__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Gcd(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_gcd(int),Xs),zero_zero(int)) ).

% Gcd_int_set_eq_fold
tff(fact_3726_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_sl(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_3727_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_gcd(A),Xs),zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_3728_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),X: product_prod(C,A),R4: set(product_prod(C,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),S2))
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert2(product_prod(C,A),X),R4),S2) = 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_sf(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X)),relcomp(C,A,B,R4,S2),S2) ) ) ).

% insert_relcomp_fold
tff(fact_3729_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),aa(list(B),set(B),set2(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_3730_min__ext__compat,axiom,
    ! [A: $tType,R4: 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))),relcomp(A,A,A,R4,S2)),R4))
     => 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,R4),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,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R4))) ) ).

% min_ext_compat
tff(fact_3731_max__ext__compat,axiom,
    ! [A: $tType,R4: 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))),relcomp(A,A,A,R4,S2)),R4))
     => 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,R4),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,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert2(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R4))) ) ).

% max_ext_compat
tff(fact_3732_snd__sndOp,axiom,
    ! [C: $tType,A: $tType,B: $tType,Bc: product_prod(B,A),P: fun(B,fun(C,bool)),Q: fun(C,fun(A,bool))] : aa(product_prod(B,A),A,product_snd(B,A),Bc) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(C,A)),fun(product_prod(B,A),A),aa(fun(product_prod(C,A),A),fun(fun(product_prod(B,A),product_prod(C,A)),fun(product_prod(B,A),A)),comp(product_prod(C,A),A,product_prod(B,A)),product_snd(C,A)),bNF_sndOp(B,C,A,P,Q)),Bc) ).

% snd_sndOp
tff(fact_3733_fst__fstOp,axiom,
    ! [C: $tType,B: $tType,A: $tType,Bc: product_prod(A,B),P: fun(A,fun(C,bool)),Q: fun(C,fun(B,bool))] : aa(product_prod(A,B),A,product_fst(A,B),Bc) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),aa(fun(product_prod(A,C),A),fun(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)),bNF_fstOp(A,C,B,P,Q)),Bc) ).

% fst_fstOp
tff(fact_3734_max__ext__additive,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),R4: set(product_prod(A,A)),C6: set(A),D5: 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,R4)))
     => ( 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)),C6),D5)),max_ext(A,R4)))
       => 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),C6)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),D5))),max_ext(A,R4))) ) ) ).

% max_ext_additive
tff(fact_3735_max__ext_Ocases,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R4: 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,R4)))
     => ~ ( pp(aa(set(A),bool,finite_finite2(A),A1))
         => ( pp(aa(set(A),bool,finite_finite2(A),A22))
           => ( ( A22 != bot_bot(set(A)) )
             => ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,member(A,X2),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),X2),Xa4)),R4)) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_3736_max__ext_Osimps,axiom,
    ! [A: $tType,A1: set(A),A22: set(A),R4: 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,R4)))
    <=> ( pp(aa(set(A),bool,finite_finite2(A),A1))
        & pp(aa(set(A),bool,finite_finite2(A),A22))
        & ( A22 != bot_bot(set(A)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A1))
           => ? [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),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),X3),Xa2)),R4)) ) ) ) ) ).

% max_ext.simps
tff(fact_3737_max__ext_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y7: set(A),R4: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y7))
       => ( ( Y7 != bot_bot(set(A)) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),X6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),Y7))
                    & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R4)) ) )
           => 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),Y7)),max_ext(A,R4))) ) ) ) ) ).

% max_ext.max_extI
tff(fact_3738_sndOp__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(C,fun(A,bool)),Q: fun(A,fun(B,bool)),Ac: product_prod(C,B)] : aa(product_prod(C,B),product_prod(A,B),bNF_sndOp(C,A,B,P,Q),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P,Q,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).

% sndOp_def
tff(fact_3739_fstOp__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(B,fun(C,bool)),Ac: product_prod(A,C)] : aa(product_prod(A,C),product_prod(A,B),bNF_fstOp(A,B,C,P,Q),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P,Q,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).

% fstOp_def
tff(fact_3740_max__ext__def,axiom,
    ! [A: $tType,X2: set(product_prod(A,A))] : max_ext(A,X2) = 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_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),X2)))) ).

% max_ext_def
tff(fact_3741_Pow__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pow2(A,A5) = finite_fold(A,set(set(A)),aTP_Lamp_sl(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),insert2(set(A),bot_bot(set(A))),bot_bot(set(set(A)))),A5) ) ) ).

% Pow_fold
tff(fact_3742_Cantors__paradox,axiom,
    ! [A: $tType,A5: set(A)] :
      ~ ? [F7: fun(A,set(A))] : aa(set(A),set(set(A)),image2(A,set(A),F7),A5) = pow2(A,A5) ).

% Cantors_paradox
tff(fact_3743_Pow__def,axiom,
    ! [A: $tType,A5: set(A)] : pow2(A,A5) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_if(set(A),fun(set(A),bool),A5)) ).

% Pow_def
tff(fact_3744_Pow__INT__eq,axiom,
    ! [A: $tType,B: $tType,B4: fun(B,set(A)),A5: set(B)] : pow2(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),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_sn(fun(B,set(A)),fun(B,set(set(A))),B4)),A5)) ).

% Pow_INT_eq
tff(fact_3745_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_sn(fun(B,set(A)),fun(B,set(set(A))),B4)),A5))),pow2(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_3746_binomial__def,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(N),K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),bool),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),bool),aTP_Lamp_so(nat,fun(nat,fun(set(nat),bool)),N),K))) ).

% binomial_def
tff(fact_3747_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),X2: set(A),Xa3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R4)),X2),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)),X2),Xa3)),max_ext(A,R4))) ) ).

% max_extp_max_ext_eq
tff(fact_3748_max__extp__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X: set(A),Y3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),X),Y3))
    <=> 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),Y3)),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),R))))) ) ).

% max_extp_eq
tff(fact_3749_Field__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)) ).

% Field_insert
tff(fact_3750_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_3751_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : pow2(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))),pow2(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image2(set(B),set(B),insert2(B,X)),pow2(B,aa(list(B),set(B),set2(B),Xs)))) ).

% Pow_set(2)
tff(fact_3752_FieldI1,axiom,
    ! [A: $tType,I: A,J: A,R4: 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),I),J)),R4))
     => pp(aa(set(A),bool,member(A,I),field2(A,R4))) ) ).

% FieldI1
tff(fact_3753_FieldI2,axiom,
    ! [A: $tType,I: A,J: A,R4: 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),I),J)),R4))
     => pp(aa(set(A),bool,member(A,J),field2(A,R4))) ) ).

% FieldI2
tff(fact_3754_fold__Cons,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X: A,Xs: list(A)] : fold(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,Xs)),aa(A,fun(B,B),F,X)) ).

% fold_Cons
tff(fact_3755_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_3756_Field__natLeq__on,axiom,
    ! [N: nat] : field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N)))) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N)) ).

% Field_natLeq_on
tff(fact_3757_mlex__eq,axiom,
    ! [A: $tType,F: fun(A,nat),R4: set(product_prod(A,A))] : mlex_prod(A,F,R4) = 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_sq(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F),R4))) ).

% mlex_eq
tff(fact_3758_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,X),aa(list(A),list(A),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)),aa(list(A),nat,size_list(A,X),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_3759_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))] :
      ~ ! [F3: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: int,I2: int] : X != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod(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)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).

% full_exhaustive_int'.cases
tff(fact_3760_exhaustive__int_H_Ocases,axiom,
    ! [X: product_prod(fun(int,option(product_prod(bool,list(code_term)))),product_prod(int,int))] :
      ~ ! [F3: fun(int,option(product_prod(bool,list(code_term)))),D2: int,I2: 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)),F3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2)) ).

% exhaustive_int'.cases
tff(fact_3761_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))] :
      ~ ! [F3: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod(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)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I2)) ).

% full_exhaustive_integer'.cases
tff(fact_3762_exhaustive__integer_H_Ocases,axiom,
    ! [X: product_prod(fun(code_integer,option(product_prod(bool,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F3: fun(code_integer,option(product_prod(bool,list(code_term)))),D2: code_integer,I2: code_integer] : X != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod(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)),F3),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),D2),I2)) ).

% exhaustive_integer'.cases
tff(fact_3763_mlex__less,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y3: A,R4: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,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),X),Y3)),mlex_prod(A,F,R4))) ) ).

% mlex_less
tff(fact_3764_mlex__iff,axiom,
    ! [A: $tType,X: A,Y3: A,F: fun(A,nat),R4: 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),Y3)),mlex_prod(A,F,R4)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y3)))
        | ( ( aa(A,nat,F,X) = aa(A,nat,F,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),X),Y3)),R4)) ) ) ) ).

% mlex_iff
tff(fact_3765_mlex__leq,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y3: A,R4: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,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),X),Y3)),R4))
       => pp(aa(set(product_prod(A,A)),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)),mlex_prod(A,F,R4))) ) ) ).

% mlex_leq
tff(fact_3766_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R: set(product_prod(A,A)),As: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R,As)
        <=> ! [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)),R))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As,I4)),aa(A,B,As,J3))) ) ) ) ).

% relChain_def
tff(fact_3767_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_3768_cofinal__def,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( bNF_Ca7293521722713021262ofinal(A,A5,R)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),field2(A,R)))
         => ? [Xa2: A] :
              ( pp(aa(set(A),bool,member(A,Xa2),A5))
              & ( X3 != Xa2 )
              & pp(aa(set(product_prod(A,A)),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),Xa2)),R)) ) ) ) ).

% cofinal_def
tff(fact_3769_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A))))
      <=> ? [N6: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N6))
            & ( aa(nat,A,semiring_1_of_nat(A),N6) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_3770_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_3771_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_3772_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),N)) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_3773_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X4: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X4) = zero_zero(A) )
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),C2),X4)) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_3774_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N6: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N6))
           => ( aa(nat,A,semiring_1_of_nat(A),N6) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_3775_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X4: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X4))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),C2))
                   => ( aa(nat,A,semiring_1_of_nat(A),X4) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_3776_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N))
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% bit_minus_2_iff
tff(fact_3777_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))))
            & ( N != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).

% bit_double_iff
tff(fact_3778_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),N))
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% bit_mask_sub_iff
tff(fact_3779_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),bit0(one2))),N))
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_3780_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),one_one(A))),N))
        <=> bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% bit_minus_1_iff
tff(fact_3781_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_3782_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,N)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_3783_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),A3)),N))
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))),N)) ) ) ) ).

% bit_minus_iff
tff(fact_3784_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% fold_possible_bit
tff(fact_3785_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),aa(nat,fun(B,product_prod(nat,B)),product_Pair(nat,B),N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_3786_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = finite_fold(A,A,gcd_gcd(A),zero_zero(A),A5) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ) ).

% Gcd_fin.eq_fold
tff(fact_3787_fun_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,R4: 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),R4),A3),B2))
    <=> ? [Z5: fun(D,product_prod(A,B))] :
          ( pp(aa(set(fun(D,product_prod(A,B))),bool,member(fun(D,product_prod(A,B)),Z5),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_sr(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4))))
          & ( aa(fun(D,product_prod(A,B)),fun(D,A),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),comp(product_prod(A,B),A,D),product_fst(A,B)),Z5) = A3 )
          & ( aa(fun(D,product_prod(A,B)),fun(D,B),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)),Z5) = B2 ) ) ) ).

% fun.in_rel
tff(fact_3788_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),bit0(one2))),K))
     => order_mono(nat,nat,aTP_Lamp_ss(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_3789_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_3790_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_3791_is__unit__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),one_one(A)))
        <=> ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = one_one(A) ) ) ) ).

% is_unit_Gcd_fin_iff
tff(fact_3792_Gcd__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) ) ).

% Gcd_fin.insert
tff(fact_3793_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = gcd_Gcd(A,A5) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_3794_power__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & power(A) )
     => ! [R4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,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),R4,bNF_rel_fun(A,B,A,B,R4,R4)),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),R4,bNF_rel_fun(nat,nat,A,B,fequal(nat),R4)),power_power(A)),power_power(B))) ) ) ) ).

% power_transfer
tff(fact_3795_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,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),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(num,B),bool,aa(fun(num,A),fun(fun(num,B),bool),bNF_rel_fun(num,num,A,B,fequal(num),R4),numeral_numeral(A)),numeral_numeral(B))) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_3796_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_ly(num,fun(num,int))),aTP_Lamp_ly(num,fun(num,int)))) ).

% sub.rsp
tff(fact_3797_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,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),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
             => ( pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,R4,R4),uminus_uminus(A)),uminus_uminus(B)))
               => pp(aa(fun(int,B),bool,aa(fun(int,A),fun(fun(int,B),bool),bNF_rel_fun(int,int,A,B,fequal(int),R4),ring_1_of_int(A)),ring_1_of_int(B))) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_3798_id__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(fun(B,B),bool,aa(fun(A,A),fun(fun(B,B),bool),bNF_rel_fun(A,B,A,B,A5,A5),id(A)),id(B))) ).

% id_transfer
tff(fact_3799_rel__fun__def__butlast,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,F2: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(C,fun(E,bool)),T4: fun(D,fun(F2,bool)),F: fun(A,fun(C,D)),G: fun(B,fun(E,F2))] :
      ( pp(aa(fun(B,fun(E,F2)),bool,aa(fun(A,fun(C,D)),fun(fun(B,fun(E,F2)),bool),bNF_rel_fun(A,B,fun(C,D),fun(E,F2),R4,bNF_rel_fun(C,E,D,F2,S2,T4)),F),G))
    <=> ! [X3: A,Y: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,X3),Y))
         => pp(aa(fun(E,F2),bool,aa(fun(C,D),fun(fun(E,F2),bool),bNF_rel_fun(C,E,D,F2,S2,T4),aa(A,fun(C,D),F,X3)),aa(B,fun(E,F2),G,Y))) ) ) ).

% rel_fun_def_butlast
tff(fact_3800_If__transfer,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(fun(bool,fun(B,fun(B,B))),bool,aa(fun(bool,fun(A,fun(A,A))),fun(fun(bool,fun(B,fun(B,B))),bool),bNF_rel_fun(bool,bool,fun(A,fun(A,A)),fun(B,fun(B,B)),fequal(bool),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A5,bNF_rel_fun(A,B,A,B,A5,A5))),if(A)),if(B))) ).

% If_transfer
tff(fact_3801_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_3802_gcd__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),gcd_gcd(int)),gcd_gcd(int))) ).

% gcd_integer.rsp
tff(fact_3803_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : order_mono(A,A,aa(A,fun(A,A),plus_plus(A),A3)) ) ).

% mono_add
tff(fact_3804_plus__natural_Orsp,axiom,
    pp(aa(fun(nat,fun(nat,nat)),bool,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),bool),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),plus_plus(nat)),plus_plus(nat))) ).

% plus_natural.rsp
tff(fact_3805_comp__transfer,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,B4: fun(A,fun(C,bool)),C6: fun(B,fun(D,bool)),A5: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F2,C),fun(F2,D)),bNF_rel_fun(A,C,B,D,B4,C6),bNF_rel_fun(fun(E,A),fun(F2,C),fun(E,B),fun(F2,D),bNF_rel_fun(E,F2,A,C,A5,B4),bNF_rel_fun(E,F2,B,D,A5,C6))),comp(A,B,E)),comp(C,D,F2))) ).

% comp_transfer
tff(fact_3806_fun_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,G2: $tType,F2: $tType,Rb2: fun(A,fun(F2,bool)),Sd: fun(B,fun(G2,bool))] : pp(aa(fun(fun(F2,G2),fun(fun(D,F2),fun(D,G2))),bool,aa(fun(fun(A,B),fun(fun(D,A),fun(D,B))),fun(fun(fun(F2,G2),fun(fun(D,F2),fun(D,G2))),bool),bNF_rel_fun(fun(A,B),fun(F2,G2),fun(fun(D,A),fun(D,B)),fun(fun(D,F2),fun(D,G2)),bNF_rel_fun(A,F2,B,G2,Rb2,Sd),bNF_rel_fun(fun(D,A),fun(D,F2),fun(D,B),fun(D,G2),bNF_rel_fun(D,D,A,F2,fequal(D),Rb2),bNF_rel_fun(D,D,B,G2,fequal(D),Sd))),comp(A,B,D)),comp(F2,G2,D))) ).

% fun.map_transfer
tff(fact_3807_fun_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: fun(A,fun(C,bool)),X: fun(D,A),G: fun(B,C),Y3: 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),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),Y3)))
    <=> 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_st(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y3)) ) ).

% fun.rel_map(2)
tff(fact_3808_fun_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: fun(D,A),Y3: 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),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),I),X)),Y3))
    <=> 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_su(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y3)) ) ).

% fun.rel_map(1)
tff(fact_3809_Gcd__fin__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),A3)) ) ) ).

% Gcd_fin_dvd
tff(fact_3810_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_kz(int,int)),aTP_Lamp_kz(int,int))) ).

% dup.rsp
tff(fact_3811_Gcd__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) ) ) ) ).

% Gcd_fin.in_idem
tff(fact_3812_mono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A)] :
          ( order_mono(nat,A,F)
        <=> ! [N6: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,N6)),aa(nat,A,F,aa(nat,nat,suc,N6)))) ) ) ).

% mono_iff_le_Suc
tff(fact_3813_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R12: fun(A,fun(B,bool)),R23: 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),R12,bNF_rel_fun(C,D,bool,bool,R23,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),R12))))
           => ( 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),R23))))
             => ( 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_3814_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),X3)) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_3815_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5))) ) ) ) ).

% Gcd_fin_greatest
tff(fact_3816_dvd__gcd__list__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,Xs: list(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(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),dvd_dvd(A),B2),X3)) ) ) ) ).

% dvd_gcd_list_iff
tff(fact_3817_gcd__list__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A3: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,member(A,B3),aa(list(A),set(A),set2(A),Bs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Bs)))) ) ) ).

% gcd_list_greatest
tff(fact_3818_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),aa(set(A),A,semiring_gcd_Gcd_fin(A),B4)),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) ) ) ) ).

% Gcd_fin.subset
tff(fact_3819_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B4: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),aa(set(A),A,semiring_gcd_Gcd_fin(A),B4)) ) ).

% Gcd_fin.union
tff(fact_3820_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => order_mono(A,A,aa(A,fun(A,A),times_times(A),A3)) ) ) ).

% mono_mult
tff(fact_3821_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N)) ) ).

% mono_times_nat
tff(fact_3822_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F)
         => 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_sv(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5)),I5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I5))))) ) ) ).

% mono_SUP
tff(fact_3823_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F)
         => 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,F),A5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A5)))) ) ) ).

% mono_Sup
tff(fact_3824_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,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_sv(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5)),I5)))) ) ) ).

% mono_INF
tff(fact_3825_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,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,F),A5)))) ) ) ).

% mono_Inf
tff(fact_3826_num__of__integer_Orsp,axiom,
    pp(aa(fun(int,num),bool,aa(fun(int,num),fun(fun(int,num),bool),bNF_rel_fun(int,int,num,num,fequal(int),fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2))) ).

% num_of_integer.rsp
tff(fact_3827_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)),F: fun(A,C),G: 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),F),G))
     => ( 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),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)),X)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),aa(fun(B,D),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),D)),comp(B,D,product_prod(A,B)),G),product_snd(A,B)),X))) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_3828_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_3829_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_3830_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_gcd(A),Xs),zero_zero(A)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_3831_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),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),insert2(A,zero_zero(A)),bot_bot(set(A)))))
            & pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_3832_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,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),R4,bNF_rel_fun(A,B,A,B,R4,R4)),plus_plus(A)),plus_plus(B)))
             => pp(aa(fun(nat,B),bool,aa(fun(nat,A),fun(fun(nat,B),bool),bNF_rel_fun(nat,nat,A,B,fequal(nat),R4),semiring_1_of_nat(A)),semiring_1_of_nat(B))) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_3833_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R4: fun(A,fun(B,bool))] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,zero_zero(A)),zero_zero(B)))
         => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,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),R4),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B))) ) ) ) ).

% transfer_rule_of_bool
tff(fact_3834_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_sw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),plus_plus(rat))) ).

% plus_rat.transfer
tff(fact_3835_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_sx(product_prod(int,int),product_prod(int,int))),inverse_inverse(rat))) ).

% inverse_rat.transfer
tff(fact_3836_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_3837_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_3838_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_sy(product_prod(int,int),product_prod(int,int))),uminus_uminus(rat))) ).

% uminus_rat.transfer
tff(fact_3839_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_sz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),times_times(rat))) ).

% times_rat.transfer
tff(fact_3840_o__rsp_I2_J,axiom,
    ! [E: $tType,F2: $tType,H6: $tType,G2: $tType,R12: fun(E,fun(F2,bool))] : pp(aa(fun(fun(G2,H6),fun(fun(F2,G2),fun(F2,H6))),bool,aa(fun(fun(G2,H6),fun(fun(E,G2),fun(E,H6))),fun(fun(fun(G2,H6),fun(fun(F2,G2),fun(F2,H6))),bool),bNF_rel_fun(fun(G2,H6),fun(G2,H6),fun(fun(E,G2),fun(E,H6)),fun(fun(F2,G2),fun(F2,H6)),fequal(fun(G2,H6)),bNF_rel_fun(fun(E,G2),fun(F2,G2),fun(E,H6),fun(F2,H6),bNF_rel_fun(E,F2,G2,G2,R12,fequal(G2)),bNF_rel_fun(E,F2,H6,H6,R12,fequal(H6)))),comp(G2,H6,E)),comp(G2,H6,F2))) ).

% o_rsp(2)
tff(fact_3841_o__rsp_I1_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,R23: fun(A,fun(C,bool)),R32: fun(B,fun(D,bool)),R12: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F2,C),fun(F2,D)),bNF_rel_fun(A,C,B,D,R23,R32),bNF_rel_fun(fun(E,A),fun(F2,C),fun(E,B),fun(F2,D),bNF_rel_fun(E,F2,A,C,R12,R23),bNF_rel_fun(E,F2,B,D,R12,R32))),comp(A,B,E)),comp(C,D,F2))) ).

% o_rsp(1)
tff(fact_3842_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_ta(product_prod(int,int),bool)),positive)) ).

% positive.transfer
tff(fact_3843_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,member(product_prod(nat,A),P2),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3844_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_3845_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_3846_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_3847_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: 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,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3848_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),case_nat(A),X),nth(A,Xs)),N) ).

% nth_Cons
tff(fact_3849_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_3850_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: 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),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_3851_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y3: A,Xs: list(A),N: nat] :
      ( ( X != Y3 )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y3 )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) = Y3 )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3852_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_3853_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_ta(product_prod(int,int),bool)) ).

% positive_def
tff(fact_3854_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F6: fun(nat,nat)] :
          ( order_mono(nat,nat,F6)
          & ( aa(set(nat),set(nat),image2(nat,nat,F6),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
          & ! [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,Ys),aa(nat,nat,F6,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,F6,I4) = aa(nat,nat,F6,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_3855_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_jl(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).

% times_int.transfer
tff(fact_3856_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_3857_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_tb(nat,product_prod(nat,nat))),semiring_1_of_nat(int))) ).

% int_transfer
tff(fact_3858_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_jv(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).

% uminus_int.transfer
tff(fact_3859_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_3860_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_3861_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_jw(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).

% of_int.transfer
tff(fact_3862_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_jy(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).

% less_int.transfer
tff(fact_3863_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_ka(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).

% less_eq_int.transfer
tff(fact_3864_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_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).

% plus_int.transfer
tff(fact_3865_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_ki(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).

% minus_int.transfer
tff(fact_3866_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_sw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% plus_rat_def
tff(fact_3867_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_sx(product_prod(int,int),product_prod(int,int))) ).

% inverse_rat_def
tff(fact_3868_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_sz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))) ).

% times_rat_def
tff(fact_3869_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_sy(product_prod(int,int),product_prod(int,int))) ).

% uminus_rat_def
tff(fact_3870_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_3871_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_3872_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_tc(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_3873_plus__rat_Oabs__eq,axiom,
    ! [Xa: 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,Xa),Xa))
     => ( 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,Xa)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),X)),aa(product_prod(int,int),int,product_snd(int,int),Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_3874_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,aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),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_3875_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_td(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3876_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( zero_zero(A) = aa(rat,A,field_char_0_of_rat(A),A3) )
        <=> ( zero_zero(rat) = A3 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_3877_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_3878_of__rat__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),zero_zero(rat)) = zero_zero(A) ) ) ).

% of_rat_0
tff(fact_3879_one__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( one_one(A) = aa(rat,A,field_char_0_of_rat(A),A3) )
        <=> ( one_one(rat) = A3 ) ) ) ).

% one_eq_of_rat_iff
tff(fact_3880_of__rat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( aa(rat,A,field_char_0_of_rat(A),A3) = one_one(A) )
        <=> ( A3 = one_one(rat) ) ) ) ).

% of_rat_eq_1_iff
tff(fact_3881_of__rat__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),one_one(rat)) = one_one(A) ) ) ).

% of_rat_1
tff(fact_3882_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A3: A,X: B,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs))) ) ).

% horner_sum_simps(2)
tff(fact_3883_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R)) ) ) ).

% zero_less_of_rat_iff
tff(fact_3884_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_3885_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R)) ) ) ).

% zero_le_of_rat_iff
tff(fact_3886_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_3887_of__rat__neg__one,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( aa(rat,A,field_char_0_of_rat(A),aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% of_rat_neg_one
tff(fact_3888_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: 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),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R)) ) ) ).

% one_less_of_rat_iff
tff(fact_3889_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R),one_one(rat))) ) ) ).

% of_rat_less_1_iff
tff(fact_3890_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R)) ) ) ).

% one_le_of_rat_iff
tff(fact_3891_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_3892_of__rat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_te(fun(B,rat),fun(B,A),F)),A5) ) ).

% of_rat_sum
tff(fact_3893_of__rat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [F: 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),F),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_te(fun(B,rat),fun(B,A),F)),A5) ) ).

% of_rat_prod
tff(fact_3894_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_3895_of__rat__add,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),plus_plus(rat),A3),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(rat,A,field_char_0_of_rat(A),A3)),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_add
tff(fact_3896_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_3897_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_tf(int,fun(int,product_prod(int,int)))),aTP_Lamp_tf(int,fun(int,product_prod(int,int))))) ).

% Fract.rsp
tff(fact_3898_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_tc(product_prod(int,int),A)),aTP_Lamp_tc(product_prod(int,int),A))) ) ).

% of_rat.rsp
tff(fact_3899_ratrel__def,axiom,
    ! [X2: 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,X2),Xa3))
    <=> ( ( aa(product_prod(int,int),int,product_snd(int,int),X2) != 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),X2)),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),X2)) ) ) ) ).

% ratrel_def
tff(fact_3900_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_sz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_sz(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% times_rat.rsp
tff(fact_3901_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_sy(product_prod(int,int),product_prod(int,int))),aTP_Lamp_sy(product_prod(int,int),product_prod(int,int)))) ).

% uminus_rat.rsp
tff(fact_3902_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_ta(product_prod(int,int),bool)),aTP_Lamp_ta(product_prod(int,int),bool))) ).

% positive.rsp
tff(fact_3903_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_sx(product_prod(int,int),product_prod(int,int))),aTP_Lamp_sx(product_prod(int,int),product_prod(int,int)))) ).

% inverse_rat.rsp
tff(fact_3904_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_sw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)))),aTP_Lamp_sw(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))))) ).

% plus_rat.rsp
tff(fact_3905_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_3906_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_tc(product_prod(int,int),A)),field_char_0_of_rat(A))) ) ).

% of_rat.transfer
tff(fact_3907_times__rat_Oabs__eq,axiom,
    ! [Xa: 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,Xa),Xa))
     => ( 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,Xa)),aa(product_prod(int,int),rat,abs_Rat,X)) = aa(product_prod(int,int),rat,abs_Rat,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_fst(int,int),X))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),X)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_3908_cr__rat__def,axiom,
    ! [X2: product_prod(int,int),Xa3: rat] :
      ( cr_rat(X2,Xa3)
    <=> ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X2),X2))
        & ( aa(product_prod(int,int),rat,abs_Rat,X2) = Xa3 ) ) ) ).

% cr_rat_def
tff(fact_3909_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_tg(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3910_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_jl(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_jl(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% times_int.rsp
tff(fact_3911_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_3912_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_3913_funpow__0,axiom,
    ! [A: $tType,F: 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)),F),X) = X ).

% funpow_0
tff(fact_3914_id__funpow,axiom,
    ! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).

% id_funpow
tff(fact_3915_surj__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_3916_intrel__iff,axiom,
    ! [X: nat,Y3: nat,U: nat,V: 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),Y3)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),U),V)))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),V) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),U),Y3) ) ) ).

% intrel_iff
tff(fact_3917_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_3918_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),N: nat] :
          ( order_mono(A,A,F)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ) ) ).

% mono_pow
tff(fact_3919_funpow__mult,axiom,
    ! [A: $tType,N: nat,M: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F)) = 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),N)),F) ).

% funpow_mult
tff(fact_3920_funpow__swap1,axiom,
    ! [A: $tType,F: fun(A,A),N: nat,X: A] : aa(A,A,F,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),aa(A,A,F,X)) ).

% funpow_swap1
tff(fact_3921_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),F)) = aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).

% comp_funpow
tff(fact_3922_int_Oabs__eq__iff,axiom,
    ! [X: product_prod(nat,nat),Y3: product_prod(nat,nat)] :
      ( ( aa(product_prod(nat,nat),int,abs_Integ,X) = aa(product_prod(nat,nat),int,abs_Integ,Y3) )
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X),Y3)) ) ).

% int.abs_eq_iff
tff(fact_3923_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_th(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_3924_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).

% funpow.simps(2)
tff(fact_3925_funpow__Suc__right,axiom,
    ! [A: $tType,N: nat,F: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),F) ).

% funpow_Suc_right
tff(fact_3926_funpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,F: 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),N)),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(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),F)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)) ).

% funpow_add
tff(fact_3927_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F: 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)),F) = id(A) ).

% funpow_simps_right(1)
tff(fact_3928_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(A,A),A5: A,B4: A,N: nat] :
          ( order_mono(A,A,F)
         => ( 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)),N),F),A5)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),B4))) ) ) ) ).

% funpow_mono
tff(fact_3929_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_3930_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F,P2)),P2))
           => 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),F),bot_bot(A))),P2)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_3931_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,F,P2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_3932_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(A,A),I: nat,J: nat,X: A,Y3: A] :
          ( order_mono(A,A,F)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F,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)),I),F),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),F),Y3))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_3933_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_jv(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_jv(nat,fun(nat,product_prod(nat,nat)))))) ).

% uminus_int.rsp
tff(fact_3934_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_3935_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_3936_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_3937_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_3938_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_mono(nat,A,aTP_Lamp_ti(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_3939_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( order_mono(A,A,F)
           => 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),F),bot_bot(A))),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_3940_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( order_mono(A,A,F)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),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),F),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_3941_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_jw(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_jw(nat,fun(nat,A))))) ) ).

% of_int.rsp
tff(fact_3942_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_tk(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).

% intrel_def
tff(fact_3943_numeral__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),K) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% numeral_unfold_funpow
tff(fact_3944_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_jy(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_jy(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_int.rsp
tff(fact_3945_int_Orel__eq__transfer,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))),intrel),fequal(int))) ).

% int.rel_eq_transfer
tff(fact_3946_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_ka(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_ka(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).

% less_eq_int.rsp
tff(fact_3947_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_ki(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_ki(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% minus_int.rsp
tff(fact_3948_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_kc(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_kc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).

% plus_int.rsp
tff(fact_3949_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_3950_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y3))
    <=> ? [F6: fun(nat,A)] :
          ( ( aa(nat,A,F6,zero_zero(nat)) = X )
          & ( aa(nat,A,F6,N) = Y3 )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F6,I4)),aa(nat,A,F6,aa(nat,nat,suc,I4)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3951_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),one_one(nat)),P) = P ).

% relpowp_1
tff(fact_3952_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_3953_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A] : pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),X)) ).

% relpowp_0_I
tff(fact_3954_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),P),X),Y3))
     => ( X = Y3 ) ) ).

% relpowp_0_E
tff(fact_3955_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R4: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),zero_zero(nat)),R4) = fequal(A) ).

% relpowp.simps(1)
tff(fact_3956_relpowp__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( 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))),M4),P),X),Y4))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z)) ) ) ) ) ).

% relpowp_E
tff(fact_3957_relpowp__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
               => ~ 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))),M4),P),Y4),Z)) ) ) ) ) ).

% relpowp_E2
tff(fact_3958_relpow__finite__bounded1,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => ( 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),R4)),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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tm(set(product_prod(A,A)),fun(nat,bool),R4)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_3959_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: 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),Y3)),lexord(A,R)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y3)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y3) = 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)),Y3))))
            & ( take(A,I4,X) = take(A,I4,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),aa(nat,A,nth(A,X),I4)),aa(nat,A,nth(A,Y3),I4))),R)) ) ) ) ).

% lexord_take_index_conv
tff(fact_3960_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)))
     => ~ ! [I2: 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),I2),aa(list(A),nat,size_size(list(A)),Ys)))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ pp(aa(set(product_prod(A,A)),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),I2)),aa(nat,A,nth(A,Ys),I2))),R)) ) ) ) ) ).

% lex_take_index
tff(fact_3961_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_3962_relpow__1,axiom,
    ! [A: $tType,R4: 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))),one_one(nat)),R4) = R4 ).

% relpow_1
tff(fact_3963_lexord__cons__cons,axiom,
    ! [A: $tType,A3: A,X: list(A),B2: A,Y3: list(A),R: 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),Y3))),lexord(A,R)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
        | ( ( A3 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),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),Y3)),lexord(A,R))) ) ) ) ).

% lexord_cons_cons
tff(fact_3964_finite__relpow,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R4))) ) ) ).

% finite_relpow
tff(fact_3965_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y3: A,Ys: list(A),R: 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),Y3),Ys))),lex(A,R)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( X = Y3 )
          & 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),Ys)),lex(A,R))) ) ) ) ).

% Cons_in_lex
tff(fact_3966_relpow__Suc__D2_H,axiom,
    ! [A: $tType,N: nat,R4: set(product_prod(A,A)),X2: 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),X2),Y5)),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),R4)))
        & pp(aa(set(product_prod(A,A)),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)),R4)) )
     => ? [W2: 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),X2),W2)),R4))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W2),Z6)),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),R4))) ) ) ).

% relpow_Suc_D2'
tff(fact_3967_lexord__linear,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: list(A),Y3: list(A)] :
      ( ! [A4: A,B3: 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),A4),B3)),R))
          | ( A4 = B3 )
          | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4)),R)) )
     => ( 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),Y3)),lexord(A,R)))
        | ( X = Y3 )
        | 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)),Y3),X)),lexord(A,R))) ) ) ).

% lexord_linear
tff(fact_3968_lexord__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: 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),X4)),R))
     => ~ 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,R))) ) ).

% lexord_irreflexive
tff(fact_3969_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: 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),Y3)),lex(A,R)))
    <=> ( 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),Y3)),lexord(A,R)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y3) ) ) ) ).

% lexord_lex
tff(fact_3970_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R4: 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),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R4)))
     => ~ ! [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),X),Y4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),R4)))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R4)) ) ) ).

% relpow_Suc_E
tff(fact_3971_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y3: A,N: nat,R4: set(product_prod(A,A)),Z: 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))),N),R4)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R4))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R4))) ) ) ).

% relpow_Suc_I
tff(fact_3972_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R4: 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),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R4)))
     => ? [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),X),Y4)),R4))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),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),R4))) ) ) ).

% relpow_Suc_D2
tff(fact_3973_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R4: 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),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R4)))
     => ~ ! [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),X),Y4)),R4))
           => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),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),R4))) ) ) ).

% relpow_Suc_E2
tff(fact_3974_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y3: A,R4: set(product_prod(A,A)),Z: 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),X),Y3)),R4))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),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),R4)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,suc,N)),R4))) ) ) ).

% relpow_Suc_I2
tff(fact_3975_relpow__0__I,axiom,
    ! [A: $tType,X: A,R4: 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)),R4))) ).

% relpow_0_I
tff(fact_3976_relpow__0__E,axiom,
    ! [A: $tType,X: A,Y3: A,R4: 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),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))),zero_zero(nat)),R4)))
     => ( X = Y3 ) ) ).

% relpow_0_E
tff(fact_3977_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X4: A,Y4: A,Z4: 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),Y4)),R))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R))
             => pp(aa(set(product_prod(A,A)),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),Z4)),R)) ) ) )
     => ( 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),Ys)),lexord(A,R)))
       => ( 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)),Ys),Zs)),lexord(A,R)))
         => 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,R))) ) ) ) ).

% lexord_partial_trans
tff(fact_3978_relpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,R4: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R4) = relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),M),R4),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),R4)) ).

% relpow_add
tff(fact_3979_relpowp__relpow__eq,axiom,
    ! [A: $tType,N: nat,R4: set(product_prod(A,A)),X2: 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))),N),aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R4)),X2),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),X2),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))),N),R4))) ) ).

% relpowp_relpow_eq
tff(fact_3980_relpow__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R4: 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),Z)),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),R4)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( pp(aa(set(product_prod(A,A)),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),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))),M4),R4)))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),R4)) ) ) ) ) ).

% relpow_E
tff(fact_3981_relpow__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R4: 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),Z)),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),R4)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M4: nat] :
              ( ( N = aa(nat,nat,suc,M4) )
             => ( pp(aa(set(product_prod(A,A)),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),Y4)),R4))
               => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),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))),M4),R4))) ) ) ) ) ).

% relpow_E2
tff(fact_3982_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_3983_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(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)) ) ) ).

% find_None_iff
tff(fact_3984_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( none(A) = find(A,P,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)) ) ) ).

% find_None_iff2
tff(fact_3985_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_3986_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B2: A,N: nat,R4: 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))),N),R4)))
    <=> ? [F6: fun(nat,A)] :
          ( ( aa(nat,A,F6,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F6,N) = B2 )
          & ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
             => pp(aa(set(product_prod(A,A)),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,F6,I4)),aa(nat,A,F6,aa(nat,nat,suc,I4)))),R4)) ) ) ) ).

% relpow_fun_conv
tff(fact_3987_relpow__finite__bounded,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => 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),R4)),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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tn(set(product_prod(A,A)),fun(nat,bool),R4)))))) ) ).

% relpow_finite_bounded
tff(fact_3988_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_3989_List_Olexordp__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R,Xs,Ys)
    <=> 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),Ys)),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),R))))) ) ).

% List.lexordp_def
tff(fact_3990_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R4: set(product_prod(A,A))] : transitive_ntrancl(A,N,R4) = 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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_to(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_3991_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => ( transitive_trancl(A,R4) = 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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tm(set(product_prod(A,A)),fun(nat,bool),R4)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_3992_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N)),R)) )
        | ( ( M = N )
          & 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,R))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_3993_ntrancl__Zero,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R4) = R4 ).

% ntrancl_Zero
tff(fact_3994_trancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R: 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,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),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)),R))
       => ~ ! [B3: 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),B3)),transitive_trancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22)),R)) ) ) ) ).

% trancl.cases
tff(fact_3995_trancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R: 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,R)))
    <=> ( ? [A7: A,B7: A] :
            ( ( A1 = A7 )
            & ( A22 = B7 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),R)) )
        | ? [A7: A,B7: A,C4: A] :
            ( ( A1 = A7 )
            & ( A22 = C4 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),transitive_trancl(A,R)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4)),R)) ) ) ) ).

% trancl.simps
tff(fact_3996_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R))) ) ).

% trancl.r_into_trancl
tff(fact_3997_tranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ~ ! [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,R)))
             => ~ pp(aa(set(product_prod(A,A)),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)),R)) ) ) ) ).

% tranclE
tff(fact_3998_trancl__trans,axiom,
    ! [A: $tType,X: A,Y3: A,R: set(product_prod(A,A)),Z: 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)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),transitive_trancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_trancl(A,R))) ) ) ).

% trancl_trans
tff(fact_3999_trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( ! [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),A3),Y4)),R))
           => pp(aa(A,bool,P,Y4)) )
       => ( ! [Y4: A,Z4: 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),Y4)),transitive_trancl(A,R)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R))
               => ( pp(aa(A,bool,P,Y4))
                 => pp(aa(A,bool,P,Z4)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% trancl_induct
tff(fact_4000_r__r__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R4: 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)),R4))
     => ( pp(aa(set(product_prod(A,A)),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)),R4))
       => pp(aa(set(product_prod(A,A)),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,R4))) ) ) ).

% r_r_into_trancl
tff(fact_4001_converse__tranclE,axiom,
    ! [A: $tType,X: A,Z: A,R: 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),Z)),transitive_trancl(A,R)))
     => ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R))
       => ~ ! [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),X),Y4)),R))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),transitive_trancl(A,R))) ) ) ) ).

% converse_tranclE
tff(fact_4002_irrefl__trancl__rD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y3: A] :
      ( ! [X4: 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),X4)),transitive_trancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
       => ( X != Y3 ) ) ) ).

% irrefl_trancl_rD
tff(fact_4003_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(set(product_prod(A,A)),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)),R))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R))) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_4004_trancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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)),R))
     => ( pp(aa(set(product_prod(A,A)),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,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R))) ) ) ).

% trancl_into_trancl2
tff(fact_4005_trancl__trans__induct,axiom,
    ! [A: $tType,X: A,Y3: A,R: 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),Y3)),transitive_trancl(A,R)))
     => ( ! [X4: 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),X4),Y4)),R))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Y4)) )
       => ( ! [X4: A,Y4: A,Z4: 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),Y4)),transitive_trancl(A,R)))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X4),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),Y4),Z4)),transitive_trancl(A,R)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z4))
                   => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Z4)) ) ) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y3)) ) ) ) ).

% trancl_trans_induct
tff(fact_4006_converse__trancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( ! [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),B2)),R))
           => pp(aa(A,bool,P,Y4)) )
       => ( ! [Y4: A,Z4: 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),Z4)),R))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B2)),transitive_trancl(A,R)))
               => ( pp(aa(A,bool,P,Z4))
                 => pp(aa(A,bool,P,Y4)) ) ) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% converse_trancl_induct
tff(fact_4007_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),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),R)))
     => ( ! [A4: A,B3: 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),A4),B3))),R))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3)) )
       => ( ! [A4: A,B3: 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),A4),B3))),transitive_trancl(product_prod(A,B),R)))
             => ( 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),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3))
                 => 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_4008_finite__trancl__ntranl,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => ( transitive_trancl(A,R4) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),R4)),one_one(nat)),R4) ) ) ).

% finite_trancl_ntranl
tff(fact_4009_trancl__set__ntrancl,axiom,
    ! [A: $tType,Xs: list(product_prod(A,A))] : transitive_trancl(A,aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) = transitive_ntrancl(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs))),one_one(nat)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),Xs)) ).

% trancl_set_ntrancl
tff(fact_4010_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P2),transitive_trancl(A,R4)))
    <=> ? [N6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N6))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P2),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))),N6),R4))) ) ) ).

% trancl_power
tff(fact_4011_lenlex__irreflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X4: 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),X4)),R))
     => ~ 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,R))) ) ).

% lenlex_irreflexive
tff(fact_4012_less__eq,axiom,
    ! [M: nat,N: 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),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% less_eq
tff(fact_4013_trancl__insert2,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_tp(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).

% trancl_insert2
tff(fact_4014_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_4015_lenlex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_tq(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lenlex_conv
tff(fact_4016_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite2(product_prod(A,A)),R4))
     => ( transitive_rtrancl(A,R4) = 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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tn(set(product_prod(A,A)),fun(nat,bool),R4)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_4017_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_4018_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_4019_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( order_mono(A,A,Q)
         => order_antimono(nat,A,aTP_Lamp_tr(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_4020_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_4021_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_4022_rotate__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_4023_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y3: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_4024_converse__rtrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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)),R))
     => ( pp(aa(set(product_prod(A,A)),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,R)))
       => pp(aa(set(product_prod(A,A)),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,R))) ) ) ).

% converse_rtrancl_into_rtrancl
tff(fact_4025_converse__rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(A,bool,P,B2))
       => ( ! [Y4: A,Z4: 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),Z4)),R))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),B2)),transitive_rtrancl(A,R)))
               => ( pp(aa(A,bool,P,Z4))
                 => pp(aa(A,bool,P,Y4)) ) ) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% converse_rtrancl_induct
tff(fact_4026_converse__rtranclE,axiom,
    ! [A: $tType,X: A,Z: A,R: 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),Z)),transitive_rtrancl(A,R)))
     => ( ( X != Z )
       => ~ ! [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),X),Y4)),R))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z)),transitive_rtrancl(A,R))) ) ) ) ).

% converse_rtranclE
tff(fact_4027_rtrancl__induct,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(A,bool,P,A3))
       => ( ! [Y4: A,Z4: 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),Y4)),transitive_rtrancl(A,R)))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R))
               => ( pp(aa(A,bool,P,Y4))
                 => pp(aa(A,bool,P,Z4)) ) ) )
         => pp(aa(A,bool,P,B2)) ) ) ) ).

% rtrancl_induct
tff(fact_4028_rtrancl__trans,axiom,
    ! [A: $tType,X: A,Y3: A,R: set(product_prod(A,A)),Z: 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)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),transitive_rtrancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_rtrancl(A,R))) ) ) ).

% rtrancl_trans
tff(fact_4029_rtranclE,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( ( A3 != B2 )
       => ~ ! [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),A3),Y4)),transitive_rtrancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),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),B2)),R)) ) ) ) ).

% rtranclE
tff(fact_4030_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(set(product_prod(A,A)),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)),R))
       => pp(aa(set(product_prod(A,A)),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,R))) ) ) ).

% rtrancl.rtrancl_into_rtrancl
tff(fact_4031_rtrancl_Ortrancl__refl,axiom,
    ! [A: $tType,A3: A,R: 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,R))) ).

% rtrancl.rtrancl_refl
tff(fact_4032_rtrancl_Osimps,axiom,
    ! [A: $tType,A1: A,A22: A,R: 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,R)))
    <=> ( ? [A7: A] :
            ( ( A1 = A7 )
            & ( A22 = A7 ) )
        | ? [A7: A,B7: A,C4: A] :
            ( ( A1 = A7 )
            & ( A22 = C4 )
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),transitive_rtrancl(A,R)))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C4)),R)) ) ) ) ).

% rtrancl.simps
tff(fact_4033_rtrancl_Ocases,axiom,
    ! [A: $tType,A1: A,A22: A,R: 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,R)))
     => ( ( A22 != A1 )
       => ~ ! [B3: 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),B3)),transitive_rtrancl(A,R)))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A22)),R)) ) ) ) ).

% rtrancl.cases
tff(fact_4034_rotate__rotate,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,M),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),Xs) ).

% rotate_rotate
tff(fact_4035_rotate__add,axiom,
    ! [A: $tType,M: nat,N: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),rotate(A,M)),rotate(A,N)) ).

% rotate_add
tff(fact_4036_trancl__rtrancl__trancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(set(product_prod(A,A)),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,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R))) ) ) ).

% trancl_rtrancl_trancl
tff(fact_4037_rtrancl__trancl__trancl,axiom,
    ! [A: $tType,X: A,Y3: A,R: set(product_prod(A,A)),Z: 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)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),transitive_trancl(A,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_trancl(A,R))) ) ) ).

% rtrancl_trancl_trancl
tff(fact_4038_rtrancl__into__trancl2,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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)),R))
     => ( pp(aa(set(product_prod(A,A)),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,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R))) ) ) ).

% rtrancl_into_trancl2
tff(fact_4039_rtrancl__into__trancl1,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(set(product_prod(A,A)),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)),R))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R))) ) ) ).

% rtrancl_into_trancl1
tff(fact_4040_rtrancl__eq__or__trancl,axiom,
    ! [A: $tType,X: A,Y3: A,R4: 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),Y3)),transitive_rtrancl(A,R4)))
    <=> ( ( X = Y3 )
        | ( ( X != 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),X),Y3)),transitive_trancl(A,R4))) ) ) ) ).

% rtrancl_eq_or_trancl
tff(fact_4041_trancl__into__rtrancl,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R))) ) ).

% trancl_into_rtrancl
tff(fact_4042_tranclD2,axiom,
    ! [A: $tType,X: A,Y3: A,R4: 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),Y3)),transitive_trancl(A,R4)))
     => ? [Z4: 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),Z4)),transitive_rtrancl(A,R4)))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y3)),R4)) ) ) ).

% tranclD2
tff(fact_4043_rtranclD,axiom,
    ! [A: $tType,A3: A,B2: A,R4: 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,R4)))
     => ( ( 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,R4))) ) ) ) ).

% rtranclD
tff(fact_4044_tranclD,axiom,
    ! [A: $tType,X: A,Y3: A,R4: 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),Y3)),transitive_trancl(A,R4)))
     => ? [Z4: 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),Z4)),R4))
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Y3)),transitive_rtrancl(A,R4))) ) ) ).

% tranclD
tff(fact_4045_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))))
     => ( ! [X4: 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),X4),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),Y4),X4)),Q))
             => ( Y4 = 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),A3),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separator_converseE
tff(fact_4046_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))))
     => ( ! [X4: 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),A3),X4)),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),X4),Y4)),Q))
             => ( X4 = 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),A3),B2)),transitive_rtrancl(A,P))) ) ) ).

% rtrancl_Un_separatorE
tff(fact_4047_converse__rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),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),R)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
       => ( ! [A4: A,B3: 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),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
             => ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).

% converse_rtrancl_induct2
tff(fact_4048_converse__rtranclE2,axiom,
    ! [B: $tType,A: $tType,Xa: A,Xb2: B,Za2: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
      ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb2) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
       => ~ ! [A4: A,B3: 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),Xa),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),R))
             => ~ 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),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R))) ) ) ) ).

% converse_rtranclE2
tff(fact_4049_rtrancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set(product_prod(product_prod(A,B),product_prod(A,B))),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),R)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
       => ( ! [A4: A,B3: 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),A4),B3))),transitive_rtrancl(product_prod(A,B),R)))
             => ( 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),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3))
                 => 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_4050_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: B,Xs: list(A),Ys: 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),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(B),bool,member(B,Y3),aa(list(B),set(B),set2(B),Ys))) ) ).

% set_zip_rightD
tff(fact_4051_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y3: B,Xs: list(A),Ys: 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),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_4052_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: B,Xs: list(A),Ys: 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),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
     => ~ ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,member(B,Y3),aa(list(B),set(B),set2(B),Ys))) ) ) ).

% in_set_zipE
tff(fact_4053_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_4054_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X4: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2) )
           => ! [Y4: B,Ys2: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys2) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) )
                 => ( Xys != zip(A,B,Xs2,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_4055_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: 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),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_4056_antimono__iff__le__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A)] :
          ( order_antimono(nat,A,F)
        <=> ! [N6: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N6))),aa(nat,A,F,N6))) ) ) ).

% antimono_iff_le_Suc
tff(fact_4057_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y3: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(B),bool,member(B,Y3),aa(list(B),set(B),set2(B),Ys)))
       => ~ ! [X4: 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),X4),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_4058_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y4: 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),Y4)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_4059_rtrancl__insert,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ts(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).

% rtrancl_insert
tff(fact_4060_rtrancl__is__UN__relpow,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] : transitive_rtrancl(A,R4) = 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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),top_top(set(nat)))) ).

% rtrancl_is_UN_relpow
tff(fact_4061_rtrancl__imp__UN__relpow,axiom,
    ! [A: $tType,P2: product_prod(A,A),R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P2),transitive_rtrancl(A,R4)))
     => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P2),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_tl(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R4)),top_top(set(nat)))))) ) ).

% rtrancl_imp_UN_relpow
tff(fact_4062_trancl__insert,axiom,
    ! [A: $tType,Y3: A,X: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_ts(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y3),X),R)))) ).

% trancl_insert
tff(fact_4063_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_tt(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).

% set_zip
tff(fact_4064_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( butlast(A,take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_4065_Image__fold,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),S2: set(A)] :
      ( pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),R4))
     => ( aa(set(A),set(B),image(A,B,R4),S2) = 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_sg(set(A),fun(A,fun(B,fun(set(B),set(B)))),S2)),bot_bot(set(B)),R4) ) ) ).

% Image_fold
tff(fact_4066_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F5: fun(A,set(B)),F: fun(B,C),G5: fun(D,set(C)),J5: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,member(A,I2),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),F5,I2)),aa(A,set(B),F5,J2)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F5,J2)),aa(A,set(B),F5,I2))) ) ) )
       => ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_tu(fun(D,set(C)),fun(D,filter(C)),G5)),J5)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_rw(fun(A,set(B)),fun(A,filter(B)),F5)),I5)))
        <=> ! [X3: D] :
              ( pp(aa(set(D),bool,member(D,X3),J5))
             => ? [Xa2: A] :
                  ( pp(aa(set(A),bool,member(A,Xa2),I5))
                  & ! [Xb3: B] :
                      ( pp(aa(set(B),bool,member(B,Xb3),aa(A,set(B),F5,Xa2)))
                     => pp(aa(set(C),bool,member(C,aa(B,C,F,Xb3)),aa(D,set(C),G5,X3))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_4067_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: 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)),R))
     => ( 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,R),A5))) ) ) ).

% ImageI
tff(fact_4068_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_tv(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P),Q))))
      <=> ! [Y: A] :
            ( pp(aa(A,bool,P,Y))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),Y)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_4069_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A3: B] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),insert2(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)),R)) ) ).

% Image_singleton_iff
tff(fact_4070_length__butlast,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),butlast(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_4071_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: 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_tx(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),F),P))) = aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_ty(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),F),P)) ).

% Union_SetCompr_eq
tff(fact_4072_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),B2: B,R: 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)),R))
       => pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R),A5))) ) ) ).

% rev_ImageI
tff(fact_4073_Image__iff,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),A5)))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),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),X3),B2)),R)) ) ) ).

% Image_iff
tff(fact_4074_ImageE,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),A5)))
     => ~ ! [X4: 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),X4),B2)),R))
           => ~ pp(aa(set(B),bool,member(B,X4),A5)) ) ) ).

% ImageE
tff(fact_4075_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A5: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_tz(fun(B,A),fun(set(B),fun(A,bool)),F),A5)) = aa(set(B),set(A),image2(B,A,F),A5) ).

% Setcompr_eq_image
tff(fact_4076_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_ua(fun(B,A),fun(fun(B,bool),fun(A,bool)),F),P)) = aa(set(B),set(A),image2(B,A,F),aa(fun(B,bool),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_4077_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F: 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_ub(fun(A,bool),fun(fun(A,B),fun(B,bool)),P),F)))) ) ).

% finite_image_set
tff(fact_4078_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),F: 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_uc(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P),Q),F)))) ) ) ).

% finite_image_set2
tff(fact_4079_filterlim__ident,axiom,
    ! [A: $tType,F5: filter(A)] : filterlim(A,A,aTP_Lamp_ak(A,A),F5,F5) ).

% filterlim_ident
tff(fact_4080_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F33: filter(B),F23: filter(A),F: fun(C,A),F12: filter(C)] :
      ( filterlim(A,B,G,F33,F23)
     => ( filterlim(C,A,F,F23,F12)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ud(fun(A,B),fun(fun(C,A),fun(C,B)),G),F),F33,F12) ) ) ).

% filterlim_compose
tff(fact_4081_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_ue(fun(set(A),bool),fun(set(A),bool))) ).

% finite.mono
tff(fact_4082_tranclp_Omono,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] : order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_uf(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R)) ).

% tranclp.mono
tff(fact_4083_rtranclp_Omono,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] : order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_ug(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R)) ).

% rtranclp.mono
tff(fact_4084_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F23: filter(B),F33: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F23),F33),F12)
    <=> ( filterlim(A,B,F,F23,F12)
        & filterlim(A,B,F,F33,F12) ) ) ).

% filterlim_inf
tff(fact_4085_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_uh(fun(B,A),fun(A,bool),F)) = aa(set(B),set(A),image2(B,A,F),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_4086_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_ui(A,fun(set(A),fun(A,bool)),A3),A5))) ) ).

% finite_inf_Sup
tff(fact_4087_Image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(B,A)),B4: fun(C,set(B)),A5: set(C)] : aa(set(B),set(A),image(B,A,R),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_uj(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B4)),A5)) ).

% Image_UN
tff(fact_4088_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G5: fun(C,filter(B)),B4: set(C),F5: filter(A)] :
      ( filterlim(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G5),B4)),F5)
    <=> ! [X3: C] :
          ( pp(aa(set(C),bool,member(C,X3),B4))
         => filterlim(A,B,F,aa(C,filter(B),G5,X3),F5) ) ) ).

% filterlim_INF
tff(fact_4089_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,X: A,A5: set(A),F: fun(B,C),F5: filter(C),G5: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,X),A5))
     => ( filterlim(B,C,F,F5,aa(A,filter(B),G5,X))
       => filterlim(B,C,F,F5,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G5),A5))) ) ) ).

% filterlim_INF'
tff(fact_4090_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G5: filter(B),F5: filter(A),P: fun(A,bool),G: fun(A,B)] :
      ( filterlim(A,B,F,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),principal(A,aa(fun(A,bool),set(A),collect(A),P))))
     => ( filterlim(A,B,G,G5,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),principal(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_lv(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_uk(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),F),P),G),G5,F5) ) ) ).

% filterlim_If
tff(fact_4091_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_ul(set(product_prod(B,A)),fun(B,fun(A,bool)),R),A3)) ).

% Image_singleton
tff(fact_4092_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: 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,R),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_uj(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B4)),A5)))) ).

% Image_INT_subset
tff(fact_4093_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_um(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_uo(fun(A,fun(B,bool)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_4094_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R,S) = 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_up(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R),S))) ).

% relcomp_unfold
tff(fact_4095_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set(A),I: fun(A,C),I5: set(C),F5: fun(C,set(D)),F: fun(D,E),G5: fun(A,set(E))] :
      ( ! [M4: A,X4: B] :
          ( pp(aa(set(A),bool,member(A,M4),J5))
         => pp(aa(set(C),bool,member(C,aa(A,C,I,M4)),I5)) )
     => ( ! [M4: A,X4: D] :
            ( pp(aa(set(A),bool,member(A,M4),J5))
           => ( pp(aa(set(D),bool,member(D,X4),aa(C,set(D),F5,aa(A,C,I,M4))))
             => pp(aa(set(E),bool,member(E,aa(D,E,F,X4)),aa(A,set(E),G5,M4))) ) )
       => filterlim(D,E,F,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),image2(A,filter(E),aTP_Lamp_uq(fun(A,set(E)),fun(A,filter(E)),G5)),J5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_ur(fun(C,set(D)),fun(C,filter(D)),F5)),I5))) ) ) ).

% filterlim_base
tff(fact_4096_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),B4: set(B)] : aa(set(B),set(A),image(B,A,R),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_us(set(product_prod(B,A)),fun(B,set(A)),R)),B4)) ).

% Image_eq_UN
tff(fact_4097_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_ut(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_4098_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),lattic5882676163264333800up_fin(A,A5)) = lattic5882676163264333800up_fin(A,aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_uu(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_4099_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),lattic5882676163264333800up_fin(A,A5)),lattic5882676163264333800up_fin(A,B4)) = lattic5882676163264333800up_fin(A,aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_uv(set(A),fun(set(A),fun(A,bool)),A5),B4))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_4100_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),lattic7752659483105999362nf_fin(A,A5)) = lattic7752659483105999362nf_fin(A,aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_uw(set(A),fun(A,fun(A,bool)),A5),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_4101_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),lattic7752659483105999362nf_fin(A,A5)),lattic7752659483105999362nf_fin(A,B4)) = lattic7752659483105999362nf_fin(A,aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ux(set(A),fun(set(A),fun(A,bool)),A5),B4))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_4102_UN__Image,axiom,
    ! [B: $tType,A: $tType,C: $tType,X6: fun(C,set(product_prod(B,A))),I5: set(C),S2: 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))),S2) = 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_uy(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X6),S2)),I5)) ).

% UN_Image
tff(fact_4103_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : butlast(A,Xs) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_4104_quotient__def,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] : equiv_quotient(A,A5,R) = 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_uz(set(product_prod(A,A)),fun(A,set(set(A))),R)),A5)) ).

% quotient_def
tff(fact_4105_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A5,F,G) = 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_va(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A5),F),G)) ).

% image2_def
tff(fact_4106_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_vb(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_4107_lex__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_vc(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lex_conv
tff(fact_4108_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_4109_fold__append,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : fold(B,A,F,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),fold(B,A,F,Ys)),fold(B,A,F,Xs)) ).

% fold_append
tff(fact_4110_size__list__append,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F),Xs)),aa(list(A),nat,size_list(A,F),Ys)) ).

% size_list_append
tff(fact_4111_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_4112_nths__append,axiom,
    ! [A: $tType,L: list(A),L6: list(A),A5: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L6),A5) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A5)),nths(A,L6,aa(fun(nat,bool),set(nat),collect(nat),aa(set(nat),fun(nat,bool),aTP_Lamp_vd(list(A),fun(set(nat),fun(nat,bool)),L),A5)))) ).

% nths_append
tff(fact_4113_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_4114_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V: list(A),R: 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),V)),lexord(A,R)))
     => 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),V))),lexord(A,R))) ) ).

% lexord_append_leftI
tff(fact_4115_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R: 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)),Ys),Zs)),lex(A,R)))
     => 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),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R))) ) ).

% lex_append_leftI
tff(fact_4116_Pow__Compl,axiom,
    ! [A: $tType,A5: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A5)) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_ve(set(A),fun(set(A),bool),A5)) ).

% Pow_Compl
tff(fact_4117_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V: list(A),R: 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),V))),lexord(A,R)))
     => ( ! [A4: 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),A4),A4)),R))
       => 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),V)),lexord(A,R))) ) ) ).

% lexord_append_leftD
tff(fact_4118_lexord__append__rightI,axiom,
    ! [A: $tType,Y3: list(A),X: list(A),R: set(product_prod(A,A))] :
      ( ? [B11: A,Z6: list(A)] : Y3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B11),Z6)
     => 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),Y3))),lexord(A,R))) ) ).

% lexord_append_rightI
tff(fact_4119_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs))),lexord(A,R)))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R))) ) ) ) ) ).

% lexord_sufE
tff(fact_4120_lex__append__leftD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: 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),X4)),R))
     => ( 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),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R)))
       => 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)),Ys),Zs)),lex(A,R))) ) ) ).

% lex_append_leftD
tff(fact_4121_lex__append__left__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X4: 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),X4)),R))
     => ( 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),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R)))
      <=> 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)),Ys),Zs)),lex(A,R))) ) ) ).

% lex_append_left_iff
tff(fact_4122_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),lex(A,R))) ) ) ).

% lex_append_rightI
tff(fact_4123_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R4: set(product_prod(A,A)),Vs: list(A),Ys: 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,R4)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))),lenlex(A,R4))) ) ) ).

% lenlex_append1
tff(fact_4124_image2__eqI,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: A,F: fun(B,A),X: B,C2: C,G: fun(B,C),A5: set(B)] :
      ( ( B2 = aa(B,A,F,X) )
     => ( ( C2 = aa(B,C,G,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,F,G))) ) ) ) ).

% image2_eqI
tff(fact_4125_lexord__append__left__rightI,axiom,
    ! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),U: list(A),X: list(A),Y3: 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)),R))
     => 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),Y3)))),lexord(A,R))) ) ).

% lexord_append_left_rightI
tff(fact_4126_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R)))
    <=> ( ? [X3: 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),X3)),R)) )
        | 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)),Ys),Zs)),lexord(A,R))) ) ) ).

% lexord_same_pref_iff
tff(fact_4127_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R: set(product_prod(A,A)),V: list(A),Z: 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),W)),lexord(A,R)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),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),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z))),lexord(A,R))) ) ) ).

% lexord_sufI
tff(fact_4128_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_vf(A,bool)) ) ) ).

% Nats_altdef1
tff(fact_4129_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_vg(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_4130_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Ys))) ) ).

% horner_sum_append
tff(fact_4131_lexord__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_vh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% lexord_def
tff(fact_4132_lexn__conv,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),N) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_vi(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R),N))) ).

% lexn_conv
tff(fact_4133_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y3: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y3),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs)) )
        & pp(aa(A,bool,P,Y3))
        & ~ ? [X2: A] :
              ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X2)) ) ) ) ).

% extract_SomeE
tff(fact_4134_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y3: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y3),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs)) )
        & pp(aa(A,bool,P,Y3))
        & ~ ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X3)) ) ) ) ).

% extract_Some_iff
tff(fact_4135_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_vj(set(A),fun(set(list(A)),fun(list(A),bool)),A5),XS)) ).

% set_Cons_def
tff(fact_4136_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_4137_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),N)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_4138_lex__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = 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,R)),top_top(set(nat)))) ).

% lex_def
tff(fact_4139_listrel1__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_vk(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).

% listrel1_def
tff(fact_4140_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)) = aa(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))),aa(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A))))),aa(option(product_prod(list(A),product_prod(A,list(A)))),fun(fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),fun(option(product_prod(list(A),product_prod(A,list(A)))),option(product_prod(list(A),product_prod(A,list(A)))))),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_vm(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_4141_lenlex__append2,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R4)
     => ( 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),Ys))),lenlex(A,R4)))
      <=> 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),Ys)),lenlex(A,R4))) ) ) ).

% lenlex_append2
tff(fact_4142_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R)
     => ( 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),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R)))
      <=> 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)),Ys),Zs)),lexord(A,R))) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_4143_length__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(A) ) ) ).

% length_0_conv
tff(fact_4144_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,N,Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_4145_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_4146_take0,axiom,
    ! [A: $tType,X2: list(A)] : take(A,zero_zero(nat),X2) = nil(A) ).

% take0
tff(fact_4147_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A3: A] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_4148_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R: 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,R)))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_4149_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_4150_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A5: set(A)] : set_Cons(A,A5,aa(set(list(A)),set(list(A)),insert2(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image2(A,list(A),aTP_Lamp_vn(A,list(A))),A5) ).

% set_Cons_sing_Nil
tff(fact_4151_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_4152_nths__singleton,axiom,
    ! [A: $tType,A5: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A5) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_4153_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y3: A,Ys: list(A),R: 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),Y3),Ys))),listrel1(A,R)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
          & ( Xs = Ys ) )
        | ( ( X = Y3 )
          & 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),Ys)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_4154_lexord__Nil__left,axiom,
    ! [A: $tType,Y3: list(A),R: 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)),Y3)),lexord(A,R)))
    <=> ? [A7: A,X3: list(A)] : Y3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),X3) ) ).

% lexord_Nil_left
tff(fact_4155_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R: 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,R))) ).

% not_Nil_listrel1
tff(fact_4156_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R: 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,R))) ).

% not_listrel1_Nil
tff(fact_4157_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P4: 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)),P4),nil(A))
     => ( ! [P4: fun(A,fun(A,bool)),X4: 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)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)))
       => ~ ! [P4: fun(A,fun(A,bool)),X4: A,Y4: A,Xs3: 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)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs3))) ) ) ).

% successively.cases
tff(fact_4158_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F3: fun(A,B),X4: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)))
         => ( ! [F3: fun(A,B),X4: A,Y4: 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)),F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))
           => ~ ! [A4: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_4159_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P4: 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)),P4),nil(A))
     => ~ ! [P4: fun(A,fun(A,bool)),X4: 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)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_4160_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_4161_irreflI,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( ! [A4: 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),A4),A4)),R4))
     => irrefl(A,R4) ) ).

% irreflI
tff(fact_4162_irrefl__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
    <=> ! [A7: 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),A7),A7)),R)) ) ).

% irrefl_def
tff(fact_4163_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y3: A,R: 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),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))))),listrel1(A,R)))
    <=> ( ( 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),Ys)),listrel1(A,R)))
          & ( X = Y3 ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4164_list_Osize_I3_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% list.size(3)
tff(fact_4165_splice_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Ys3)
     => ~ ! [X4: A,Xs3: 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),X4),Xs3)),Ys3) ) ).

% splice.cases
tff(fact_4166_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)
     => ( ! [Xs3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs3),nil(A))
       => ~ ! [X4: A,Xs3: list(A),Y4: 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),X4),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)) ) ) ).

% shuffles.cases
tff(fact_4167_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F3: 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))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs2))
     => ~ ! [F3: fun(A,B),A4: A,As2: 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))),F3),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As2)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_4168_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_4169_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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),Ys)),transitive_rtrancl(list(A),listrel1(A,R))))
     => 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),Ys))),transitive_rtrancl(list(A),listrel1(A,R)))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_4170_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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),Ys)),listrel1(A,R)))
     => 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),Ys))),listrel1(A,R))) ) ).

% listrel1I2
tff(fact_4171_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: 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),Y3)),transitive_rtrancl(list(A),listrel1(A,R))))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y3) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_4172_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_4173_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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),Ys)),listrel1(A,R)))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & 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,R))) ) )
     => 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),Ys),Vs))),listrel1(A,R))) ) ).

% append_listrel1I
tff(fact_4174_lexord__Nil__right,axiom,
    ! [A: $tType,X: list(A),R: 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,R))) ).

% lexord_Nil_right
tff(fact_4175_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(list(A),nat,size_list(A,X),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_4176_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R: 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)),Ys)),lex(A,R))) ).

% Nil_notin_lex
tff(fact_4177_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R: 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,R))) ).

% Nil2_notin_lex
tff(fact_4178_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R: 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,R))) ).

% Nil_lenlex_iff2
tff(fact_4179_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,bool)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_4180_take__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),nil(A)),aa(list(A),fun(nat,list(A)),aTP_Lamp_vo(A,fun(list(A),fun(nat,list(A))),X),Xs)),N) ).

% take_Cons
tff(fact_4181_listrel1I1,axiom,
    ! [A: $tType,X: A,Y3: A,R: 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),Y3)),R))
     => 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),Y3),Xs))),listrel1(A,R))) ) ).

% listrel1I1
tff(fact_4182_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)),listrel1(A,R)))
     => ( ! [Y4: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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),Y4)),R)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1E1
tff(fact_4183_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y3: A,Ys: list(A),R: 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),Y3),Ys))),listrel1(A,R)))
     => ( ! [X4: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys) )
           => ~ pp(aa(set(product_prod(A,A)),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),Y3)),R)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),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),Ys)),listrel1(A,R))) ) ) ) ).

% Cons_listrel1E2
tff(fact_4184_lexl__not__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: list(A)] :
      ( irrefl(A,R)
     => ~ 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,R))) ) ).

% lexl_not_refl
tff(fact_4185_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_vp(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_4186_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
     => ~ ! [X4: 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),X4),Y4)),R))
           => ! [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),X4),Vs2)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_4187_listrel1I,axiom,
    ! [A: $tType,X: A,Y3: A,R: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),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),Ys)),listrel1(A,R))) ) ) ) ).

% listrel1I
tff(fact_4188_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y3: A,R: set(product_prod(A,A)),Xs: list(A),Ys: 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),Y3)),transitive_rtrancl(A,R)))
     => ( 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),Ys)),transitive_rtrancl(list(A),listrel1(A,R))))
       => 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),Y3),Ys))),transitive_rtrancl(list(A),listrel1(A,R)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_4189_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_vq(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ).

% lexordp.mono
tff(fact_4190_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_4191_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs)) ) ) ) ).

% take_Cons'
tff(fact_4192_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),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(bool,fun(list(A),fun(list(A),list(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_vr(set(nat),fun(nat,bool),A5)))) ).

% nths_Cons
tff(fact_4193_listrel1p__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R,Xs,Ys)
    <=> 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),Ys)),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),R))))) ) ).

% listrel1p_def
tff(fact_4194_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_vs(set(list(A)),fun(list(A),fun(A,bool)),Kl),Kl2)) ).

% Succ_def
tff(fact_4195_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y3: list(int)] :
      ( ( upto(X,Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y3 = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y3 = 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),Xa))) ) ) ) ).

% upto.pelims
tff(fact_4196_upto_Opsimps,axiom,
    ! [I: 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),I),J)))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I),J))
         => ( upto(I,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_4197_list__encode_Oelims,axiom,
    ! [X: list(nat),Y3: nat] :
      ( ( nat_list_encode(X) = Y3 )
     => ( ( ( X = nil(nat) )
         => ( Y3 != zero_zero(nat) ) )
       => ~ ! [X4: nat,Xs3: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X4),Xs3) )
             => ( Y3 != aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X4),nat_list_encode(Xs3)))) ) ) ) ) ).

% list_encode.elims
tff(fact_4198_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),bool),aTP_Lamp_ue(fun(set(A),bool),fun(set(A),bool))) ).

% finite_def
tff(fact_4199_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_preorder_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(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)),R)) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_4200_card__quotient__disjoint,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(A,set(set(A)),aTP_Lamp_vt(set(product_prod(A,A)),fun(A,set(set(A))),R),A5)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A5,R)) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_quotient_disjoint
tff(fact_4201_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : inj_on(A,B,F,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_4202_inj__uminus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A5: set(A)] : inj_on(A,A,uminus_uminus(A),A5) ) ).

% inj_uminus
tff(fact_4203_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_4204_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_vu(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_4205_inj__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),top_top(set(product_prod(A,B))))
    <=> inj_on(A,C,F,top_top(set(A))) ) ).

% inj_apfst
tff(fact_4206_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_4207_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: A,A5: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert2(A,A3),A5))
    <=> ( inj_on(A,B,F,A5)
        & ~ pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))))) ) ) ).

% inj_on_insert
tff(fact_4208_inj__on__image__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(A),G: fun(A,B),F: fun(A,A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ! [Xa4: A] :
              ( pp(aa(set(A),bool,member(A,Xa4),A5))
             => ( ( aa(A,B,G,aa(A,A,F,X4)) = aa(A,B,G,aa(A,A,F,Xa4)) )
              <=> ( aa(A,B,G,X4) = aa(A,B,G,Xa4) ) ) ) )
     => ( inj_on(A,A,F,A5)
       => ( inj_on(A,B,G,aa(set(A),set(A),image2(A,A,F),A5))
        <=> inj_on(A,B,G,A5) ) ) ) ).

% inj_on_image_iff
tff(fact_4209_inj__on__image__Pow,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
     => inj_on(set(A),set(B),image2(A,B,F),pow2(A,A5)) ) ).

% inj_on_image_Pow
tff(fact_4210_inj__on__imageI2,axiom,
    ! [B: $tType,C: $tType,A: $tType,F4: fun(C,B),F: fun(A,C),A5: set(A)] :
      ( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F4),F),A5)
     => inj_on(A,C,F,A5) ) ).

% inj_on_imageI2
tff(fact_4211_inj__on__id,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,A,id(A),A5) ).

% inj_on_id
tff(fact_4212_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(B,A),D5: set(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(B,A,F,D5)
       => 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_vv(set(A),fun(fun(B,A),fun(set(B),fun(B,bool))),A5),F),D5)))) ) ) ).

% finite_inverse_image_gen
tff(fact_4213_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_vw(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_4214_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,A5)
     => inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B4)) ) ).

% inj_on_diff
tff(fact_4215_inj__on__inverseI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),G: fun(B,A),F: fun(A,B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ( aa(B,A,G,aa(A,B,F,X4)) = X4 ) )
     => inj_on(A,B,F,A5) ) ).

% inj_on_inverseI
tff(fact_4216_inj__on__contraD,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A,Y3: A] :
      ( inj_on(A,B,F,A5)
     => ( ( X != Y3 )
       => ( pp(aa(set(A),bool,member(A,X),A5))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( aa(A,B,F,X) != aa(A,B,F,Y3) ) ) ) ) ) ).

% inj_on_contraD
tff(fact_4217_inj__on__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A,Y3: A] :
      ( inj_on(A,B,F,A5)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y3),A5))
         => ( ( aa(A,B,F,X) = aa(A,B,F,Y3) )
          <=> ( X = Y3 ) ) ) ) ) ).

% inj_on_eq_iff
tff(fact_4218_inj__on__cong,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [A4: A] :
          ( pp(aa(set(A),bool,member(A,A4),A5))
         => ( aa(A,B,F,A4) = aa(A,B,G,A4) ) )
     => ( inj_on(A,B,F,A5)
      <=> inj_on(A,B,G,A5) ) ) ).

% inj_on_cong
tff(fact_4219_inj__on__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => ! [Xa2: A] :
              ( pp(aa(set(A),bool,member(A,Xa2),A5))
             => ( ( aa(A,B,F,X3) = aa(A,B,F,Xa2) )
               => ( X3 = Xa2 ) ) ) ) ) ).

% inj_on_def
tff(fact_4220_inj__onI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ( pp(aa(set(A),bool,member(A,Y4),A5))
           => ( ( aa(A,B,F,X4) = aa(A,B,F,Y4) )
             => ( X4 = Y4 ) ) ) )
     => inj_on(A,B,F,A5) ) ).

% inj_onI
tff(fact_4221_inj__onD,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A,Y3: A] :
      ( inj_on(A,B,F,A5)
     => ( ( aa(A,B,F,X) = aa(A,B,F,Y3) )
       => ( pp(aa(set(A),bool,member(A,X),A5))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( X = Y3 ) ) ) ) ) ).

% inj_onD
tff(fact_4222_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A5: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),A5) ) ).

% inj_on_add
tff(fact_4223_inj__on__id2,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,A,aTP_Lamp_ak(A,A),A5) ).

% inj_on_id2
tff(fact_4224_inj__on__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( ( inj_on(A,B,F,A5)
        | inj_on(A,B,F,B4) )
     => inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)) ) ).

% inj_on_Int
tff(fact_4225_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_vx(fun(A,B),fun(A,fun(C,B)),F),top_top(set(A))) ) ).

% inj_fun
tff(fact_4226_injD,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Y3: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,B,F,X) = aa(A,B,F,Y3) )
       => ( X = Y3 ) ) ) ).

% injD
tff(fact_4227_injI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( ! [X4: A,Y4: A] :
          ( ( aa(A,B,F,X4) = aa(A,B,F,Y4) )
         => ( X4 = Y4 ) )
     => inj_on(A,B,F,top_top(set(A))) ) ).

% injI
tff(fact_4228_inj__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A,Y3: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,B,F,X) = aa(A,B,F,Y3) )
      <=> ( X = Y3 ) ) ) ).

% inj_eq
tff(fact_4229_inj__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
    <=> ! [X3: A,Y: A] :
          ( ( aa(A,B,F,X3) = aa(A,B,F,Y) )
         => ( X3 = Y ) ) ) ).

% inj_def
tff(fact_4230_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(A),A5: set(A)] :
      ( inj_on(A,B,F,B4)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
       => inj_on(A,B,F,A5) ) ) ).

% subset_inj_on
tff(fact_4231_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,A5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),A5))
       => inj_on(A,B,F,B4) ) ) ).

% inj_on_subset
tff(fact_4232_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A5: set(A)] : inj_on(A,A,aTP_Lamp_vy(A,fun(A,A),A3),A5) ) ).

% inj_on_add'
tff(fact_4233_fun_Oinj__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),top_top(set(fun(D,A)))) ) ).

% fun.inj_map
tff(fact_4234_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A5: set(A),F: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
             => ( pp(aa(set(A),bool,member(A,X4),A5))
               => ( pp(aa(set(A),bool,member(A,Y4),A5))
                 => ( aa(A,B,F,X4) != aa(A,B,F,Y4) ) ) ) )
         => ( ! [X4: A,Y4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A5))
               => ( pp(aa(set(A),bool,member(A,Y4),A5))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y4))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X4)) ) ) )
           => inj_on(A,B,F,A5) ) ) ) ).

% linorder_inj_onI
tff(fact_4235_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_4236_list__encode_Osimps_I1_J,axiom,
    nat_list_encode(nil(nat)) = zero_zero(nat) ).

% list_encode.simps(1)
tff(fact_4237_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F: fun(A,B)] :
          ( ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y4))
             => ( aa(A,B,F,X4) != aa(A,B,F,Y4) ) )
         => inj_on(A,B,F,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_4238_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C6: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C6))
         => ( ( aa(set(A),set(B),image2(A,B,F),A5) = aa(set(A),set(B),image2(A,B,F),B4) )
          <=> ( A5 = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_4239_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(A),A3: A,A5: set(A)] :
      ( inj_on(A,B,F,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,F,A3)),aa(set(A),set(B),image2(A,B,F),A5)))
          <=> pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_4240_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A3),top_top(set(A))) ) ).

% inj_add_left
tff(fact_4241_inj__img__insertE,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: B,B4: set(B)] :
      ( inj_on(A,B,F,A5)
     => ( ~ pp(aa(set(B),bool,member(B,X),B4))
       => ( ( aa(set(B),set(B),insert2(B,X),B4) = aa(set(A),set(B),image2(A,B,F),A5) )
         => ~ ! [X8: A,A8: set(A)] :
                ( ~ pp(aa(set(A),bool,member(A,X8),A8))
               => ( ( A5 = aa(set(A),set(A),insert2(A,X8),A8) )
                 => ( ( X = aa(A,B,F,X8) )
                   => ( B4 != aa(set(A),set(B),image2(A,B,F),A8) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_4242_inj__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: A,A5: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),aa(set(A),set(B),image2(A,B,F),A5)))
      <=> pp(aa(set(A),bool,member(A,A3),A5)) ) ) ).

% inj_image_mem_iff
tff(fact_4243_inj__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(set(A),set(B),image2(A,B,F),A5) = aa(set(A),set(B),image2(A,B,F),B4) )
      <=> ( A5 = B4 ) ) ) ).

% inj_image_eq_iff
tff(fact_4244_range__ex1__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B2: B] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image2(A,B,F),top_top(set(A)))))
      <=> ? [X3: A] :
            ( ( B2 = aa(A,B,F,X3) )
            & ! [Y: A] :
                ( ( B2 = aa(A,B,F,Y) )
               => ( Y = X3 ) ) ) ) ) ).

% range_ex1_eq
tff(fact_4245_comp__inj__on__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A5: set(A),F4: fun(B,C)] :
      ( inj_on(A,B,F,A5)
     => ( inj_on(B,C,F4,aa(set(A),set(B),image2(A,B,F),A5))
      <=> inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F),A5) ) ) ).

% comp_inj_on_iff
tff(fact_4246_inj__on__imageI,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F: fun(A,C),A5: set(A)] :
      ( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),F),A5)
     => inj_on(C,B,G,aa(set(A),set(C),image2(A,C,F),A5)) ) ).

% inj_on_imageI
tff(fact_4247_comp__inj__on,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A5: set(A),G: fun(B,C)] :
      ( inj_on(A,B,F,A5)
     => ( inj_on(B,C,G,aa(set(A),set(B),image2(A,B,F),A5))
       => inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A5) ) ) ).

% comp_inj_on
tff(fact_4248_inj__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( inj_on(C,A,G,top_top(set(C)))
       => inj_on(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G),top_top(set(C))) ) ) ).

% inj_compose
tff(fact_4249_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,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,F),A5) = aa(set(A),set(B),image2(A,B,F),B4) )
      <=> ( A5 = B4 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_4250_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(A),A5: set(A)] :
      ( inj_on(A,B,F,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,F),A5)),aa(set(A),set(B),image2(A,B,F),B4))) ) ) ).

% inj_on_strict_subset
tff(fact_4251_inj__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( inj_on(A,A,F,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(set(A))) ) ).

% inj_fn
tff(fact_4252_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_mk(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_4253_inj__singleton,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,set(A),aTP_Lamp_nl(A,set(A)),A5) ).

% inj_singleton
tff(fact_4254_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( inj_on(B,A,F,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_vz(set(A),fun(fun(B,A),fun(B,bool)),A5),F)))) ) ) ).

% finite_inverse_image
tff(fact_4255_sum_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A5: set(B)] :
          ( inj_on(B,A,G,A5)
         => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_wa(A,A)),aa(set(B),set(A),image2(B,A,G),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A5) ) ) ) ).

% sum.image_eq
tff(fact_4256_prod_Oimage__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A5: set(B)] :
          ( inj_on(B,A,G,A5)
         => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_wb(A,A)),aa(set(B),set(A),image2(B,A,G),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A5) ) ) ) ).

% prod.image_eq
tff(fact_4257_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,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,F),A5)),aa(set(A),set(B),image2(A,B,F),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_4258_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C6: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C6))
         => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),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,F),A5)),aa(set(A),set(B),image2(A,B,F),B4)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_4259_sum_Oreindex,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [H: fun(B,C),A5: set(B),G: fun(C,A)] :
          ( inj_on(B,C,H,A5)
         => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ).

% sum.reindex
tff(fact_4260_image__Int,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),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,F),A5)),aa(set(A),set(B),image2(A,B,F),B4)) ) ) ).

% image_Int
tff(fact_4261_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C6: set(A),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,C6)
     => ( 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)),C6))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),C6))
         => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),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,F),A5)),aa(set(A),set(B),image2(A,B,F),B4)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_4262_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),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,F),A5)),aa(set(A),set(B),image2(A,B,F),B4)) ) ) ).

% image_set_diff
tff(fact_4263_prod_Oreindex,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [H: fun(B,C),A5: set(B),G: fun(C,A)] :
          ( inj_on(B,C,H,A5)
         => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ).

% prod.reindex
tff(fact_4264_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A5: set(A),F: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A5)
     => ( finite_fold(B,C,F,Z,aa(set(A),set(B),image2(A,B,G),A5)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),aa(fun(B,fun(C,C)),fun(fun(A,B),fun(A,fun(C,C))),comp(B,fun(C,C),A),F),G),Z,A5) ) ) ).

% fold_image
tff(fact_4265_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),F: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( pp(aa(set(A),bool,member(A,I2),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,I2)),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,I2))) ) ) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,member(A,I2),I5))
           => inj_on(B,C,F,aa(A,set(B),A5,I2)) )
       => inj_on(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_4266_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F: fun(B,C),A5: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,member(A,I2),I5))
           => inj_on(B,C,F,aa(A,set(B),A5,I2)) )
       => inj_on(B,C,F,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5))) ) ) ).

% inj_on_INTER
tff(fact_4267_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,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,F),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,F),A5)))) ) ).

% inj_image_Compl_subset
tff(fact_4268_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),N: nat] :
          ( order_mono(A,A,F)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F)) = complete_lattice_lfp(A,F) ) ) ) ).

% lfp_funpow
tff(fact_4269_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),G: fun(A,B),B4: set(A)] :
      ( inj_on(A,B,F,A5)
     => ( inj_on(A,B,G,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,F),A5)),aa(set(A),set(B),image2(A,B,G),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_wc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A5),G),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_4270_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),C6: set(A),A5: set(C),B4: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F,C6)
     => ( ! [X4: C] :
            ( pp(aa(set(C),bool,member(C,X4),A5))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B4,X4)),C6)) )
       => ( pp(aa(set(C),bool,member(C,J),A5))
         => ( aa(set(A),set(B),image2(A,B,F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),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_wd(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B4)),A5)) ) ) ) ) ).

% image_INT
tff(fact_4271_quotient__diff1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_vt(set(product_prod(A,A)),fun(A,set(set(A))),R),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),insert2(A,A3),bot_bot(set(A)))),R) = 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,R)),equiv_quotient(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))),R)) ) ) ) ).

% quotient_diff1
tff(fact_4272_subset__Image__Image__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_preorder_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),field2(A,R)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),field2(A,R)))
         => ( 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,R),A5)),aa(set(A),set(A),image(A,A,R),B4)))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => ? [Xa2: A] :
                    ( pp(aa(set(A),bool,member(A,Xa2),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),Xa2),X3)),R)) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_4273_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4))
    <=> ( inj_on(A,B,F,A5)
        & inj_on(A,B,F,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,F),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,F),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_4274_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),K: nat] :
          ( order_mono(A,A,F)
         => ( ( 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)),F),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),F),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_4275_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list(nat)] : nat_list_encode(aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_4276_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),X: A] :
      ( inj_on(A,A,P2,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_we(fun(A,A),fun(A,fun(A,bool)),P2),X))))
       => ~ ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),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),P2),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_4277_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A)] : complete_lattice_lfp(A,F) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_wf(fun(A,A),fun(A,bool),F))) ) ).

% lfp_def
tff(fact_4278_lfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G: fun(A,B),F: fun(B,A)] :
          ( order_mono(A,B,G)
         => ( order_mono(B,A,F)
           => ( aa(A,B,G,complete_lattice_lfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_wg(fun(A,B),fun(fun(B,A),fun(A,A)),G),F))) = complete_lattice_lfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_wh(fun(A,B),fun(fun(B,A),fun(B,B)),G),F)) ) ) ) ) ).

% lfp_rolling
tff(fact_4279_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,fun(A,A))] :
          ( ! [X4: A,Y4: A,W2: A,Z4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X4),W2)),aa(A,A,aa(A,fun(A,A),F,Y4),Z4))) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_wi(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_lfp(A,aTP_Lamp_wj(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).

% lfp_lfp
tff(fact_4280_linear__order__on__singleton,axiom,
    ! [A: $tType,X: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_4281_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_wk(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_4282_lfp__induct__set,axiom,
    ! [A: $tType,A3: A,F: fun(set(A),set(A)),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),complete_lattice_lfp(set(A),F)))
     => ( order_mono(set(A),set(A),F)
       => ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),complete_lattice_lfp(set(A),F)),aa(fun(A,bool),set(A),collect(A),P)))))
             => pp(aa(A,bool,P,X4)) )
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% lfp_induct_set
tff(fact_4283_def__lfp__induct__set,axiom,
    ! [A: $tType,A5: set(A),F: fun(set(A),set(A)),A3: A,P: fun(A,bool)] :
      ( ( A5 = complete_lattice_lfp(set(A),F) )
     => ( order_mono(set(A),set(A),F)
       => ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(set(A),set(A),F,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,X4)) )
           => pp(aa(A,bool,P,A3)) ) ) ) ) ).

% def_lfp_induct_set
tff(fact_4284_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_wl(fun(A,B),fun(A,product_prod(A,B)),F),X6) ).

% inj_on_convol_ident
tff(fact_4285_inj__Some,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,option(A),some(A),A5) ).

% inj_Some
tff(fact_4286_inj__Suc,axiom,
    ! [N5: set(nat)] : inj_on(nat,nat,suc,N5) ).

% inj_Suc
tff(fact_4287_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_4288_inj__on__diff__nat,axiom,
    ! [N5: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,member(nat,N2),N5))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_lz(nat,fun(nat,nat),K),N5) ) ).

% inj_on_diff_nat
tff(fact_4289_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_fw(A,fun(B,product_prod(B,A)))),A5) ).

% swap_inj_on
tff(fact_4290_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_wn(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_4291_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [N2: nat,F3: fun(nat,A)] :
          ( ( 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_cq(nat,fun(nat,bool)),N2))) )
          & inj_on(nat,A,F3,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_4292_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ? [F3: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F3),A5) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N2)) )
          & inj_on(A,nat,F3,A5) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_4293_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,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)),F)))
     => ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
       => ( ! [A4: A,B3: 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),A4),B3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F)),aa(fun(product_prod(A,B),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,A4),B3)) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,A3),B2)) ) ) ) ).

% lfp_induct2
tff(fact_4294_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F: fun(A,A),S: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),S) = S )
     => ( ! [M4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M4))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M4),F),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_wo(fun(A,A),fun(A,fun(nat,A)),F),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_4295_coinduct3__mono__lemma,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [F: fun(A,set(B)),X6: set(B),B4: set(B)] :
          ( order_mono(A,set(B),F)
         => order_mono(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_wp(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),F),X6),B4)) ) ) ).

% coinduct3_mono_lemma
tff(fact_4296_lfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T2: A] : complete_lattice_lfp(A,aTP_Lamp_wq(A,fun(A,A),T2)) = T2 ) ).

% lfp_const
tff(fact_4297_Total__subset__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( total_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
       => ( ( R = bot_bot(set(product_prod(A,A))) )
          | ? [A4: A] : R = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_4298_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),bool)),aTP_Lamp_vq(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ) ).

% ord_class.lexordp_def
tff(fact_4299_list__encode_Opelims,axiom,
    ! [X: list(nat),Y3: nat] :
      ( ( nat_list_encode(X) = Y3 )
     => ( pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),X))
       => ( ( ( X = nil(nat) )
           => ( ( Y3 = zero_zero(nat) )
             => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),nil(nat))) ) )
         => ~ ! [X4: nat,Xs3: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X4),Xs3) )
               => ( ( Y3 = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X4),nat_list_encode(Xs3)))) )
                 => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X4),Xs3))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_4300_If__the__inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,I: A,C6: set(A),G: fun(A,B),X: A] :
      ( pp(aa(set(A),bool,member(A,I),C6))
     => ( inj_on(A,B,G,C6)
       => ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(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_wr(set(A),fun(fun(A,B),fun(A,fun(B,A))),C6),G),X)),G),I) = aa(A,A,id(A),I) ) ) ) ).

% If_the_inv_into_f_f
tff(fact_4301_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_4302_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_4303_the__inv__into__onto,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
     => ( aa(set(B),set(A),image2(B,A,the_inv_into(A,B,A5,F)),aa(set(A),set(B),image2(A,B,F),A5)) = A5 ) ) ).

% the_inv_into_onto
tff(fact_4304_IdD,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 ) ) ).

% IdD
tff(fact_4305_IdE,axiom,
    ! [A: $tType,P2: product_prod(A,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P2),id2(A)))
     => ~ ! [X4: A] : P2 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X4) ) ).

% IdE
tff(fact_4306_the__inv__into__f__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A,Y3: B] :
      ( inj_on(A,B,F,A5)
     => ( ( aa(A,B,F,X) = Y3 )
       => ( pp(aa(set(A),bool,member(A,X),A5))
         => ( aa(B,A,the_inv_into(A,B,A5,F),Y3) = X ) ) ) ) ).

% the_inv_into_f_eq
tff(fact_4307_the__inv__into__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: A] :
      ( inj_on(A,B,F,A5)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( aa(B,A,the_inv_into(A,B,A5,F),aa(A,B,F,X)) = X ) ) ) ).

% the_inv_into_f_f
tff(fact_4308_the__inv__into__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X2: B] : aa(B,A,the_inv_into(A,B,A5,F),X2) = the(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F),X2)) ).

% the_inv_into_def
tff(fact_4309_relpow_Osimps_I1_J,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R4) = id2(A) ).

% relpow.simps(1)
tff(fact_4310_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_ws(product_prod(A,A),bool)) ).

% Id_def
tff(fact_4311_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_wt(product_prod(A,A),bool)) ).

% Id_fstsnd_eq
tff(fact_4312_f__the__inv__into__f,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),Y3: B] :
      ( inj_on(A,B,F,A5)
     => ( pp(aa(set(B),bool,member(B,Y3),aa(set(A),set(B),image2(A,B,F),A5)))
       => ( aa(A,B,F,aa(B,A,the_inv_into(A,B,A5,F),Y3)) = Y3 ) ) ) ).

% f_the_inv_into_f
tff(fact_4313_inj__on__the__inv__into,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
     => inj_on(B,A,the_inv_into(A,B,A5,F),aa(set(A),set(B),image2(A,B,F),A5)) ) ).

% inj_on_the_inv_into
tff(fact_4314_the__inv__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(B,A,the_inv_into(A,B,top_top(set(A)),F),aa(A,B,F,X)) = X ) ) ).

% the_inv_f_f
tff(fact_4315_Linear__order__in__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
          <=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) ) ) ) ) ).

% Linear_order_in_diff_Id
tff(fact_4316_reflcl__set__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X2: 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_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),fequal(A)),X2),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),X2),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))),R),id2(A)))) ) ).

% reflcl_set_eq
tff(fact_4317_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),X: B,B4: set(A)] :
      ( inj_on(A,B,F,A5)
     => ( pp(aa(set(B),bool,member(B,X),aa(set(A),set(B),image2(A,B,F),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,aa(B,A,the_inv_into(A,B,A5,F),X)),B4)) ) ) ) ).

% the_inv_into_into
tff(fact_4318_the__inv__into__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A5: set(C),X: B] :
      ( inj_on(A,B,F,aa(set(C),set(A),image2(C,A,G),A5))
     => ( inj_on(C,A,G,A5)
       => ( pp(aa(set(B),bool,member(B,X),aa(set(A),set(B),image2(A,B,F),aa(set(C),set(A),image2(C,A,G),A5))))
         => ( aa(B,C,the_inv_into(C,B,A5,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),the_inv_into(C,A,A5,G)),the_inv_into(A,B,aa(set(C),set(A),image2(C,A,G),A5),F)),X) ) ) ) ) ).

% the_inv_into_comp
tff(fact_4319_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),ord_lexordp(A),Xs),Ys))
        <=> 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),Ys)),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_4320_the__inv__f__o__f__id,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Z: A] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),the_inv_into(A,B,top_top(set(A)),F)),F),Z) = aa(A,A,id(A),Z) ) ) ).

% the_inv_f_o_f_id
tff(fact_4321_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C6: set(A),B4: set(A),X: A] :
      ( inj_on(A,B,G,C6)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C6),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B4),aa(set(A),set(A),insert2(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_wu(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C6),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),insert2(A,X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_4322_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_vp(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_4323_lists__length__Suc__eq,axiom,
    ! [A: $tType,A5: set(A),N: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_wv(set(A),fun(nat,fun(list(A),bool)),A5),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_wk(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_jf(set(A),fun(nat,fun(list(A),bool)),A5),N)),aTP_Lamp_ww(set(A),fun(list(A),set(A)),A5))) ).

% lists_length_Suc_eq
tff(fact_4324_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
    <=> ? [Y: A,N6: 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),N6)),Y)),R))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N6),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys = list_update(A,Xs,N6,Y) ) ) ) ).

% listrel1_iff_update
tff(fact_4325_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_4326_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_4327_Field__square,axiom,
    ! [A: $tType,X: set(A)] : field2(A,product_Sigma(A,A,X,aTP_Lamp_wx(set(A),fun(A,set(A)),X))) = X ).

% Field_square
tff(fact_4328_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_wy(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_wz(fun(B,bool),fun(A,set(B)),Q)) ).

% Collect_case_prod
tff(fact_4329_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_4330_Times__empty,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_xa(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_4331_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A5: set(A)] : product_Sigma(A,B,A5,aTP_Lamp_xb(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_4332_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_xc(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A5),aTP_Lamp_xc(A,set(B))) ).

% Compl_Times_UNIV2
tff(fact_4333_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_xa(set(B),fun(A,set(B)),A5))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xd(set(B),fun(A,set(B)),A5)) ).

% Compl_Times_UNIV1
tff(fact_4334_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,member(A,A4),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A4))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B4))) ) ) ).

% finite_SigmaI
tff(fact_4335_UNIV__Times__UNIV,axiom,
    ! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xc(A,set(B))) = top_top(set(product_prod(A,B))) ).

% UNIV_Times_UNIV
tff(fact_4336_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_xa(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_xa(set(B),fun(A,set(B)),B4))) = A5 ) ) ) ).

% fst_image_times
tff(fact_4337_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_nh(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_nh(set(A),fun(B,set(A)),B4))) = B4 ) ) ) ).

% snd_image_times
tff(fact_4338_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_xe(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_4339_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),insert2(A,A3),A5),aa(set(B),fun(A,set(B)),aTP_Lamp_xf(B,fun(set(B),fun(A,set(B))),B2),B4)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A5,aa(set(B),fun(A,set(B)),aTP_Lamp_xf(B,fun(set(B),fun(A,set(B))),B2),B4))),product_Sigma(A,B,aa(set(A),set(A),insert2(A,A3),A5),aTP_Lamp_xa(set(B),fun(A,set(B)),B4)))) ).

% insert_Times_insert
tff(fact_4340_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,X4))) )
       => ( 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_ox(fun(A,set(B)),fun(A,nat),B4)),A5) ) ) ) ).

% card_SigmaI
tff(fact_4341_inj__on__apfst,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C),A5: set(A)] :
      ( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),product_Sigma(A,B,A5,aTP_Lamp_xc(A,set(B))))
    <=> inj_on(A,C,F,A5) ) ).

% inj_on_apfst
tff(fact_4342_inj__on__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_xa(set(B),fun(A,set(B)),A5)))
    <=> inj_on(B,C,F,A5) ) ).

% inj_on_apsnd
tff(fact_4343_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_xg(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_xh(fun(A,fun(B,bool)),fun(A,set(B)),Q)) ).

% Collect_case_prod_Sigma
tff(fact_4344_Sigma__cong,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(A),C6: fun(A,set(B)),D5: fun(A,set(B))] :
      ( ( A5 = B4 )
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),B4))
           => ( aa(A,set(B),C6,X4) = aa(A,set(B),D5,X4) ) )
       => ( product_Sigma(A,B,A5,C6) = product_Sigma(A,B,B4,D5) ) ) ) ).

% Sigma_cong
tff(fact_4345_Times__eq__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C6: set(A),A5: set(B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C6))
     => ( ( product_Sigma(B,A,A5,aTP_Lamp_nh(set(A),fun(B,set(A)),C6)) = product_Sigma(B,A,B4,aTP_Lamp_nh(set(A),fun(B,set(A)),C6)) )
      <=> ( A5 = B4 ) ) ) ).

% Times_eq_cancel2
tff(fact_4346_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,X: A,Ys: list(B),Y3: B] : zip(A,B,list_update(A,Xs,I,X),list_update(B,Ys,I,Y3)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) ).

% zip_update
tff(fact_4347_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_4348_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_4349_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_4350_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)))
     => ~ ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => ! [Y4: B] :
                ( pp(aa(set(B),bool,member(B,Y4),aa(A,set(B),B4,X4)))
               => ( C2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y4) ) ) ) ) ).

% SigmaE
tff(fact_4351_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),C6: set(A),D5: set(B)] :
      ( ( product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4)) = product_Sigma(A,B,C6,aTP_Lamp_xa(set(B),fun(A,set(B)),D5)) )
    <=> ( ( ( A5 = C6 )
          & ( B4 = D5 ) )
        | ( ( ( A5 = bot_bot(set(A)) )
            | ( B4 = bot_bot(set(B)) ) )
          & ( ( C6 = bot_bot(set(A)) )
            | ( D5 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_4352_Sigma__Diff__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C6: 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),C6) = 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,C6)),product_Sigma(A,B,J5,C6)) ).

% Sigma_Diff_distrib1
tff(fact_4353_Times__Diff__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C6: 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_xa(set(B),fun(A,set(B)),C6)) = 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_xa(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),C6))) ).

% Times_Diff_distrib1
tff(fact_4354_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_xi(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_4355_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C6: set(A),B4: fun(A,set(B)),D5: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),C6))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B4,X4)),aa(A,set(B),D5,X4))) )
       => 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,C6,D5))) ) ) ).

% Sigma_mono
tff(fact_4356_Sigma__Int__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C6: 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),C6) = 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,C6)),product_Sigma(A,B,J5,C6)) ).

% Sigma_Int_distrib1
tff(fact_4357_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J5: set(A),C6: 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),C6) = 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,C6)),product_Sigma(A,B,J5,C6)) ).

% Sigma_Un_distrib1
tff(fact_4358_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y3: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),zero_zero(nat),Y3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) ).

% list_update_code(2)
tff(fact_4359_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C6: set(A),A5: set(B),B4: set(B)] :
      ( pp(aa(set(A),bool,member(A,X),C6))
     => ( 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_nh(set(A),fun(B,set(A)),C6))),product_Sigma(B,A,B4,aTP_Lamp_nh(set(A),fun(B,set(A)),C6))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A5),B4)) ) ) ).

% Times_subset_cancel2
tff(fact_4360_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_xa(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_4361_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_fx(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A5,aTP_Lamp_nh(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),A5)) ).

% swap_product
tff(fact_4362_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))) )
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),I5))
         => ( aa(A,set(B),X6,X3) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_4363_Times__Int__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B),C6: set(A),D5: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4))),product_Sigma(A,B,C6,aTP_Lamp_xa(set(B),fun(A,set(B)),D5))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C6),aa(set(B),fun(A,set(B)),aTP_Lamp_xj(set(B),fun(set(B),fun(A,set(B))),B4),D5)) ).

% Times_Int_Times
tff(fact_4364_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_xk(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_4365_Times__Int__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C6: 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_xa(set(B),fun(A,set(B)),C6)) = 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_xa(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),C6))) ).

% Times_Int_distrib1
tff(fact_4366_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_xa(set(B),fun(A,set(B)),B4)))) ) ) ).

% infinite_cartesian_product
tff(fact_4367_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_xa(set(B),fun(A,set(B)),B4)))) ) ) ).

% finite_cartesian_product
tff(fact_4368_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_xl(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_4369_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(A),C6: 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_xa(set(B),fun(A,set(B)),C6)) = 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_xa(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),C6))) ).

% Times_Un_distrib1
tff(fact_4370_trancl__subset__Sigma,axiom,
    ! [A: $tType,R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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,R)),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% trancl_subset_Sigma
tff(fact_4371_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,B)),A5: set(A),B4: set(B),S: set(product_prod(B,C)),C6: 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))),R),product_Sigma(A,B,A5,aTP_Lamp_xa(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))),S),product_Sigma(B,C,B4,aTP_Lamp_xm(set(C),fun(B,set(C)),C6))))
       => 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,R,S)),product_Sigma(A,C,A5,aTP_Lamp_xn(set(C),fun(A,set(C)),C6)))) ) ) ).

% relcomp_subset_Sigma
tff(fact_4372_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_xo(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B4)),X6)) ).

% Sigma_Union
tff(fact_4373_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_wx(set(A),fun(A,set(A)),A5)))) ).

% Id_on_subset_Times
tff(fact_4374_total__on__lex__prod,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: set(product_prod(A,A)),B4: set(B),S: set(product_prod(B,B))] :
      ( total_on(A,A5,R)
     => ( total_on(B,B4,S)
       => total_on(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4)),lex_prod(A,B,R,S)) ) ) ).

% total_on_lex_prod
tff(fact_4375_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C6: set(B),B4: set(A),D5: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),D5))))
    <=> ( ( A5 = bot_bot(set(A)) )
        | ( C6 = 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)),C6),D5)) ) ) ) ).

% times_subset_iff
tff(fact_4376_image__paired__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,A),G: 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_xp(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),product_Sigma(C,D,A5,aTP_Lamp_xq(set(D),fun(C,set(D)),B4))) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A5),aa(set(D),fun(A,set(B)),aTP_Lamp_xr(fun(D,B),fun(set(D),fun(A,set(B))),G),B4)) ).

% image_paired_Times
tff(fact_4377_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B2: A,R: 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,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))))
       => ( ( A3 = B2 )
          | pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_4378_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_xa(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_4379_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_xa(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_4380_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_xa(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_4381_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_xs(set(A),fun(fun(A,set(B)),fun(A,bool)),A5),B4))))
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,member(A,A4),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),B4,A4))) )
       => pp(aa(set(product_prod(A,B)),bool,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A5,B4))) ) ) ).

% finite_SigmaI2
tff(fact_4382_Image__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,B)),A5: set(A),B4: set(B),C6: 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))),R),product_Sigma(A,B,A5,aTP_Lamp_xa(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,R),C6)),B4)) ) ).

% Image_subset
tff(fact_4383_trancl__subset__Field2,axiom,
    ! [A: $tType,R: 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,R)),product_Sigma(A,A,field2(A,R),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),R)))) ).

% trancl_subset_Field2
tff(fact_4384_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_xs(set(A),fun(fun(A,set(B)),fun(A,bool)),A5),B4)) ).

% fst_image_Sigma
tff(fact_4385_finite__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))))
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
         => pp(aa(set(A),bool,finite_finite2(A),X6)) ) ) ) ).

% finite_equiv_class
tff(fact_4386_UN__Times__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,E5: fun(C,set(A)),F5: 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_xv(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E5),F5))),product_Sigma(C,D,A5,aTP_Lamp_xq(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_xw(fun(D,set(B)),fun(set(D),fun(A,set(B))),F5),B4)) ).

% UN_Times_distrib
tff(fact_4387_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),I,V) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Xs)),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_xx(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V)),I) ).

% list_update.simps(2)
tff(fact_4388_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_xa(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_4389_sum_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: 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_da(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),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),G)),product_Sigma(B,C,A5,aTP_Lamp_xm(set(C),fun(B,set(C)),B4))) ) ).

% sum.cartesian_product
tff(fact_4390_prod_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: 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_fg(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),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),G)),product_Sigma(B,C,A5,aTP_Lamp_xm(set(C),fun(B,set(C)),B4))) ) ).

% prod.cartesian_product
tff(fact_4391_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_4392_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_4393_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_xy(set(product_prod(A,B)),fun(A,set(B)),A5)))) ).

% subset_fst_snd
tff(fact_4394_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),insert2(A,X),bot_bot(set(A))),aTP_Lamp_xa(set(B),fun(A,set(B)),A5))) = aa(set(B),nat,finite_card(B),A5) ).

% card_cartesian_product_singleton
tff(fact_4395_butlast__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),X: A] :
      ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( butlast(A,list_update(A,Xs,K,X)) = butlast(A,Xs) ) )
      & ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( butlast(A,list_update(A,Xs,K,X)) = list_update(A,butlast(A,Xs),K,X) ) ) ) ).

% butlast_list_update
tff(fact_4396_finite__quotient,axiom,
    ! [A: $tType,A5: set(A),R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))))
       => pp(aa(set(set(A)),bool,finite_finite2(set(A)),equiv_quotient(A,A5,R))) ) ) ).

% finite_quotient
tff(fact_4397_sum_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: fun(B,set(C)),G: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),A5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B4,X4))) )
           => ( 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_xz(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B4),G)),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),G)),product_Sigma(B,C,A5,B4)) ) ) ) ) ).

% sum.Sigma
tff(fact_4398_prod_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: fun(B,set(C)),G: fun(B,fun(C,A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),A5))
               => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B4,X4))) )
           => ( 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_ya(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B4),G)),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),G)),product_Sigma(B,C,A5,B4)) ) ) ) ) ).

% prod.Sigma
tff(fact_4399_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_yc(fun(A,set(B)),fun(A,set(product_prod(A,B))),B4)),A5)) ).

% Sigma_def
tff(fact_4400_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_xa(set(B),fun(A,set(B)),B4)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_ye(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_4401_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),I5: set(B)] :
    ? [F3: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F3,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),F3),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_4402_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F: fun(A,B),G: 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_df(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F),G)),product_Sigma(A,C,A5,aTP_Lamp_xn(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),F),A5)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),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_yf(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F),G),A5),B4))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_4403_Preorder__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_preorder_on(A,field2(A,R),R)
     => order_preorder_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Preorder_Restr
tff(fact_4404_Linear__order__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => order_679001287576687338der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Linear_order_Restr
tff(fact_4405_Total__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( total_on(A,field2(A,R),R)
     => total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Total_Restr
tff(fact_4406_total__on__imp__Total__Restr,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( total_on(A,A5,R)
     => total_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% total_on_imp_Total_Restr
tff(fact_4407_Restr__subset,axiom,
    ! [A: $tType,A5: set(A),B4: set(A),R: 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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4)))),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))) ) ) ).

% Restr_subset
tff(fact_4408_Restr__Field,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,field2(A,R),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),R))) = R ).

% Restr_Field
tff(fact_4409_Field__Restr__subset,axiom,
    ! [A: $tType,R: 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)),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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))))),A5)) ).

% Field_Restr_subset
tff(fact_4410_linear__order__on__Restr,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A] :
      ( order_679001287576687338der_on(A,A5,R)
     => 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,R,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))),R),product_Sigma(A,A,order_above(A,R,X),aa(A,fun(A,set(A)),aTP_Lamp_yg(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X)))) ) ).

% linear_order_on_Restr
tff(fact_4411_bsqr__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_yi(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R)))) ).

% bsqr_def
tff(fact_4412_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: 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,F)),product_Sigma(A,B,A5,aTP_Lamp_xa(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,F),A5)),B4)) ) ).

% Gr_incl
tff(fact_4413_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))
      <=> ! [A9: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R)))
           => ( ( A9 != bot_bot(set(A)) )
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A9))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,member(A,Xa2),A9))
                     => pp(aa(set(product_prod(A,A)),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),Xa2)),R)) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_4414_wf__insert,axiom,
    ! [A: $tType,Y3: A,X: A,R: set(product_prod(A,A))] :
      ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X)),R))
    <=> ( wf(A,R)
        & ~ pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,R))) ) ) ).

% wf_insert
tff(fact_4415_wf__induct__rule,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( wf(A,R)
     => ( ! [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)),R))
               => pp(aa(A,bool,P,Y5)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wf_induct_rule
tff(fact_4416_wf__eq__minimal,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [Q7: set(A)] :
          ( ? [X3: A] : pp(aa(set(A),bool,member(A,X3),Q7))
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Q7))
              & ! [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),Y),X3)),R))
                 => ~ pp(aa(set(A),bool,member(A,Y),Q7)) ) ) ) ) ).

% wf_eq_minimal
tff(fact_4417_wf__not__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(A,R)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).

% wf_not_refl
tff(fact_4418_wf__not__sym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R)) ) ) ).

% wf_not_sym
tff(fact_4419_wf__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( wf(A,R)
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).

% wf_irrefl
tff(fact_4420_wf__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( wf(A,R)
     => ( ! [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)),R))
               => pp(aa(A,bool,P,Y5)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wf_induct
tff(fact_4421_wf__asym,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,X: A] :
      ( wf(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R))
       => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),A3)),R)) ) ) ).

% wf_asym
tff(fact_4422_wfUNIVI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [P4: fun(A,bool),X4: A] :
          ( ! [Xa3: 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),Xa3)),R))
                 => pp(aa(A,bool,P4,Y4)) )
             => pp(aa(A,bool,P4,Xa3)) )
         => pp(aa(A,bool,P4,X4)) )
     => wf(A,R) ) ).

% wfUNIVI
tff(fact_4423_wfI__min,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( ! [X4: A,Q8: set(A)] :
          ( pp(aa(set(A),bool,member(A,X4),Q8))
         => ? [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),Q8))
              & ! [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),Xa3)),R4))
                 => ~ pp(aa(set(A),bool,member(A,Y4),Q8)) ) ) )
     => wf(A,R4) ) ).

% wfI_min
tff(fact_4424_wfE__min,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),X: A,Q: set(A)] :
      ( wf(A,R4)
     => ( pp(aa(set(A),bool,member(A,X),Q))
       => ~ ! [Z4: A] :
              ( pp(aa(set(A),bool,member(A,Z4),Q))
             => ~ ! [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),Z4)),R4))
                   => ~ pp(aa(set(A),bool,member(A,Y5),Q)) ) ) ) ) ).

% wfE_min
tff(fact_4425_wf__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [P5: fun(A,bool)] :
          ( ! [X3: 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),Y),X3)),R))
                 => pp(aa(A,bool,P5,Y)) )
             => pp(aa(A,bool,P5,X3)) )
         => ! [X_1: A] : pp(aa(A,bool,P5,X_1)) ) ) ).

% wf_def
tff(fact_4426_GrD2,axiom,
    ! [A: $tType,B: $tType,X: A,Fx: B,A5: set(A),F: 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,F)))
     => ( aa(A,B,F,X) = Fx ) ) ).

% GrD2
tff(fact_4427_GrD1,axiom,
    ! [B: $tType,A: $tType,X: A,Fx: B,A5: set(A),F: 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,F)))
     => pp(aa(set(A),bool,member(A,X),A5)) ) ).

% GrD1
tff(fact_4428_wfE__min_H,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R4)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z4: A] :
              ( pp(aa(set(A),bool,member(A,Z4),Q))
             => ~ ! [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),Z4)),R4))
                   => ~ pp(aa(set(A),bool,member(A,Y5),Q)) ) ) ) ) ).

% wfE_min'
tff(fact_4429_wf__no__infinite__down__chainE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),F: fun(nat,A)] :
      ( wf(A,R)
     => ~ ! [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,F,aa(nat,nat,suc,K2))),aa(nat,A,F,K2))),R)) ) ).

% wf_no_infinite_down_chainE
tff(fact_4430_wf__iff__no__infinite__down__chain,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ~ ? [F6: 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,F6,aa(nat,nat,suc,I4))),aa(nat,A,F6,I4))),R)) ) ).

% wf_iff_no_infinite_down_chain
tff(fact_4431_wf__int__ge__less__than,axiom,
    ! [D3: int] : wf(int,int_ge_less_than(D3)) ).

% wf_int_ge_less_than
tff(fact_4432_wf__int__ge__less__than2,axiom,
    ! [D3: int] : wf(int,int_ge_less_than2(D3)) ).

% wf_int_ge_less_than2
tff(fact_4433_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_4434_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,bool),F: fun(A,nat),G: fun(A,A)] :
      ( ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,aa(A,A,G,X4))),aa(A,nat,F,X4))) )
     => 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_yj(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P),G)))) ) ).

% wf_if_measure
tff(fact_4435_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_4436_wf__bounded__measure,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
      ( ! [A4: A,B3: 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),B3),A4)),R))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B3)),aa(A,nat,Ub,A4)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,B3)),aa(A,nat,Ub,A4)))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,A4)),aa(A,nat,F,B3))) ) )
     => wf(A,R) ) ).

% wf_bounded_measure
tff(fact_4437_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P: fun(B,bool),K: B,M: fun(B,A)] :
      ( wf(A,R)
     => ( ! [X4: 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),X4),Y4)),transitive_trancl(A,R)))
          <=> ~ pp(aa(set(product_prod(A,A)),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),X4)),transitive_rtrancl(A,R))) )
       => ( pp(aa(B,bool,P,K))
         => ? [X4: B] :
              ( pp(aa(B,bool,P,X4))
              & ! [Y5: B] :
                  ( pp(aa(B,bool,P,Y5))
                 => pp(aa(set(product_prod(A,A)),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,X4)),aa(B,A,M,Y5))),transitive_rtrancl(A,R))) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_4438_wfI,axiom,
    ! [A: $tType,R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))))
     => ( ! [X4: A,P4: fun(A,bool)] :
            ( ! [Xa3: 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),Xa3)),R))
                   => pp(aa(A,bool,P4,Y4)) )
               => pp(aa(A,bool,P4,Xa3)) )
           => ( pp(aa(set(A),bool,member(A,X4),A5))
             => ( pp(aa(set(A),bool,member(A,X4),B4))
               => pp(aa(A,bool,P4,X4)) ) ) )
       => wf(A,R) ) ) ).

% wfI
tff(fact_4439_wf__eq__minimal2,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wf(A,R)
    <=> ! [A9: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R)))
            & ( A9 != bot_bot(set(A)) ) )
         => ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A9))
              & ! [Xa2: A] :
                  ( pp(aa(set(A),bool,member(A,Xa2),A9))
                 => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X3)),R)) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_4440_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
      ( ! [A4: A,B3: 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),B3),A4)),R))
         => ( pp(aa(set(B),bool,finite_finite2(B),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B3)),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,B3)),aa(A,set(B),Ub,A4)))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F,A4)),aa(A,set(B),F,B3))) ) )
     => wf(A,R) ) ).

% wf_bounded_set
tff(fact_4441_Gr__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] : bNF_Gr(A,B,A5,F) = 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_yk(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A5),F)) ).

% Gr_def
tff(fact_4442_Field__bsqr,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : field2(product_prod(A,A),bNF_Wellorder_bsqr(A,R)) = product_Sigma(A,A,field2(A,R),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),R)) ).

% Field_bsqr
tff(fact_4443_above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_above(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% above_def
tff(fact_4444_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_yl(set(A),fun(set(A),fun(set(A),bool)),A5)))) ) ).

% finite_subset_wf
tff(fact_4445_reduction__pairI,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( wf(A,R4)
     => ( 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,R4,S2)),R4))
       => 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))),R4),S2)) ) ) ).

% reduction_pairI
tff(fact_4446_dependent__wf__choice,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,A)),P: fun(fun(A,B),fun(A,fun(B,bool)))] :
      ( wf(A,R4)
     => ( ! [F3: fun(A,B),G3: fun(A,B),X4: A,R2: B] :
            ( ! [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),Z6),X4)),R4))
               => ( aa(A,B,F3,Z6) = aa(A,B,G3,Z6) ) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),R2))
            <=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,G3),X4),R2)) ) )
       => ( ! [X4: A,F3: fun(A,B)] :
              ( ! [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)),R4))
                 => pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),Y5),aa(A,B,F3,Y5))) )
             => ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X4),X_12)) )
         => ? [F3: fun(A,B)] :
            ! [X2: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P,F3),X2),aa(A,B,F3,X2))) ) ) ) ).

% dependent_wf_choice
tff(fact_4447_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: 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_ym(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),A5) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F),A5),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_yn(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),A5)) ).

% image_split_eq_Sigma
tff(fact_4448_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F: fun(D,B)] :
          ( order_mono(A,fun(B,C),Q)
         => order_mono(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_yo(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F)) ) ) ).

% mono_compose
tff(fact_4449_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),P: fun(B,bool)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),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_yp(fun(A,B),fun(fun(B,bool),fun(A,bool)),F),P)) ).

% vimage_Collect_eq
tff(fact_4450_vimage__ident,axiom,
    ! [A: $tType,Y7: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_ak(A,A)),Y7) = Y7 ).

% vimage_ident
tff(fact_4451_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_mj(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_mj(B,fun(A,B),C2)),A5) = bot_bot(set(A)) ) ) ) ).

% vimage_const
tff(fact_4452_vimage__id,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),id(A)) = id(set(A)) ).

% vimage_id
tff(fact_4453_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_yq(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_yq(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_yq(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_yq(B,fun(B,fun(set(A),fun(A,B))),C2),D3),B4)),A5) = bot_bot(set(A)) ) ) ) ) ) ).

% vimage_if
tff(fact_4454_set_Ocompositionality,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(B,A),Set: set(A)] : aa(set(B),set(C),aa(fun(C,B),fun(set(B),set(C)),vimage(C,B),F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),G),Set)) = aa(set(A),set(C),aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),Set) ).

% set.compositionality
tff(fact_4455_vimage__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C),X: set(C)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(C),set(B),aa(fun(B,C),fun(set(C),set(B)),vimage(B,C),G),X)) = aa(set(C),set(A),aa(fun(A,C),fun(set(C),set(A)),vimage(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),X) ).

% vimage_comp
tff(fact_4456_vimage__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B4) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_yr(fun(A,B),fun(set(B),fun(A,bool)),F),B4)) ).

% vimage_def
tff(fact_4457_surj__image__vimage__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A5: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),set(A),image2(B,A,F),aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A5)) = A5 ) ) ).

% surj_image_vimage_eq
tff(fact_4458_set_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),vimage(A,A),aTP_Lamp_ak(A,A)) = id(set(A)) ).

% set.identity
tff(fact_4459_vimage__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: 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),F),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_ys(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B4)),A5)) ).

% vimage_UN
tff(fact_4460_vimage__INT,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: 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),F),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_ys(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B4)),A5)) ).

% vimage_INT
tff(fact_4461_vimage__Union,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(set(B))] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),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),F)),A5)) ).

% vimage_Union
tff(fact_4462_vimage__Times,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: 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)),F),product_Sigma(B,C,A5,aTP_Lamp_xm(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),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F)),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),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)),B4)) ).

% vimage_Times
tff(fact_4463_set_Ocomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(B,A)] : aa(fun(set(A),set(B)),fun(set(A),set(C)),aa(fun(set(B),set(C)),fun(fun(set(A),set(B)),fun(set(A),set(C))),comp(set(B),set(C),set(A)),aa(fun(C,B),fun(set(B),set(C)),vimage(C,B),F)),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),G)) = aa(fun(C,A),fun(set(A),set(C)),vimage(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)) ).

% set.comp
tff(fact_4464_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A5: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => ( ( aa(set(A),set(B),aa(fun(B,A),fun(set(A),set(B)),vimage(B,A),F),A5) = bot_bot(set(B)) )
      <=> ( A5 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_4465_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),B4: set(A),A5: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F),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),F),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,F),A5))) ) ) ).

% vimage_subsetD
tff(fact_4466_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,X: B,A5: set(B),F: 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,F)) = aa(B,set(A),F,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,F)) = bot_bot(set(A)) ) ) ) ).

% Pair_vimage_Sigma
tff(fact_4467_inj__vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(A),set(B),image2(A,B,F),A5)) = A5 ) ) ).

% inj_vimage_image_eq
tff(fact_4468_vimage__Suc__insert__0,axiom,
    ! [A5: set(nat)] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),vimage(nat,nat),suc),aa(set(nat),set(nat),insert2(nat,zero_zero(nat)),A5)) = aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),vimage(nat,nat),suc),A5) ).

% vimage_Suc_insert_0
tff(fact_4469_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_xc(A,set(B))) ).

% vimage_fst
tff(fact_4470_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_xa(set(B),fun(A,set(B)),A5)) ).

% vimage_snd
tff(fact_4471_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(B),A5: set(A)] :
      ( inj_on(A,B,F,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,F),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),F),B4)),A5)) ) ) ).

% vimage_subsetI
tff(fact_4472_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),B4: set(B)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),B4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_yt(fun(A,B),fun(B,set(A)),F)),B4)) ).

% vimage_eq_UN
tff(fact_4473_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: B] :
      ( inj_on(A,B,F,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),F),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,bool),aTP_Lamp_yu(fun(A,B),fun(B,fun(A,bool)),F),A3))),bot_bot(set(A))))) ) ).

% inj_vimage_singleton
tff(fact_4474_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),A3: B] :
      ( inj_on(A,B,F,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),F),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))),A5)),aa(set(A),set(A),insert2(A,the(A,aa(B,fun(A,bool),aa(set(A),fun(B,fun(A,bool)),aTP_Lamp_yv(fun(A,B),fun(set(A),fun(B,fun(A,bool))),F),A5),A3))),bot_bot(set(A))))) ) ).

% inj_on_vimage_singleton
tff(fact_4475_inv__image__partition,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
     => ( ! [Y4: A] :
            ( pp(aa(set(A),bool,member(A,Y4),aa(list(A),set(A),set2(A),Ys)))
           => ~ pp(aa(A,bool,P,Y4)) )
       => ( 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))),insert2(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),bot_bot(set(product_prod(list(A),list(A)))))) = shuffles(A,Xs,Ys) ) ) ) ).

% inv_image_partition
tff(fact_4476_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)))
         => ( aa(list(A),A,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),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_4477_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_yw(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).

% zip_Cons1
tff(fact_4478_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y3: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_yx(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y3),Ys)),Xs) ).

% zip_Cons
tff(fact_4479_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( aa(list(A),A,groups8242544230860333062m_list(A),nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_4480_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( aa(list(A),A,groups8242544230860333062m_list(A),Ns) = zero_zero(A) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Ns)))
             => ( X3 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_4481_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [X: A,Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_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),plus_plus(A),X),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ).

% sum_list.Cons
tff(fact_4482_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),aa(list(A),A,groups8242544230860333062m_list(A),Ys)) ) ).

% sum_list_append
tff(fact_4483_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_yy(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F22)),List) ).

% list.case_distrib
tff(fact_4484_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_4485_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X4)) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),Xs)))
               => ( X3 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_4486_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X4)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_4487_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_4488_partition__P,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No: 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)),Yes),No) )
     => ( ! [X2: A] :
            ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),Yes)))
           => pp(aa(A,bool,P,X2)) )
        & ! [X2: A] :
            ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),No)))
           => ~ pp(aa(A,bool,P,X2)) ) ) ) ).

% partition_P
tff(fact_4489_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_wa(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_4490_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_yz(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_4491_card__length__sum__list__rec,axiom,
    ! [M: nat,N5: 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_za(nat,fun(nat,fun(list(nat),bool)),M),N5))) = 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_zb(nat,fun(nat,fun(list(nat),bool)),M),N5)))),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_zc(nat,fun(nat,fun(list(nat),bool)),M),N5)))) ) ) ).

% card_length_sum_list_rec
tff(fact_4492_card__length__sum__list,axiom,
    ! [M: nat,N5: 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_za(nat,fun(nat,fun(list(nat),bool)),M),N5))) = 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),N5),M)),one_one(nat))),N5) ).

% card_length_sum_list
tff(fact_4493_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_zd(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_4494_partition__set,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No: 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)),Yes),No) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes)),aa(list(A),set(A),set2(A),No)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_4495_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : aa(list(B),B,groups8242544230860333062m_list(B),Xs) = aa(set(nat),B,aa(fun(nat,B),fun(set(nat),B),groups7311177749621191930dd_sum(nat,B),nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_4496_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_ze(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_4497_lexn_Osimps_I2_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),aa(nat,nat,suc,N)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),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),R,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),N)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_zf(nat,fun(list(A),fun(list(A),bool)),N)))) ).

% lexn.simps(2)
tff(fact_4498_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_zg(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P),A5))) ).

% relation_of_def
tff(fact_4499_acyclic__insert,axiom,
    ! [A: $tType,Y3: A,X: A,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X)),R))
    <=> ( transitive_acyclic(A,R)
        & ~ pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,R))) ) ) ).

% acyclic_insert
tff(fact_4500_map__prod__ident,axiom,
    ! [B: $tType,A: $tType,X2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)),X2) = X2 ).

% map_prod_ident
tff(fact_4501_map__prod__simp,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,A3)),aa(D,B,G,B2)) ).

% map_prod_simp
tff(fact_4502_fst__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,A),G: 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,F,G),X)) = aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X)) ).

% fst_map_prod
tff(fact_4503_snd__map__prod,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,B),G: 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,F,G),X)) = aa(D,A,G,aa(product_prod(C,D),D,product_snd(C,D),X)) ).

% snd_map_prod
tff(fact_4504_map__prod__imageI,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R4: set(product_prod(A,B)),F: fun(A,C),G: 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)),R4))
     => 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,F,A3)),aa(B,D,G,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),image2(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F,G)),R4))) ) ).

% map_prod_imageI
tff(fact_4505_fst__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),aa(fun(product_prod(C,D),C),fun(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,F,G)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).

% fst_comp_map_prod
tff(fact_4506_snd__comp__map__prod,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,D),G: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),aa(fun(product_prod(D,C),C),fun(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,F,G)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),G),product_snd(A,B)) ).

% snd_comp_map_prod
tff(fact_4507_map__prod_Ocomp,axiom,
    ! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F2)),aa(fun(product_prod(C,D),product_prod(E,F2)),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F2))),comp(product_prod(C,D),product_prod(E,F2),product_prod(A,B)),product_map_prod(C,E,D,F2,F,G)),product_map_prod(A,C,B,D,H,I)) = product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).

% map_prod.comp
tff(fact_4508_map__prod_Ocompositionality,axiom,
    ! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,F,G),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I),Prod)) = aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Prod) ).

% map_prod.compositionality
tff(fact_4509_map__prod__compose,axiom,
    ! [D: $tType,C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F2,D),G22: fun(B,F2)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),F1),F22),aa(fun(B,F2),fun(B,D),aa(fun(F2,D),fun(fun(B,F2),fun(B,D)),comp(F2,D,B),G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F2)),fun(product_prod(A,B),product_prod(C,D)),aa(fun(product_prod(E,F2),product_prod(C,D)),fun(fun(product_prod(A,B),product_prod(E,F2)),fun(product_prod(A,B),product_prod(C,D))),comp(product_prod(E,F2),product_prod(C,D),product_prod(A,B)),product_map_prod(E,C,F2,D,F1,G1)),product_map_prod(A,E,B,F2,F22,G22)) ).

% map_prod_compose
tff(fact_4510_prod_Omap__comp,axiom,
    ! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F2),F1: fun(A,C),F22: fun(B,D),V: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,G1,G22),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),V)) = aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G22),F22)),V) ).

% prod.map_comp
tff(fact_4511_prod_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)),T2) = T2 ).

% prod.map_ident
tff(fact_4512_case__prod__map__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: fun(B,fun(C,A)),F: fun(D,B),G: fun(E,C),X: product_prod(D,E)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H),aa(product_prod(D,E),product_prod(B,C),product_map_prod(D,B,E,C,F,G),X)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_case_prod(D,E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_zh(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),H),F),G)),X) ).

% case_prod_map_prod
tff(fact_4513_map__prod_Oidentity,axiom,
    ! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)) = id(product_prod(A,B)) ).

% map_prod.identity
tff(fact_4514_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_zi(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_4515_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_zj(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_4516_apfst__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,C)] : product_apfst(A,C,B,F) = product_map_prod(A,C,B,B,F,id(B)) ).

% apfst_def
tff(fact_4517_prod__fun__imageE,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod(A,B),F: fun(C,A),G: fun(D,B),R4: 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,F,G)),R4)))
     => ~ ! [X4: C,Y4: D] :
            ( ( C2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X4)),aa(D,B,G,Y4)) )
           => ~ 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),X4),Y4)),R4)) ) ) ).

% prod_fun_imageE
tff(fact_4518_apsnd__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F) = product_map_prod(A,A,B,C,id(A),F) ).

% apsnd_def
tff(fact_4519_map__prod__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_zk(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F),G)) ).

% map_prod_def
tff(fact_4520_case__prod__o__map__prod,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,fun(E,C)),G1: fun(A,D),G22: fun(B,E)] : aa(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C),aa(fun(product_prod(D,E),C),fun(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C)),comp(product_prod(D,E),C,product_prod(A,B)),aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F)),product_map_prod(A,D,B,E,G1,G22)) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_zl(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),F),G1),G22)) ).

% case_prod_o_map_prod
tff(fact_4521_acyclicI__order,axiom,
    ! [A: $tType,B: $tType] :
      ( preorder(A)
     => ! [R: set(product_prod(B,B)),F: fun(B,A)] :
          ( ! [A4: B,B3: B] :
              ( 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),A4),B3)),R))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,B3)),aa(B,A,F,A4))) )
         => transitive_acyclic(B,R) ) ) ).

% acyclicI_order
tff(fact_4522_acyclic__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( transitive_acyclic(A,R)
    <=> ! [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,R))) ) ).

% acyclic_def
tff(fact_4523_acyclicI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X4: 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),X4)),transitive_trancl(A,R)))
     => transitive_acyclic(A,R) ) ).

% acyclicI
tff(fact_4524_prod_Osize__gen__o__map,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),nat),aa(fun(product_prod(C,D),nat),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),nat)),comp(product_prod(C,D),nat,product_prod(A,B)),basic_BNF_size_prod(C,D,F,Fa)),product_map_prod(A,C,B,D,G,Ga)) = basic_BNF_size_prod(A,B,aa(fun(A,C),fun(A,nat),aa(fun(C,nat),fun(fun(A,C),fun(A,nat)),comp(C,nat,A),F),G),aa(fun(B,D),fun(B,nat),aa(fun(D,nat),fun(fun(B,D),fun(B,nat)),comp(D,nat,B),Fa),Ga)) ).

% prod.size_gen_o_map
tff(fact_4525_map__prod__surj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(B,A),A5: set(B),A10: set(A),G: fun(D,C),B4: set(D),B12: set(C)] :
      ( ( aa(set(B),set(A),image2(B,A,F),A5) = A10 )
     => ( ( aa(set(D),set(C),image2(D,C,G),B4) = B12 )
       => ( aa(set(product_prod(B,D)),set(product_prod(A,C)),image2(product_prod(B,D),product_prod(A,C),product_map_prod(B,A,D,C,F,G)),product_Sigma(B,D,A5,aTP_Lamp_zm(set(D),fun(B,set(D)),B4))) = product_Sigma(A,C,A10,aTP_Lamp_xn(set(C),fun(A,set(C)),B12)) ) ) ) ).

% map_prod_surj_on
tff(fact_4526_map__prod__inj__on,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(A,B),A5: set(A),G: fun(C,D),B4: set(C)] :
      ( inj_on(A,B,F,A5)
     => ( inj_on(C,D,G,B4)
       => inj_on(product_prod(A,C),product_prod(B,D),product_map_prod(A,B,C,D,F,G),product_Sigma(A,C,A5,aTP_Lamp_xn(set(C),fun(A,set(C)),B4))) ) ) ).

% map_prod_inj_on
tff(fact_4527_map__prod__surj,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,B),G: fun(C,D)] :
      ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
     => ( ( aa(set(C),set(D),image2(C,D,G),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,F,G)),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).

% map_prod_surj
tff(fact_4528_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y3: A] :
          ( ( min_list(A,X) = Y3 )
         => ( ! [X4: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
               => ( Y3 != aa(list(A),A,case_list(A,A,X4,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ze(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs3)),Xs3) ) )
           => ~ ( ( X = nil(A) )
               => ( Y3 != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_4529_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y3: A] :
          ( ( min_list(A,X) = Y3 )
         => ( pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),X))
           => ( ! [X4: A,Xs3: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                 => ( ( Y3 = aa(list(A),A,case_list(A,A,X4,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_ze(A,fun(list(A),fun(A,fun(list(A),A))),X4),Xs3)),Xs3) )
                   => ~ 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),X4),Xs3))) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y3 = undefined(A) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),nil(A))) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_4530_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_cp(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_4531_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y3: A,Xs: list(A)] :
      ( ( ( X = Y3 )
       => ( count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Y3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),count_list(A,Xs,Y3)),one_one(nat)) ) )
      & ( ( X != Y3 )
       => ( count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Y3) = count_list(A,Xs,Y3) ) ) ) ).

% count_list.simps(2)
tff(fact_4532_nth__upt,axiom,
    ! [I: 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),I),K)),J))
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_4533_take__upt,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),N))
     => ( take(nat,M,upt(I,N)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_4534_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
     => ( count_list(A,Xs,X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_4535_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I)) ) ) ).

% upt_eq_Nil_conv
tff(fact_4536_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),J))
     => ( upt(I,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(I,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_4537_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_4538_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)),insert2(fun(A,B),aTP_Lamp_zn(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_4539_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),undefined(A)),aTP_Lamp_ak(A,A)),Option) ).

% option.the_def
tff(fact_4540_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_4541_atLeast__upt,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_4542_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y3: A] : count_list(A,nil(A),Y3) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_4543_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
        & ( I = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_4544_atMost__upto,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_4545_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y3: A] :
          ( ( arg_min_list(A,B,X,Xa) = Y3 )
         => ( ! [X4: A] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
               => ( Y3 != X4 ) )
           => ( ! [X4: A,Y4: A,Zs2: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                 => ( Y3 != aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X4)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))))),X4),arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))) ) )
             => ~ ( ( Xa = nil(A) )
                 => ( Y3 != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_4546_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),X6))
       => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_zo(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_4547_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(list(bool),A,aa(A,fun(list(bool),A),aa(fun(bool,A),fun(A,fun(list(bool),A)),groups4207007520872428315er_sum(bool,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),aa(list(nat),list(bool),map(nat,bool,bit_se5641148757651400278ts_bit(A,A3)),upt(zero_zero(nat),N))) = bit_se2584673776208193580ke_bit(A,N,A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_4548_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)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(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_4549_swap__swap,axiom,
    ! [B: $tType,A: $tType,P2: 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),P2)) = P2 ).

% swap_swap
tff(fact_4550_map__ident,axiom,
    ! [A: $tType,X2: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_ak(A,A)),X2) = X2 ).

% map_ident
tff(fact_4551_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),Xs) ).

% map_map
tff(fact_4552_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),map(B,C,F),aa(list(A),list(B),map(A,B,G),List)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),List) ).

% List.map.compositionality
tff(fact_4553_list_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: list(A)] : aa(list(B),list(C),map(B,C,G),aa(list(A),list(B),map(A,B,F),V)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).

% list.map_comp
tff(fact_4554_swap__simp,axiom,
    ! [A: $tType,B: $tType,X: B,Y3: 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),Y3)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y3),X) ).

% swap_simp
tff(fact_4555_case__swap,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,fun(B,A)),P2: 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_zp(fun(C,fun(B,A)),fun(B,fun(C,A)),F)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P2)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F),P2) ).

% case_swap
tff(fact_4556_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_4557_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_4558_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aTP_Lamp_zq(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_4559_List_Omap_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),list(C)),aa(fun(list(B),list(C)),fun(fun(list(A),list(B)),fun(list(A),list(C))),comp(list(B),list(C),list(A)),map(B,C,F)),map(A,B,G)) = map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).

% List.map.comp
tff(fact_4560_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),aa(fun(list(C),list(B)),fun(fun(list(A),list(C)),fun(list(A),list(B))),comp(list(C),list(B),list(A)),map(C,B,F)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G)) ).

% map_comp_map
tff(fact_4561_size__list__map,axiom,
    ! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F),aa(list(B),list(A),map(B,A,G),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),aa(fun(A,nat),fun(fun(B,A),fun(B,nat)),comp(A,nat,B),F),G)),Xs) ).

% size_list_map
tff(fact_4562_pair__in__swap__image,axiom,
    ! [A: $tType,B: $tType,Y3: 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),Y3),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),Y3)),A5)) ) ).

% pair_in_swap_image
tff(fact_4563_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_4564_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_4565_list_Omap__ident,axiom,
    ! [A: $tType,T2: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_ak(A,A)),T2) = T2 ).

% list.map_ident
tff(fact_4566_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),aa(list(D),list(B),map(D,B,G),Ys)) = 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_xp(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_4567_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_zr(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_4568_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_zs(fun(C,A),fun(C,fun(B,product_prod(A,B))),F))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_4569_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_zt(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_4570_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_zu(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_4571_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H)),zip(B,C,aa(list(D),list(B),map(D,B,F),Xs),aa(list(D),list(C),map(D,C,G),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_zv(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F),G)),Xs) ).

% map2_map_map
tff(fact_4572_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),map(nat,A,F),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_zw(fun(nat,A),fun(nat,product_prod(nat,A)),F)),upt(N,M)) ).

% enumerate_map_upt
tff(fact_4573_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_ak(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_4574_fold__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C)] : fold(B,A,G,aa(list(C),list(B),map(C,B,F),Xs)) = fold(C,A,aa(fun(C,B),fun(C,fun(A,A)),aa(fun(B,fun(A,A)),fun(fun(C,B),fun(C,fun(A,A))),comp(B,fun(A,A),C),G),F),Xs) ).

% fold_map
tff(fact_4575_sum__list__mult__const,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),C2: A,Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_de(fun(B,A),fun(A,fun(B,A)),F),C2)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))),C2) ) ).

% sum_list_mult_const
tff(fact_4576_sum__list__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [C2: A,F: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dd(A,fun(fun(B,A),fun(B,A)),C2),F)),Xs)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))) ) ).

% sum_list_const_mult
tff(fact_4577_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_dh(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_addf
tff(fact_4578_sum__list__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_di(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),Xs)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_subtractf
tff(fact_4579_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),aa(fun(list(B),nat),fun(fun(list(A),list(B)),fun(list(A),nat)),comp(list(B),nat,list(A)),size_list(B,F)),map(A,B,G)) = size_list(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).

% list.size_gen_o_map
tff(fact_4580_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_4581_uminus__sum__list__map,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),uminus_uminus(A)),F)),Xs)) ) ).

% uminus_sum_list_map
tff(fact_4582_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X4)),aa(A,B,G,X4))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,F),Xs))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(A),list(B),map(A,B,G),Xs)))) ) ) ).

% sum_list_mono
tff(fact_4583_map__add__upt,axiom,
    ! [N: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_4584_map__upt__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),N: nat] : aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_zy(fun(nat,A),fun(nat,A),F)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_4585_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R: B,Xs: list(C)] : aa(list(B),B,groups8242544230860333062m_list(B),aa(list(C),list(B),map(C,B,aTP_Lamp_zz(B,fun(C,B),R)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R) ) ).

% sum_list_triv
tff(fact_4586_map__decr__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_dx(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_4587_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_4588_size__list__conv__sum__list,axiom,
    ! [B: $tType,F: fun(B,nat),Xs: list(B)] : aa(list(B),nat,size_list(B,F),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(B),list(nat),map(B,nat,F),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_4589_sum__list__Suc,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,aTP_Lamp_ja(fun(A,nat),fun(A,nat),F)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_4590_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),X: A,Y3: A,Zs: list(A)] : arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs))) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,F,arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs))))),X),arg_min_list(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_4591_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_nh(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),A5)) ).

% product_swap
tff(fact_4592_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M: nat,F: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M)))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F),upt(M,N))),I) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_4593_prod_Oswap__def,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P2) = 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),P2)),aa(product_prod(A,B),A,product_fst(A,B),P2)) ).

% prod.swap_def
tff(fact_4594_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M) )
     => ( ! [I2: nat] :
            ( 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,Xs),I2) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F),upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_4595_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] : aa(list(nat),nat,groups8242544230860333062m_list(nat),aa(list(A),list(nat),map(A,nat,F),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(list(A),fun(A,nat),aTP_Lamp_aaa(fun(A,nat),fun(list(A),fun(A,nat)),F),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_4596_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( 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(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = N ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_aac(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_4597_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F: fun(nat,B),Ns: list(nat)] :
          ( ! [X4: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F,X4)),aa(nat,B,F,Y4))) )
         => ( 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),F),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(B),B,groups8242544230860333062m_list(B),aa(list(nat),list(B),map(nat,B,F),Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_4598_map__prod__o__convol__id,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),X: C] : aa(C,product_prod(A,B),aa(fun(C,product_prod(C,B)),fun(C,product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(fun(C,product_prod(C,B)),fun(C,product_prod(A,B))),comp(product_prod(C,B),product_prod(A,B),C),product_map_prod(C,A,B,B,F,id(B))),aa(fun(C,B),fun(C,product_prod(C,B)),aa(fun(C,C),fun(fun(C,B),fun(C,product_prod(C,B))),bNF_convol(C,C,B),id(C)),G)),X) = aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aa(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),bNF_convol(C,A,B),aa(fun(C,A),fun(C,A),aa(fun(A,A),fun(fun(C,A),fun(C,A)),comp(A,A,C),id(A)),F)),G),X) ).

% map_prod_o_convol_id
tff(fact_4599_map__rec,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),Xs) = aa(list(B),list(A),rec_list(list(A),B,nil(A),aTP_Lamp_aad(fun(B,A),fun(B,fun(list(B),fun(list(A),list(A)))),F)),Xs) ).

% map_rec
tff(fact_4600_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),F: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R4,aa(list(B),list(A),map(B,A,F),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_aae(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R4),F),Xs) ) ).

% sorted_wrt_map
tff(fact_4601_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_aaf(A,fun(A,bool)),Xs) ).

% sorted_wrt_true
tff(fact_4602_convol__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C),X2: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),F),G),X2) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,F,X2)),aa(A,C,G,X2)) ).

% convol_def
tff(fact_4603_fst__convol_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),X: C] : aa(product_prod(A,B),A,product_fst(A,B),aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aa(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),bNF_convol(C,A,B),F),G),X)) = aa(C,A,F,X) ).

% fst_convol'
tff(fact_4604_snd__convol_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(C,B),G: fun(C,A),X: C] : aa(product_prod(B,A),A,product_snd(B,A),aa(C,product_prod(B,A),aa(fun(C,A),fun(C,product_prod(B,A)),aa(fun(C,B),fun(fun(C,A),fun(C,product_prod(B,A))),bNF_convol(C,B,A),F),G),X)) = aa(C,A,G,X) ).

% snd_convol'
tff(fact_4605_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F),Xs))
        <=> sorted_wrt(B,aTP_Lamp_aag(fun(B,A),fun(B,fun(B,bool)),F),Xs) ) ) ).

% sorted_map
tff(fact_4606_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_dm(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_4607_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_4608_rec__list__Cons__imp,axiom,
    ! [B: $tType,A: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X: A,Xs: list(A)] :
      ( ( F = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X),Xs),aa(list(A),B,F,Xs)) ) ) ).

% rec_list_Cons_imp
tff(fact_4609_rec__list__Nil__imp,axiom,
    ! [A: $tType,B: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
      ( ( F = rec_list(B,A,F1,F22) )
     => ( aa(list(A),B,F,nil(A)) = F1 ) ) ).

% rec_list_Nil_imp
tff(fact_4610_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_aai(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_4611_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_aaj(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_4612_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_fx(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_4613_fst__convol,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] : aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),F),G)) = F ).

% fst_convol
tff(fact_4614_snd__convol,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(A,B)] : aa(fun(A,product_prod(C,B)),fun(A,B),aa(fun(product_prod(C,B),B),fun(fun(A,product_prod(C,B)),fun(A,B)),comp(product_prod(C,B),B,A),product_snd(C,B)),aa(fun(A,B),fun(A,product_prod(C,B)),aa(fun(A,C),fun(fun(A,B),fun(A,product_prod(C,B))),bNF_convol(A,C,B),F),G)) = G ).

% snd_convol
tff(fact_4615_convol__o,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F: fun(D,B),G: fun(D,C),H: fun(A,D)] : aa(fun(A,D),fun(A,product_prod(B,C)),aa(fun(D,product_prod(B,C)),fun(fun(A,D),fun(A,product_prod(B,C))),comp(D,product_prod(B,C),A),aa(fun(D,C),fun(D,product_prod(B,C)),aa(fun(D,B),fun(fun(D,C),fun(D,product_prod(B,C))),bNF_convol(D,B,C),F),G)),H) = aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),H)),aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),G),H)) ).

% convol_o
tff(fact_4616_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_4617_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_dm(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_4618_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_4619_convol__expand__snd_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B),H: fun(A,C)] :
      ( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
     => ( ( H = aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F) )
      <=> ( aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),G),H) = F ) ) ) ).

% convol_expand_snd'
tff(fact_4620_convol__expand__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B)] :
      ( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
     => ( aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),G),aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)) = F ) ) ).

% convol_expand_snd
tff(fact_4621_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X2: A] :
                ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P,X2)) )
           => ( 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_aak(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_4622_list_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,fun(list(B),fun(C,C))),F: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),C),aa(fun(list(B),C),fun(fun(list(A),list(B)),fun(list(A),C)),comp(list(B),C,list(A)),rec_list(C,B,G,Ga)),map(A,B,F)) = rec_list(C,A,G,aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_aal(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F)) ).

% list.rec_o_map
tff(fact_4623_map__prod__o__convol,axiom,
    ! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: fun(D,B),H22: fun(E,C),F: fun(A,D),G: fun(A,E)] : aa(fun(A,product_prod(D,E)),fun(A,product_prod(B,C)),aa(fun(product_prod(D,E),product_prod(B,C)),fun(fun(A,product_prod(D,E)),fun(A,product_prod(B,C))),comp(product_prod(D,E),product_prod(B,C),A),product_map_prod(D,B,E,C,H1,H22)),aa(fun(A,E),fun(A,product_prod(D,E)),aa(fun(A,D),fun(fun(A,E),fun(A,product_prod(D,E))),bNF_convol(A,D,E),F),G)) = aa(fun(A,C),fun(A,product_prod(B,C)),aa(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),bNF_convol(A,B,C),aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),H1),F)),aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),H22),G)) ).

% map_prod_o_convol
tff(fact_4624_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_4625_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_aan(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_4626_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_4627_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_4628_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_aao(fun(B,A),fun(B,list(A)),F)),Xs)) = aa(list(B),list(A),map(B,A,F),Xs) ).

% concat_map_singleton
tff(fact_4629_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_aap(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_4630_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,member(A,Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4)) ) ) )
     => ( fold(A,B,F,rev(A,Xs)) = fold(A,B,F,Xs) ) ) ).

% fold_rev
tff(fact_4631_fold__plus__sum__list__rev,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : fold(A,A,plus_plus(A),Xs) = aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),rev(A,Xs))) ) ).

% fold_plus_sum_list_rev
tff(fact_4632_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_aaq(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_aar(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_4633_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y3: list(list(A))] :
      ( ( transpose(A,X) = Y3 )
     => ( ( ( X = nil(list(A)) )
         => ( Y3 != 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) )
             => ( Y3 != transpose(A,Xss2) ) )
         => ~ ! [X4: A,Xs3: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3)),Xss2) )
               => ( Y3 != 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),X4),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_aaq(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs3),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aar(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_4634_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aas(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_4635_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y3: 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,Y3)) = 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)),Y3) ) ) ).

% rev_update
tff(fact_4636_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_aat(A,fun(list(A),fun(set(A),set(A))))),Xs) ).

% set_rec
tff(fact_4637_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_aas(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_4638_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_aaq(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_aar(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_4639_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),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))),filter2(list(A),aTP_Lamp_aau(nat,fun(list(A),bool),I),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_4640_filter__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : filter2(A,P,filter2(A,Q,Xs)) = filter2(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aav(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),Xs) ).

% filter_filter
tff(fact_4641_set__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),filter2(A,P,Xs)) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_aaw(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).

% set_filter
tff(fact_4642_partition__filter__conv,axiom,
    ! [A: $tType,F: fun(A,bool),Xs: list(A)] : aa(list(A),product_prod(list(A),list(A)),partition(A,F),Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,F,Xs)),filter2(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),F),Xs)) ).

% partition_filter_conv
tff(fact_4643_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),filter2(A,P,aa(list(B),list(A),map(B,A,F),Xs))) = aa(list(B),nat,size_size(list(B)),filter2(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P),F),Xs)) ).

% length_filter_map
tff(fact_4644_partition__filter2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),partition(A,P),Xs)) = filter2(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs) ).

% partition_filter2
tff(fact_4645_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,filter2(A,P,Xs),filter2(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P),Xs)))) ).

% partition_in_shuffles
tff(fact_4646_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : filter2(A,P,aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),filter2(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P),F),Xs)) ).

% filter_map
tff(fact_4647_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)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_4648_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),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5),Xs)) ).

% inter_set_filter
tff(fact_4649_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_aax(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs),Xs)) ) ).

% sorted_same
tff(fact_4650_fold__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : fold(B,A,F,filter2(B,P,Xs)) = fold(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_aay(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F),P),Xs) ).

% fold_filter
tff(fact_4651_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y3: A,Xs: list(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert2(A,Y3),aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,bool),aTP_Lamp_aaz(fun(A,B),fun(A,fun(A,bool)),F),Y3),Xs) = filter2(A,aa(A,fun(A,bool),fequal(A),Y3),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_4652_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F),filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_aba(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F),G),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_4653_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_abb(fun(B,A),fun(fun(B,bool),fun(B,A)),F),P)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4654_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))) ) )
         => ( ! [X4: A,Xs3: 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),X4),Xs3)),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)),Xs3),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aar(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),X4),Xs3)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_4655_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S: set(nat)] :
      ( distinct(A,Xs)
     => ( filter2(A,aa(set(nat),fun(A,bool),aTP_Lamp_abc(list(A),fun(set(nat),fun(A,bool)),Xs),S),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_4656_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P: fun(B,bool),F: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P,X4))
               => ( aa(B,A,F,X4) = zero_zero(A) ) ) )
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),filter2(B,P,Xs))) = aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_4657_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y3: 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),insert2(A,Y3),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_abd(A,fun(A,bool)),Y3),Xs)) ).

% set_minus_filter_out
tff(fact_4658_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys)))
       => ( filter2(A,aTP_Lamp_abe(list(A),fun(A,bool),Xs),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4659_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys)))
       => ( filter2(A,aTP_Lamp_abf(list(A),fun(A,bool),Xs),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4660_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys)))
       => ( filter2(A,aTP_Lamp_abe(list(A),fun(A,bool),Ys),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4661_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,member(list(A),Zs),shuffles(A,Xs,Ys)))
       => ( filter2(A,aTP_Lamp_abf(list(A),fun(A,bool),Ys),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4662_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: 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_abg(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% filter_eq_nths
tff(fact_4663_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,bool),set(nat),collect(nat),aa(list(A),fun(nat,bool),aTP_Lamp_abg(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_4664_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(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_4665_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_xa(set(B),fun(A,set(B)),B4)) ).

% Product_Type.product_def
tff(fact_4666_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_xa(set(B),fun(A,set(B)),B4)))) ) ).

% member_product
tff(fact_4667_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_abh(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_4668_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_abi(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_aau(nat,fun(list(A),bool),I),Xs)) ) ) ).

% nth_transpose
tff(fact_4669_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_aau(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_4670_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_abi(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_aau(nat,fun(list(A),bool),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_4671_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),aa(fun(option(B),B),fun(fun(A,option(B)),fun(A,B)),comp(option(B),B,A),the2(B)),F)),filter2(A,aTP_Lamp_abj(fun(A,option(B)),fun(A,bool),F),Xs)) ).

% map_filter_def
tff(fact_4672_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_abk(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_4673_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_aar(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),filter2(list(A),aTP_Lamp_abl(list(A),bool),Xss)) ).

% transpose_aux_filter_tail
tff(fact_4674_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_4675_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_abm(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_4676_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_abn(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_4677_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(option(A),list(A),aa(fun(A,list(A)),fun(option(A),list(A)),aa(list(A),fun(fun(A,list(A)),fun(option(A),list(A))),case_option(list(A),A),map_filter(B,A,F,Xs)),aa(list(B),fun(A,list(A)),aTP_Lamp_abo(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs)),aa(B,option(A),F,X)) ).

% map_filter_simps(1)
tff(fact_4678_tl__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),take(A,N,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_4679_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))) ) ) ) ).

% Nitpick.size_list_simp(2)
tff(fact_4680_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)),filter2(product_prod(A,nat),aTP_Lamp_abp(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)),filter2(product_prod(A,nat),aTP_Lamp_abq(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_4681_nths__shift__lemma,axiom,
    ! [A: $tType,A5: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_abr(set(nat),fun(product_prod(A,nat),bool),A5),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_abs(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A5),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_4682_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)),filter2(product_prod(A,nat),aTP_Lamp_abr(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_4683_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),filter2(B,P,Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_abt(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F),P),Xs) ).

% map_filter_map_filter
tff(fact_4684_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_abu(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_abl(list(A),bool),Xs)) ).

% transpose_max_length
tff(fact_4685_length__remdups__concat,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),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_4686_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_abv(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_abw(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_abx(list(B),bool),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4687_foldr__Cons,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X: A,Xs: list(A)] : foldr(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),foldr(A,B,F,Xs)) ).

% foldr_Cons
tff(fact_4688_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C),A3: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F),Xs)),A3) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),aa(fun(B,fun(A,A)),fun(fun(C,B),fun(C,fun(A,A))),comp(B,fun(A,A),C),G),F),Xs),A3) ).

% foldr_map
tff(fact_4689_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : foldr(B,A,F,filter2(B,P,Xs)) = foldr(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_aay(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F),P),Xs) ).

% foldr_filter
tff(fact_4690_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_4691_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,member(A,Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4)) ) ) )
     => ( foldr(A,B,F,Xs) = fold(A,B,F,Xs) ) ) ).

% foldr_fold
tff(fact_4692_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F),A3),Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_aby(fun(B,A),fun(A,fun(B,fun(A,A))),F),A3),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_4693_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_abu(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_4694_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y3: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( ( ( Xs = nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y3) = Y3 ) )
            & ( ( Xs != nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y3) ) ) ) ) ) ).

% foldr_max_sorted
tff(fact_4695_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_abl(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_4696_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_4697_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_aaq(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),filter2(list(A),aTP_Lamp_abl(list(A),bool),Xss)) ).

% transpose_aux_filter_head
tff(fact_4698_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_4699_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),Xs: list(A),Ys: list(B)] : zip(A,B,takeWhile(A,P,Xs),Ys) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),bool),aa(fun(A,bool),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),bool)),comp(A,bool,product_prod(A,B)),P),product_fst(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_fst
tff(fact_4700_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),P: fun(B,bool),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),bool),aa(fun(B,bool),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),bool)),comp(B,bool,product_prod(A,B)),P),product_snd(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_snd
tff(fact_4701_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_4702_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : takeWhile(A,P,aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),takeWhile(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P),F),Xs)) ).

% takeWhile_map
tff(fact_4703_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_abz(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_4704_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_4705_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_list(A,F),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_list(A,F),Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F),aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).

% Nitpick.size_list_simp(1)
tff(fact_4706_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F),Xs)))
         => ( filter2(B,aa(A,fun(B,bool),aTP_Lamp_aca(fun(B,A),fun(A,fun(B,bool)),F),T2),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_aca(fun(B,A),fun(A,fun(B,bool)),F),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_4707_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_acb(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),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs)) ).

% extract_def
tff(fact_4708_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X: A,Y3: 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),Y3) )
      <=> 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)),Xys))) ) ) ).

% map_of_eq_Some_iff
tff(fact_4709_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y3: 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),Y3) = 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),Y3)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))) ) ) ).

% Some_eq_map_of_iff
tff(fact_4710_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X3: A] : aa(A,option(B),map_of(A,B,Xys),X3) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_4711_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_acc(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_4712_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = none(B) )
      <=> ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% map_of_zip_is_None
tff(fact_4713_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X: A,Y3: 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),Y3)),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),Y3) ) ) ) ).

% map_of_is_SomeI
tff(fact_4714_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),Xs: list(B)] : dropWhile(A,P,aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),dropWhile(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P),F),Xs)) ).

% dropWhile_map
tff(fact_4715_map__of__Cons__code_I1_J,axiom,
    ! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).

% map_of_Cons_code(1)
tff(fact_4716_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X2: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X2) = none(B) ).

% map_of.simps(1)
tff(fact_4717_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_kn(A,fun(A,bool),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_4718_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)))
     => ? [X4: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X4) ) ).

% weak_map_of_SomeI
tff(fact_4719_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y3: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y3) )
     => 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),Y3)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ) ).

% map_of_SomeD
tff(fact_4720_map__of__Cons__code_I2_J,axiom,
    ! [C: $tType,B: $tType,L: B,K: B,V: 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),V)),Ps)),K) = aa(C,option(C),some(C),V) ) )
      & ( ( 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),V)),Ps)),K) = aa(B,option(C),map_of(B,C,Ps),K) ) ) ) ).

% map_of_Cons_code(2)
tff(fact_4721_map__of__filter__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z: A,P: fun(B,fun(A,bool))] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z) )
     => ( pp(aa(A,bool,aa(B,fun(A,bool),P,K),Z))
       => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P),Xs)),K) = aa(A,option(A),some(A),Z) ) ) ) ).

% map_of_filter_in
tff(fact_4722_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = aa(B,option(B),some(B),Y) ) ) ).

% map_of_zip_is_Some
tff(fact_4723_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),X) = none(A) )
    <=> ~ pp(aa(set(B),bool,member(B,X),aa(set(product_prod(B,A)),set(B),image2(product_prod(B,A),B,product_fst(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xys)))) ) ).

% map_of_eq_None_iff
tff(fact_4724_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y3: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_kn(A,fun(A,bool),Y3),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_4725_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_acd(list(A),fun(A,bool),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_4726_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F: fun(A,B),X2: A] :
      ( ( pp(aa(set(A),bool,member(A,X2),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,F),Xs))),X2) = aa(B,option(B),some(B),aa(A,B,F,X2)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X2),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,F),Xs))),X2) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_4727_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_4728_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_ace(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P),Xs)) ).

% find_dropWhile
tff(fact_4729_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_abd(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_abd(A,fun(A,bool)),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_4730_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_abd(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_abd(A,fun(A,bool)),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_4731_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_4732_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_acf(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).

% set_map_of_compr
tff(fact_4733_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),T2: list(product_prod(A,C)),K: A,X: C] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T2),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_acg(fun(A,B),fun(A,fun(C,product_prod(B,C))),F))),T2)),aa(A,B,F,K)) = aa(C,option(C),some(C),X) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_4734_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_wl(fun(A,B),fun(A,product_prod(A,B)),F)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),aa(fun(B,option(B)),fun(fun(A,B),fun(A,option(B))),comp(B,option(B),A),some(B)),F),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_4735_convol__image__vimage2p,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,F: fun(C,A),G: fun(D,B),R4: fun(A,fun(B,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))),aa(set(product_prod(C,D)),set(product_prod(A,B)),image2(product_prod(C,D),product_prod(A,B),aa(fun(product_prod(C,D),B),fun(product_prod(C,D),product_prod(A,B)),aa(fun(product_prod(C,D),A),fun(fun(product_prod(C,D),B),fun(product_prod(C,D),product_prod(A,B))),bNF_convol(product_prod(C,D),A,B),aa(fun(product_prod(C,D),C),fun(product_prod(C,D),A),aa(fun(C,A),fun(fun(product_prod(C,D),C),fun(product_prod(C,D),A)),comp(C,A,product_prod(C,D)),F),product_fst(C,D))),aa(fun(product_prod(C,D),D),fun(product_prod(C,D),B),aa(fun(D,B),fun(fun(product_prod(C,D),D),fun(product_prod(C,D),B)),comp(D,B,product_prod(C,D)),G),product_snd(C,D)))),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),aa(fun(A,fun(B,bool)),fun(C,fun(D,bool)),bNF_vimage2p(C,A,D,B,bool,F,G),R4))))),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),R4)))) ).

% convol_image_vimage2p
tff(fact_4736_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A3: A] : enumerate(A,N,replicate(A,M,A3)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ach(A,fun(nat,product_prod(nat,A)),A3)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_4737_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat,Y3: A] :
      ( ( replicate(A,M,X) = replicate(A,N,Y3) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( X = Y3 ) ) ) ) ).

% replicate_eq_replicate
tff(fact_4738_restrict__out,axiom,
    ! [A: $tType,B: $tType,X: A,A5: set(A),M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,member(A,X),A5))
     => ( aa(A,option(B),restrict_map(A,B,M,A5),X) = none(B) ) ) ).

% restrict_out
tff(fact_4739_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D5: set(A),X2: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_acc(A,option(B)),D5),X2) = none(B) ).

% restrict_map_empty
tff(fact_4740_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4741_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4742_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
         => pp(aa(A,bool,P,X3)) )
    <=> ( pp(aa(A,bool,P,A3))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_4743_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(list(A),set(A),set2(A),replicate(A,N,A3))))
          & pp(aa(A,bool,P,X3)) )
    <=> ( pp(aa(A,bool,P,A3))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_4744_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y3: A] :
      ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),replicate(A,N,Y3))))
    <=> ( ( X = Y3 )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_4745_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X2: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X2) = none(B) ).

% restrict_map_to_empty
tff(fact_4746_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,N,X)) = X ) ) ).

% hd_replicate
tff(fact_4747_tl__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),tl(A),replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ).

% tl_replicate
tff(fact_4748_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y3: B] : zip(A,B,replicate(A,I,X),replicate(B,J,Y3)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) ).

% zip_replicate
tff(fact_4749_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_4750_vimage2p__mono,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,D),R4: fun(B,fun(D,bool)),X: A,Y3: C,S2: fun(B,fun(D,bool))] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(D,bool)),fun(A,fun(C,bool)),bNF_vimage2p(A,B,C,D,bool,F,G),R4),X),Y3))
     => ( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(B,fun(D,bool)),fun(fun(B,fun(D,bool)),bool),ord_less_eq(fun(B,fun(D,bool))),R4),S2))
       => pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(D,bool)),fun(A,fun(C,bool)),bNF_vimage2p(A,B,C,D,bool,F,G),S2),X),Y3)) ) ) ).

% vimage2p_mono
tff(fact_4751_vimage2p__rel__fun,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D),R4: fun(C,fun(D,bool))] : pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,aa(fun(C,fun(D,bool)),fun(A,fun(B,bool)),bNF_vimage2p(A,C,B,D,bool,F,G),R4),R4),F),G)) ).

% vimage2p_rel_fun
tff(fact_4752_vimage2p__refl,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),F: fun(B,A),X: B] :
      ( ! [X4: A] : pp(aa(A,bool,aa(A,fun(A,bool),R4,X4),X4))
     => pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(A,fun(A,bool)),fun(B,fun(B,bool)),bNF_vimage2p(B,A,B,A,bool,F,F),R4),X),X)) ) ).

% vimage2p_refl
tff(fact_4753_vimage2p__def,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(A,D),G: fun(B,E),R4: fun(D,fun(E,C)),X2: A,Xa3: B] : aa(B,C,aa(A,fun(B,C),aa(fun(D,fun(E,C)),fun(A,fun(B,C)),bNF_vimage2p(A,D,B,E,C,F,G),R4),X2),Xa3) = aa(E,C,aa(D,fun(E,C),R4,aa(A,D,F,X2)),aa(B,E,G,Xa3)) ).

% vimage2p_def
tff(fact_4754_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),M: fun(A,option(B)),X2: A] :
      ( ( pp(aa(set(A),bool,member(A,X2),A5))
       => ( aa(A,option(B),restrict_map(A,B,M,A5),X2) = aa(A,option(B),M,X2) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X2),A5))
       => ( aa(A,option(B),restrict_map(A,B,M,A5),X2) = none(B) ) ) ) ).

% restrict_map_def
tff(fact_4755_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_4756_replicate__add,axiom,
    ! [A: $tType,N: nat,M: nat,X: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),replicate(A,M,X)) ).

% replicate_add
tff(fact_4757_vimage2p__comp,axiom,
    ! [E: $tType,D: $tType,F2: $tType,A: $tType,C: $tType,B: $tType,G2: $tType,F1: fun(F2,A),F22: fun(D,F2),G1: fun(G2,B),G22: fun(E,G2)] : bNF_vimage2p(D,A,E,B,C,aa(fun(D,F2),fun(D,A),aa(fun(F2,A),fun(fun(D,F2),fun(D,A)),comp(F2,A,D),F1),F22),aa(fun(E,G2),fun(E,B),aa(fun(G2,B),fun(fun(E,G2),fun(E,B)),comp(G2,B,E),G1),G22)) = aa(fun(fun(A,fun(B,C)),fun(F2,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C))),aa(fun(fun(F2,fun(G2,C)),fun(D,fun(E,C))),fun(fun(fun(A,fun(B,C)),fun(F2,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C)))),comp(fun(F2,fun(G2,C)),fun(D,fun(E,C)),fun(A,fun(B,C))),bNF_vimage2p(D,F2,E,G2,C,F22,G22)),bNF_vimage2p(F2,A,G2,B,C,F1,G1)) ).

% vimage2p_comp
tff(fact_4758_vimage2p__id,axiom,
    ! [C: $tType,B: $tType,A: $tType,R4: fun(A,fun(B,C))] : aa(fun(A,fun(B,C)),fun(A,fun(B,C)),bNF_vimage2p(A,A,B,B,C,id(A),id(B)),R4) = R4 ).

% vimage2p_id
tff(fact_4759_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_cu(A,fun(B,A)),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_4760_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs)),X) = filter2(A,aa(A,fun(A,bool),fequal(A),X),Xs) ).

% replicate_length_filter
tff(fact_4761_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_aci(A,fun(nat,A),X)),upt(zero_zero(nat),I)) = replicate(A,I,X) ).

% map_replicate_trivial
tff(fact_4762_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_4763_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_4764_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N,X),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X)),take(B,N,Ys)) ).

% zip_replicate1
tff(fact_4765_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y3: B] : zip(A,B,Xs,replicate(B,N,Y3)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_fx(B,fun(A,product_prod(A,B))),Y3)),take(A,N,Xs)) ).

% zip_replicate2
tff(fact_4766_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y3: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y3) )
    <=> ( ( X = Y3 )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_4767_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N2))
              & ( concat(A,replicate(list(A),N2,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4768_remdups__adj__singleton__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( Xs != nil(A) )
        & ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),aa(list(A),A,hd(A),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_4769_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),A3)) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(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_4770_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R4: 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,R4),A3),B2))
    <=> ? [Z5: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,member(list(product_prod(A,B)),Z5),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_acj(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R4))))
          & ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z5) = A3 )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z5) = B2 ) ) ) ).

% list.in_rel
tff(fact_4771_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)
        <=> ! [F6: fun(B,A),Xs4: list(B),K3: A] : filter2(B,aa(A,fun(B,bool),aTP_Lamp_ack(fun(B,A),fun(A,fun(B,bool)),F6),K3),aa(list(B),list(B),aa(fun(B,A),fun(list(B),list(B)),Sk,F6),Xs4)) = filter2(B,aa(A,fun(B,bool),aTP_Lamp_ack(fun(B,A),fun(A,fun(B,bool)),F6),K3),Xs4) ) ) ).

% stable_sort_key_def
tff(fact_4772_comp__fun__commute__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite6289374366891150609ommute(A,list(A),linorder_insort_key(A,A,aTP_Lamp_vw(A,A))) ) ).

% comp_fun_commute_insort
tff(fact_4773_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y3: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),Y3),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),Y3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_4774_list_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R4: 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,R4),fequal(bool)),aTP_Lamp_acl(list(A),bool)),aTP_Lamp_acm(list(B),bool))) ).

% list.disc_transfer(1)
tff(fact_4775_list_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: 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,R4),fequal(bool)),aTP_Lamp_abl(list(A),bool)),aTP_Lamp_abx(list(B),bool))) ).

% list.disc_transfer(2)
tff(fact_4776_list_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G: fun(B,C),Y3: 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,G),Y3)))
    <=> 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_st(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y3)) ) ).

% list.rel_map(2)
tff(fact_4777_list_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: list(A),Y3: 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,I),X)),Y3))
    <=> 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_su(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y3)) ) ).

% list.rel_map(1)
tff(fact_4778_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F: fun(C,A),As3: 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,F),As3)),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_acn(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F)),As3),Bs)) ) ).

% list_all2_map1
tff(fact_4779_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As3: list(A),F: fun(C,B),Bs: list(C)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),As3),aa(list(C),list(B),map(C,B,F),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_aco(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F)),As3),Bs)) ) ).

% list_all2_map2
tff(fact_4780_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y3: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(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_vw(A,A)),Y3)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(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_vw(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),Y3)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_4781_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_vw(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_4782_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_acq(list(list(A)),fun(list(A),bool),Xss)) ).

% product_lists_set
tff(fact_4783_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A5: fun(A,fun(B,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(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),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B))) ) ) ) ).

% sum_list_transfer
tff(fact_4784_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_4785_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),insert2(A,X),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(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),insert2(A,X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4786_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A3: B,Xs: list(B),F: 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,F),Xs))
           => ( ( aa(list(B),B,hd(B),filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_acr(B,fun(fun(B,A),fun(B,bool)),A3),F),Xs)) = A3 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F),A3),remove1(B,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_4787_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_4788_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aa(list(A),A,groups5270119922927024881d_list(A),nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_4789_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),insert2(A,X),A5)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A5)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4790_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( aa(list(A),A,groups5270119922927024881d_list(A),Xs) = zero_zero(A) )
        <=> pp(aa(set(A),bool,member(A,zero_zero(A)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% prod_list_zero_iff
tff(fact_4791_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_vw(A,A)),nil(A),A5) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_4792_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B)),X: A] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,member(A,Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) ) ) )
     => ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( fold(A,B,F,Xs) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,remove1(A,X,Xs))),aa(A,fun(B,B),F,X)) ) ) ) ).

% fold_remove1_split
tff(fact_4793_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_4794_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% length_remove1
tff(fact_4795_sum__list__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [X: B,Xs: list(B),F: fun(B,A)] :
          ( pp(aa(set(B),bool,member(B,X),aa(list(B),set(B),set2(B),Xs)))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,X)),aa(list(A),A,groups8242544230860333062m_list(A),aa(list(B),list(A),map(B,A,F),remove1(B,X,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_4796_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_vw(A,A)),A3),remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_4797_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_vw(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),insert2(A,X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4798_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_vw(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_4799_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I))))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4800_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),X: A,Y3: B] : restrict_map(A,B,fun_upd(A,option(B),M,X,aa(B,option(B),some(B),Y3)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_4801_fun__upd__upd,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),X: A,Y3: B,Z: B] : fun_upd(A,B,fun_upd(A,B,F,X,Y3),X,Z) = fun_upd(A,B,F,X,Z) ).

% fun_upd_upd
tff(fact_4802_fun__upd__triv,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: A] : fun_upd(A,B,F,X,aa(A,B,F,X)) = F ).

% fun_upd_triv
tff(fact_4803_fun__upd__apply,axiom,
    ! [A: $tType,B: $tType,Z: B,X: B,F: fun(B,A),Y3: A] :
      ( ( ( Z = X )
       => ( aa(B,A,fun_upd(B,A,F,X,Y3),Z) = Y3 ) )
      & ( ( Z != X )
       => ( aa(B,A,fun_upd(B,A,F,X,Y3),Z) = aa(B,A,F,Z) ) ) ) ).

% fun_upd_apply
tff(fact_4804_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X2: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_acc(A,option(B)),X,none(B)),X2) = none(B) ).

% empty_upd_none
tff(fact_4805_image__map__upd,axiom,
    ! [B: $tType,A: $tType,X: A,A5: set(A),M: fun(A,option(B)),Y3: 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),Y3))),A5) = aa(set(A),set(option(B)),image2(A,option(B),M),A5) ) ) ).

% image_map_upd
tff(fact_4806_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,X: A,D5: set(A),M: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,member(A,X),D5))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D5),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)),D5),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X),D5))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D5),X,none(B)) = restrict_map(A,B,M,D5) ) ) ) ).

% fun_upd_None_restrict
tff(fact_4807_map__upd__Some__unfold,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),A3: B,B2: A,X: B,Y3: 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),Y3) )
    <=> ( ( ( X = A3 )
          & ( B2 = Y3 ) )
        | ( ( X != A3 )
          & ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y3) ) ) ) ) ).

% map_upd_Some_unfold
tff(fact_4808_map__upd__triv,axiom,
    ! [A: $tType,B: $tType,T2: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),T2,K) = aa(A,option(A),some(A),X) )
     => ( fun_upd(B,option(A),T2,K,aa(A,option(A),some(A),X)) = T2 ) ) ).

% map_upd_triv
tff(fact_4809_map__upd__eqD1,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,X: B,N: fun(A,option(B)),Y3: B] :
      ( ( fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),X)) = fun_upd(A,option(B),N,A3,aa(B,option(B),some(B),Y3)) )
     => ( X = Y3 ) ) ).

% map_upd_eqD1
tff(fact_4810_fun__upd__def,axiom,
    ! [B: $tType,A: $tType,A3: A,F: fun(A,B),B2: B,X2: A] :
      ( ( ( X2 = A3 )
       => ( aa(A,B,fun_upd(A,B,F,A3,B2),X2) = B2 ) )
      & ( ( X2 != A3 )
       => ( aa(A,B,fun_upd(A,B,F,A3,B2),X2) = aa(A,B,F,X2) ) ) ) ).

% fun_upd_def
tff(fact_4811_fun__upd__eqD,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),X: A,Y3: B,G: fun(A,B),Z: B] :
      ( ( fun_upd(A,B,F,X,Y3) = fun_upd(A,B,G,X,Z) )
     => ( Y3 = Z ) ) ).

% fun_upd_eqD
tff(fact_4812_fun__upd__idem,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: B,Y3: A] :
      ( ( aa(B,A,F,X) = Y3 )
     => ( fun_upd(B,A,F,X,Y3) = F ) ) ).

% fun_upd_idem
tff(fact_4813_fun__upd__same,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),X: B,Y3: A] : aa(B,A,fun_upd(B,A,F,X,Y3),X) = Y3 ).

% fun_upd_same
tff(fact_4814_fun__upd__other,axiom,
    ! [B: $tType,A: $tType,Z: A,X: A,F: fun(A,B),Y3: B] :
      ( ( Z != X )
     => ( aa(A,B,fun_upd(A,B,F,X,Y3),Z) = aa(A,B,F,Z) ) ) ).

% fun_upd_other
tff(fact_4815_fun__upd__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,C2: A,M: fun(A,B),B2: B,D3: B] :
      ( ( A3 != C2 )
     => ( fun_upd(A,B,fun_upd(A,B,M,A3,B2),C2,D3) = fun_upd(A,B,fun_upd(A,B,M,C2,D3),A3,B2) ) ) ).

% fun_upd_twist
tff(fact_4816_fun__upd__idem__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),X: A,Y3: B] :
      ( ( fun_upd(A,B,F,X,Y3) = F )
    <=> ( aa(A,B,F,X) = Y3 ) ) ).

% fun_upd_idem_iff
tff(fact_4817_fun__upd__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(A,C),X: A,Y3: C] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),fun_upd(A,C,G,X,Y3)) = fun_upd(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G),X,aa(C,B,F,Y3)) ).

% fun_upd_comp
tff(fact_4818_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F: 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_acs(fun(A,B),fun(B,fun(A,bool)),F),C2))))
     => ( pp(aa(fun(A,B),bool,P,aTP_Lamp_mj(B,fun(A,B),C2)))
       => ( ! [A4: A,B3: B,F3: 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_act(B,fun(fun(A,B),fun(A,bool)),C2),F3))))
             => ( ( aa(A,B,F3,A4) = C2 )
               => ( ( B3 != C2 )
                 => ( pp(aa(fun(A,B),bool,P,F3))
                   => pp(aa(fun(A,B),bool,P,fun_upd(A,B,F3,A4,B3))) ) ) ) )
         => pp(aa(fun(A,B),bool,P,F)) ) ) ) ).

% finite_update_induct
tff(fact_4819_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,option(B)),K: A,X: B] :
      ~ ! [X4: A] : aa(A,option(B),fun_upd(A,option(B),T2,K,aa(B,option(B),some(B),X)),X4) = none(B) ).

% map_upd_nonempty
tff(fact_4820_inj__on__fun__updI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),Y3: B,X: A] :
      ( inj_on(A,B,F,A5)
     => ( ~ pp(aa(set(B),bool,member(B,Y3),aa(set(A),set(B),image2(A,B,F),A5)))
       => inj_on(A,B,fun_upd(A,B,F,X,Y3),A5) ) ) ).

% inj_on_fun_updI
tff(fact_4821_finite__range__updI,axiom,
    ! [A: $tType,B: $tType,F: 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),F),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),F,A3,aa(A,option(A),some(A),B2))),top_top(set(B))))) ) ).

% finite_range_updI
tff(fact_4822_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X: A,Y3: B,Z: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X,aa(B,option(B),some(B),Y3)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_4823_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] : restrict_map(A,B,F,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A))))) = fun_upd(A,option(B),F,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_4824_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,X: B,A5: set(B),F: fun(B,A),Y3: A] :
      ( ( pp(aa(set(B),bool,member(B,X),A5))
       => ( aa(set(B),set(A),image2(B,A,fun_upd(B,A,F,X,Y3)),A5) = aa(set(A),set(A),insert2(A,Y3),aa(set(B),set(A),image2(B,A,F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),insert2(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,F,X,Y3)),A5) = aa(set(B),set(A),image2(B,A,F),A5) ) ) ) ).

% fun_upd_image
tff(fact_4825_map__of_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType,P2: 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)),P2),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P2),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P2))) ).

% map_of.simps(2)
tff(fact_4826_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: 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)),Ys)))
     => ( 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))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_4827_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_4828_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),insert2(A,B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_4829_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_acu(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_4830_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_acc(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_4831_map__upds__Cons,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),A3: A,As3: 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),As3),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)),As3,Bs) ).

% map_upds_Cons
tff(fact_4832_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A3: A,As3: 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),As3)))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A3,aa(B,option(B),some(B),B2)),As3,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As3,Bs),A3,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_4833_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_4834_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V) )
     => 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),V)),graph(B,A,M))) ) ).

% in_graphI
tff(fact_4835_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,V: 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),V)),graph(A,B,M)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).

% in_graphD
tff(fact_4836_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V: 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),V)),graph(A,B,restrict_map(A,B,M,A5))))
     => pp(aa(set(A),bool,member(A,K),A5)) ) ).

% graph_restrictD(1)
tff(fact_4837_ran__restrictD,axiom,
    ! [B: $tType,A: $tType,Y3: A,M: fun(B,option(A)),A5: set(B)] :
      ( pp(aa(set(A),bool,member(A,Y3),ran(B,A,restrict_map(B,A,M,A5))))
     => ? [X4: B] :
          ( pp(aa(set(B),bool,member(B,X4),A5))
          & ( aa(B,option(A),M,X4) = aa(A,option(A),some(A),Y3) ) ) ) ).

% ran_restrictD
tff(fact_4838_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_acv(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_4839_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V: 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),V)),graph(A,B,restrict_map(A,B,M,A5))))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V) ) ) ).

% graph_restrictD(2)
tff(fact_4840_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_acw(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).

% graph_def
tff(fact_4841_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_acx(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K)) ).

% graph_fun_upd_None
tff(fact_4842_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys: list(B),Xs: list(A),F: fun(A,option(B)),Y3: 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)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F,X,aa(B,option(B),some(B),Y3)),Xs,Ys) = map_upds(A,B,F,Xs,Ys) ) )
      & ( ~ 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)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F,X,aa(B,option(B),some(B),Y3)),Xs,Ys) = fun_upd(A,option(B),map_upds(A,B,F,Xs,Ys),X,aa(B,option(B),some(B),Y3)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_4843_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),X: B,Y3: A,Z: A] :
      ( ( aa(B,option(A),M,X) = aa(A,option(A),some(A),Y3) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ pp(aa(set(A),bool,member(A,Z),ran(B,A,M)))
         => ( ran(B,A,fun_upd(B,option(A),M,X,aa(A,option(A),some(A),Z))) = 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),insert2(A,Y3),bot_bot(set(A))))),aa(set(A),set(A),insert2(A,Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_4844_map__upds__fold__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),Ks: list(A),Vs: list(B)] : map_upds(A,B,M,Ks,Vs) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_acz(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_4845_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_4846_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A3: A,B2: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(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)) ) ).

% image_add_greaterThanAtMost
tff(fact_4847_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B))] :
      ( ( dom(A,B,F) = bot_bot(set(A)) )
    <=> ! [X3: A] : aa(A,option(B),F,X3) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_4848_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K: A,M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,member(A,K),dom(A,B,M)))
     => ( fun_upd(A,option(B),M,K,none(B)) = M ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_4849_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_acc(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_4850_dom__const,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : dom(A,B,aTP_Lamp_ada(fun(A,B),fun(A,option(B)),F)) = top_top(set(A)) ).

% dom_const
tff(fact_4851_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y3: option(B),F: fun(A,option(B)),X: A] :
      ( ( ( Y3 = none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F,X,Y3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),dom(A,B,F)),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) ) )
      & ( ( Y3 != none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F,X,Y3)) = aa(set(A),set(A),insert2(A,X),dom(A,B,F)) ) ) ) ).

% dom_fun_upd
tff(fact_4852_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_4853_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)))
     => ? [B3: B] : aa(A,option(B),M,A3) = aa(B,option(B),some(B),B3) ) ).

% domD
tff(fact_4854_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_abj(fun(A,option(B)),fun(A,bool),M)) ).

% dom_def
tff(fact_4855_domIff,axiom,
    ! [A: $tType,B: $tType,A3: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,member(A,A3),dom(A,B,M)))
    <=> ( aa(A,option(B),M,A3) != none(B) ) ) ).

% domIff
tff(fact_4856_insert__dom,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),X: B,Y3: A] :
      ( ( aa(B,option(A),F,X) = aa(A,option(A),some(A),Y3) )
     => ( aa(set(B),set(B),insert2(B,X),dom(B,A,F)) = dom(B,A,F) ) ) ).

% insert_dom
tff(fact_4857_foldl__map,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(A,fun(B,A)),A3: A,F: fun(C,B),Xs: list(C)] : foldl(A,B,G,A3,aa(list(C),list(B),map(C,B,F),Xs)) = foldl(A,C,aa(fun(C,B),fun(A,fun(C,A)),aTP_Lamp_adb(fun(A,fun(B,A)),fun(fun(C,B),fun(A,fun(C,A))),G),F),A3,Xs) ).

% foldl_map
tff(fact_4858_dom__if,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F: fun(A,option(B)),G: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_adc(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F),G)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F)),aa(fun(A,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,G)),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_lv(fun(A,bool),fun(A,bool),P)))) ).

% dom_if
tff(fact_4859_foldl__conv__fold,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,A)),S: A,Xs: list(B)] : foldl(A,B,F,S,Xs) = aa(A,A,fold(B,A,aTP_Lamp_add(fun(A,fun(B,A)),fun(B,fun(A,A)),F),Xs),S) ).

% foldl_conv_fold
tff(fact_4860_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,F)))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),top_top(set(A))))
       => ? [X4: A] : aa(A,option(B),F,X4) = none(B) ) ) ).

% finite_map_freshness
tff(fact_4861_dom__minus,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),X: B,A5: set(B)] :
      ( ( aa(B,option(A),F,X) = none(A) )
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F)),aa(set(B),set(B),insert2(B,X),A5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),dom(B,A,F)),A5) ) ) ).

% dom_minus
tff(fact_4862_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_ade(set(A),fun(set(B),fun(fun(A,option(B)),bool)),A5),B4)))) ) ) ).

% finite_set_of_finite_maps
tff(fact_4863_foldl__conv__foldr,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(B,A)),A3: A,Xs: list(B)] : foldl(A,B,F,A3,Xs) = aa(A,A,foldr(B,A,aTP_Lamp_add(fun(A,fun(B,A)),fun(B,fun(A,A)),F),rev(B,Xs)),A3) ).

% foldl_conv_foldr
tff(fact_4864_foldr__conv__foldl,axiom,
    ! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),A3: A] : aa(A,A,foldr(B,A,F,Xs),A3) = foldl(A,B,aTP_Lamp_adf(fun(B,fun(A,A)),fun(A,fun(B,A)),F),A3,rev(B,Xs)) ).

% foldr_conv_foldl
tff(fact_4865_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_4866_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_acc(A,option(B))))
       => ( ! [K2: A,V5: B,M4: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite2(A),dom(A,B,M4)))
             => ( ~ pp(aa(set(A),bool,member(A,K2),dom(A,B,M4)))
               => ( pp(aa(fun(A,option(B)),bool,P,M4))
                 => pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M4,K2,aa(B,option(B),some(B),V5)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P,M)) ) ) ) ).

% finite_Map_induct
tff(fact_4867_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_adg(fun(A,option(B)),fun(A,product_prod(A,B)),M)),dom(A,B,M)) ).

% graph_eq_to_snd_dom
tff(fact_4868_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) )
    <=> ? [V7: B] : F = fun_upd(A,option(B),aTP_Lamp_acc(A,option(B)),X,aa(B,option(B),some(B),V7)) ) ).

% dom_eq_singleton_conv
tff(fact_4869_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_adg(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_4870_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),G: fun(A,option(B)),A5: set(A)] : dom(A,B,override_on(A,option(B),F,G,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,F)),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adh(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A5)))),aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adi(fun(A,option(B)),fun(set(A),fun(A,bool)),G),A5))) ).

% dom_override_on
tff(fact_4871_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N6: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N6),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),N6)),aa(nat,B,nth(B,Ys),N6))),R)) ) ) ) ).

% listrel_iff_nth
tff(fact_4872_last__list__update,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( Xs != nil(A) )
     => ( ( ( K = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K,X)) = X ) )
        & ( ( K != aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
         => ( last(A,list_update(A,Xs,K,X)) = last(A,Xs) ) ) ) ) ).

% last_list_update
tff(fact_4873_override__on__apply__notin,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),F: fun(A,B),G: fun(A,B)] :
      ( ~ pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(A,B,override_on(A,B,F,G,A5),A3) = aa(A,B,F,A3) ) ) ).

% override_on_apply_notin
tff(fact_4874_override__on__apply__in,axiom,
    ! [B: $tType,A: $tType,A3: A,A5: set(A),F: fun(A,B),G: fun(A,B)] :
      ( pp(aa(set(A),bool,member(A,A3),A5))
     => ( aa(A,B,override_on(A,B,F,G,A5),A3) = aa(A,B,G,A3) ) ) ).

% override_on_apply_in
tff(fact_4875_override__on__emptyset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] : override_on(A,B,F,G,bot_bot(set(A))) = F ).

% override_on_emptyset
tff(fact_4876_last__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( last(A,replicate(A,N,X)) = X ) ) ).

% last_replicate
tff(fact_4877_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R: 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,R)))) ).

% listrel_rtrancl_refl
tff(fact_4878_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),J))
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_4879_override__on__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B),G: fun(A,B),X2: A] :
      ( ( pp(aa(set(A),bool,member(A,X2),A5))
       => ( aa(A,B,override_on(A,B,F,G,A5),X2) = aa(A,B,G,X2) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X2),A5))
       => ( aa(A,B,override_on(A,B,F,G,A5),X2) = aa(A,B,F,X2) ) ) ) ).

% override_on_def
tff(fact_4880_last__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
         => ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).

% last_zip
tff(fact_4881_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R: 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,R))) ).

% listrel.Nil
tff(fact_4882_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R: 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,R)))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_4883_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R: 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,R)))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_4884_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_4885_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: 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),Ys)),listrel(A,A,transitive_rtrancl(A,R))))
     => ( 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)),Ys),Zs)),listrel(A,A,transitive_rtrancl(A,R))))
       => 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,R)))) ) ) ).

% listrel_rtrancl_trans
tff(fact_4886_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y3: B,R: set(product_prod(A,B)),Xs: list(A),Ys: 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),Y3)),R))
     => ( 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),Ys)),listrel(A,B,R)))
       => 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),Y3),Ys))),listrel(A,B,R))) ) ) ).

% listrel.Cons
tff(fact_4887_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y3: A,Ys: list(A),Xs: list(B),R: 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),Y3),Ys)),Xs)),listrel(A,B,R)))
     => ~ ! [Y4: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),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),Y3),Y4)),R))
             => ~ 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)),Ys),Ys3)),listrel(A,B,R))) ) ) ) ).

% listrel_Cons1
tff(fact_4888_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y3: B,Ys: list(B),R: 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),Y3),Ys))),listrel(A,B,R)))
     => ~ ! [X4: A,Xs3: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
           => ( 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),Y3)),R))
             => ~ 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),Ys)),listrel(A,B,R))) ) ) ) ).

% listrel_Cons2
tff(fact_4889_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
     => 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),Ys)),listrel(A,A,transitive_rtrancl(A,R)))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_4890_override__on__insert,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B),X: A,X6: set(A)] : override_on(A,B,F,G,aa(set(A),set(A),insert2(A,X),X6)) = fun_upd(A,B,override_on(A,B,F,G,X6),X,aa(A,B,G,X)) ).

% override_on_insert
tff(fact_4891_override__on__insert_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B),X: A,X6: set(A)] : override_on(A,B,F,G,aa(set(A),set(A),insert2(A,X),X6)) = override_on(A,B,fun_upd(A,B,F,X,aa(A,B,G,X)),G,X6) ).

% override_on_insert'
tff(fact_4892_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel(A,A,R)))
     => 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),Ys)),transitive_rtrancl(list(A),listrel1(A,R)))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_4893_takeWhile__not__last,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( takeWhile(A,aTP_Lamp_adj(list(A),fun(A,bool),Xs),Xs) = butlast(A,Xs) ) ) ).

% takeWhile_not_last
tff(fact_4894_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: 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,R)))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X4: A,Y4: B,Xs3: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),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),X4),Y4)),R))
                   => ~ 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),Ys3)),listrel(A,B,R))) ) ) ) ) ) ).

% listrel.cases
tff(fact_4895_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: 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,R)))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X3: A,Y: B,Xs4: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),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),X3),Y)),R))
            & 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)),Xs4),Ys4)),listrel(A,B,R))) ) ) ) ).

% listrel.simps
tff(fact_4896_remdups__adj__append_H_H,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,Xs)),remdups_adj(A,dropWhile(A,aTP_Lamp_adk(list(A),fun(A,bool),Xs),Ys))) ) ) ).

% remdups_adj_append''
tff(fact_4897_last__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( last(A,Xs) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))) ) ) ).

% last_conv_nth
tff(fact_4898_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X3)) ) ) ) ).

% listrel_iff_zip
tff(fact_4899_listrel__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : listrel(A,B,X2) = 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),X2)))) ).

% listrel_def
tff(fact_4900_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X2),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)),X2),Xa3)),listrel(A,B,R))) ) ).

% listrelp_listrel_eq
tff(fact_4901_INF__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_adl(set(A),fun(fun(A,bool),bool)) = ball(A) ).

% INF_bool_eq
tff(fact_4902_Ball__fold,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(A,bool,P,X3)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_adm(fun(A,bool),fun(A,fun(bool,bool)),P),fTrue,A5)) ) ) ).

% Ball_fold
tff(fact_4903_Ball__image__comp,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A5: set(B),G: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),aa(set(B),set(A),image2(B,A,F),A5)))
         => pp(aa(A,bool,G,X3)) )
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),A5))
         => pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),G),F),X3)) ) ) ).

% Ball_image_comp
tff(fact_4904_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_adn(fun(B,set(A)),fun(set(B),fun(A,bool)),B4),A5)) ).

% INTER_eq
tff(fact_4905_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_ado(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_uo(fun(A,fun(B,bool)),fun(B,set(A)),P)),A5)) ).

% Collect_ball_eq
tff(fact_4906_irrefl__distinct,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
    <=> ! [X3: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),R))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_adp(A,fun(A,bool))),X3)) ) ) ).

% irrefl_distinct
tff(fact_4907_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: fun(B,set(C)),F: fun(C,bool),G: fun(A,B),X2: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),aa(fun(C,bool),fun(B,bool),aTP_Lamp_adq(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),A5),F)),G),X2))
    <=> ! [Xa2: C] :
          ( pp(aa(set(C),bool,member(C,Xa2),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),aa(fun(B,set(C)),fun(fun(A,B),fun(A,set(C))),comp(B,set(C),A),A5),G),X2)))
         => pp(aa(C,bool,F,Xa2)) ) ) ).

% Ball_comp_iff
tff(fact_4908_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_adr(set(A),fun(A,bool),A5))) ) ).

% Inf_eq_Sup
tff(fact_4909_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_ads(set(A),fun(A,bool),A5))) ) ).

% Sup_eq_Inf
tff(fact_4910_takeWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X4)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append
tff(fact_4911_dropWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X4)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P,Ys) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ) ) ).

% dropWhile_append
tff(fact_4912_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X3: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Ys))))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X3)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_4913_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X4: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X4),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
             => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adt(list(A),fun(list(A),bool))),X4)) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_4914_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X4: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X4),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set2(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
         => pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_adt(list(A),fun(list(A),bool))),X4)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( concat(A,Xs) = concat(A,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_4915_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X3: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
           => pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P),X3)) ) ) ) ).

% list_all2_iff
tff(fact_4916_Under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] : order_Under(A,R,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adu(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A5)) ).

% Under_def
tff(fact_4917_UnderS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] : order_UnderS(A,R,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adv(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A5)) ).

% UnderS_def
tff(fact_4918_Above__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] : order_Above(A,R,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_adw(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A5)) ).

% Above_def
tff(fact_4919_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_adx(set(set(A)),fun(A,bool),A5)) ).

% Inter_eq
tff(fact_4920_Sup__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X6) = the(int,aTP_Lamp_ady(set(int),fun(int,bool),X6)) ).

% Sup_int_def
tff(fact_4921_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_adz(set(set(A)),fun(set(A),bool),A5)))) ) ).

% Inf_Sup
tff(fact_4922_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_adz(set(set(A)),fun(set(A),bool),A5)))) ) ).

% Sup_Inf
tff(fact_4923_Inf__filter__def,axiom,
    ! [A: $tType,S2: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S2) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),bool),set(filter(A)),collect(filter(A)),aTP_Lamp_aea(set(filter(A)),fun(filter(A),bool),S2))) ).

% Inf_filter_def
tff(fact_4924_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_aeb(set(set(A)),fun(set(A),bool),A5)))))) ) ).

% finite_Inf_Sup
tff(fact_4925_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_aec(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_4926_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_adz(set(set(A)),fun(set(A),bool),A5)))))) ) ).

% Inf_Sup_le
tff(fact_4927_INF__SUP__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: 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_aed(fun(B,A),fun(set(B),A),G)),A5)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_aee(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_aef(set(set(B)),fun(set(B),bool),A5)))) ) ).

% INF_SUP_set
tff(fact_4928_SUP__INF__set,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [G: 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_aee(fun(B,A),fun(set(B),A),G)),A5)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),image2(set(B),A,aTP_Lamp_aed(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_aef(set(set(B)),fun(set(B),bool),A5)))) ) ).

% SUP_INF_set
tff(fact_4929_min__ext__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : min_ext(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_aeg(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R)) ).

% min_ext_def
tff(fact_4930_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F: fun(A,A)] : order_mono(fun(A,bool),fun(A,bool),aTP_Lamp_aeh(fun(A,A),fun(fun(A,bool),fun(A,bool)),F)) ) ).

% iteratesp.mono
tff(fact_4931_congruent2__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
      ( equiv_congruent2(A,B,C,R1,R22,F)
    <=> ! [X3: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),R1))
         => 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_aej(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),R22),F)),X3)) ) ) ).

% congruent2_def
tff(fact_4932_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_aek(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),Q))))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),X3)))) ) ) ) ).

% finite_Collect_bex
tff(fact_4933_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_ael(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),P),A5)) ).

% Image_Collect_case_prod
tff(fact_4934_SUP__bool__eq,axiom,
    ! [A: $tType] : aTP_Lamp_aem(set(A),fun(fun(A,bool),bool)) = bex(A) ).

% SUP_bool_eq
tff(fact_4935_image__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] : aa(set(A),set(B),image2(A,B,F),A5) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aen(fun(A,B),fun(set(A),fun(B,bool)),F),A5)) ).

% image_def
tff(fact_4936_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_aeo(set(set(A)),fun(A,bool),A5)) ).

% Union_eq
tff(fact_4937_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_aw(set(A),fun(fun(A,bool),fun(A,bool)),A5),P))) ) ).

% chain_compr
tff(fact_4938_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_aep(fun(B,set(A)),fun(set(B),fun(A,bool)),B4),A5)) ).

% UNION_eq
tff(fact_4939_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_aeq(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_uo(fun(A,fun(B,bool)),fun(B,set(A)),P)),A5)) ).

% Collect_bex_eq
tff(fact_4940_Image__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] : aa(set(A),set(B),image(A,B,R),S) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aer(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R),S)) ).

% Image_def
tff(fact_4941_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] : aa(set(B),set(A),aa(fun(A,B),fun(set(B),set(A)),vimage(A,B),F),aa(set(A),set(B),image2(A,B,F),A5)) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_aes(fun(A,B),fun(set(A),fun(A,bool)),F),A5)) ).

% vimage_image_eq
tff(fact_4942_congruent2I_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: 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)),R1))
         => ( 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)),R22))
           => ( aa(B,C,aa(A,fun(B,C),F,Y12),Y22) = aa(B,C,aa(A,fun(B,C),F,Z12),Z23) ) ) )
     => equiv_congruent2(A,B,C,R1,R22,F) ) ).

% congruent2I'
tff(fact_4943_congruent2D,axiom,
    ! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: fun(A,fun(B,C)),Y1: A,Z1: A,Y2: B,Z22: B] :
      ( equiv_congruent2(A,B,C,R1,R22,F)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z1)),R1))
       => ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y2),Z22)),R22))
         => ( aa(B,C,aa(A,fun(B,C),F,Y1),Y2) = aa(B,C,aa(A,fun(B,C),F,Z1),Z22) ) ) ) ) ).

% congruent2D
tff(fact_4944_Bex__fold,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
            & pp(aa(A,bool,P,X3)) )
      <=> pp(finite_fold(A,bool,aTP_Lamp_aet(fun(A,bool),fun(A,fun(bool,bool)),P),fFalse,A5)) ) ) ).

% Bex_fold
tff(fact_4945_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_aev(set(nat),fun(set(nat),fun(nat,bool)),A5),B4))) ).

% nths_nths
tff(fact_4946_max__extp_Ocases,axiom,
    ! [A: $tType,R4: 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,R4),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))) )
             => ~ ! [X2: A] :
                    ( pp(aa(set(A),bool,member(A,X2),A1))
                   => ? [Xa4: A] :
                        ( pp(aa(set(A),bool,member(A,Xa4),A22))
                        & pp(aa(A,bool,aa(A,fun(A,bool),R4,X2),Xa4)) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_4947_max__extp_Osimps,axiom,
    ! [A: $tType,R4: 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,R4),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))) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A1))
           => ? [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),A22))
                & pp(aa(A,bool,aa(A,fun(A,bool),R4,X3),Xa2)) ) ) ) ) ).

% max_extp.simps
tff(fact_4948_max__extp_Omax__extI,axiom,
    ! [A: $tType,X6: set(A),Y7: set(A),R4: fun(A,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,finite_finite2(A),Y7))
       => ( ( Y7 != aa(fun(A,bool),set(A),collect(A),bot_bot(fun(A,bool))) )
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),X6))
               => ? [Xa3: A] :
                    ( pp(aa(set(A),bool,member(A,Xa3),Y7))
                    & pp(aa(A,bool,aa(A,fun(A,bool),R4,X4),Xa3)) ) )
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R4),X6),Y7)) ) ) ) ) ).

% max_extp.max_extI
tff(fact_4949_map__project__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),A5: set(A)] : map_project(A,B,F,A5) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_aew(fun(A,option(B)),fun(set(A),fun(B,bool)),F),A5)) ).

% map_project_def
tff(fact_4950_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X2: fun(A,A)] : comple7512665784863727008ratesp(A,X2) = complete_lattice_lfp(fun(A,bool),aTP_Lamp_aeh(fun(A,A),fun(fun(A,bool),fun(A,bool)),X2)) ) ).

% iteratesp_def
tff(fact_4951_max__ext__eq,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] : max_ext(A,R4) = 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_aex(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R4))) ).

% max_ext_eq
tff(fact_4952_rel__fun__eq__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R4: fun(B,fun(C,bool)),X2: 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),R4),X2),Xa3))
    <=> ! [Xb3: A] : pp(aa(C,bool,aa(B,fun(C,bool),R4,aa(A,B,X2,Xb3)),aa(A,C,Xa3,Xb3))) ) ).

% rel_fun_eq_rel
tff(fact_4953_rel__fun__def,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: fun(A,fun(C,bool)),B4: fun(B,fun(D,bool)),X2: 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),X2),Xa3))
    <=> ! [Xb3: A,Y: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),A5,Xb3),Y))
         => pp(aa(D,bool,aa(B,fun(D,bool),B4,aa(A,B,X2,Xb3)),aa(C,D,Xa3,Y))) ) ) ).

% rel_fun_def
tff(fact_4954_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_aey(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),A5),B4),D3)))) ) ) ).

% finite_set_of_finite_funs
tff(fact_4955_Collect__all__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aez(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_uo(fun(A,fun(B,bool)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_all_eq
tff(fact_4956_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_afa(set(A),fun(set(B),fun(fun(A,B),bool)),A5),B4)) ).

% Func_def
tff(fact_4957_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_afb(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_4958_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),P: fun(A,bool)] : least(A,Less_eq,P) = the(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afc(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Less_eq),P)) ).

% ord.Least_def
tff(fact_4959_transfer__bforall__def,axiom,
    ! [A: $tType,X2: fun(A,bool),Xa3: fun(A,bool)] :
      ( transfer_bforall(A,X2,Xa3)
    <=> ! [Xb3: A] :
          ( pp(aa(A,bool,X2,Xb3))
         => pp(aa(A,bool,Xa3,Xb3)) ) ) ).

% transfer_bforall_def
tff(fact_4960_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_4961_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_4962_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_4963_congruent__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
      ( equiv_congruent(A,B,R,F)
    <=> ! [X3: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),R))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_aaz(fun(A,B),fun(A,fun(A,bool)),F)),X3)) ) ) ).

% congruent_def
tff(fact_4964_type__copy__set__map0,axiom,
    ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F2: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: fun(B,set(D)),M2: fun(C,B),F: fun(E,D),S5: fun(C,set(E)),G: fun(F2,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( aa(fun(C,B),fun(C,set(D)),aa(fun(B,set(D)),fun(fun(C,B),fun(C,set(D))),comp(B,set(D),C),S2),M2) = aa(fun(C,set(E)),fun(C,set(D)),aa(fun(set(E),set(D)),fun(fun(C,set(E)),fun(C,set(D))),comp(set(E),set(D),C),image2(E,D,F)),S5) )
       => ( aa(fun(F2,A),fun(F2,set(D)),aa(fun(A,set(D)),fun(fun(F2,A),fun(F2,set(D))),comp(A,set(D),F2),aa(fun(A,B),fun(A,set(D)),aa(fun(B,set(D)),fun(fun(A,B),fun(A,set(D))),comp(B,set(D),A),S2),Rep)),aa(fun(F2,C),fun(F2,A),aa(fun(C,A),fun(fun(F2,C),fun(F2,A)),comp(C,A,F2),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M2)),G)) = aa(fun(F2,set(E)),fun(F2,set(D)),aa(fun(set(E),set(D)),fun(fun(F2,set(E)),fun(F2,set(D))),comp(set(E),set(D),F2),image2(E,D,F)),aa(fun(F2,C),fun(F2,set(E)),aa(fun(C,set(E)),fun(fun(F2,C),fun(F2,set(E))),comp(C,set(E),F2),S5),G)) ) ) ) ).

% type_copy_set_map0
tff(fact_4965_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R4))
    <=> irrefl(A,R4) ) ).

% irreflp_irrefl_eq
tff(fact_4966_type__copy__obj__one__point__absE,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S: A] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ~ ! [X4: B] : S != aa(B,A,Abs,X4) ) ).

% type_copy_obj_one_point_absE
tff(fact_4967_type__copy__map__comp0__undo,axiom,
    ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F2: $tType,Rep: fun(A,B),Abs: fun(B,A),Rep2: fun(C,D),Abs2: fun(D,C),Rep3: fun(E,F2),Abs3: fun(F2,E),M2: fun(F2,D),M1: fun(B,D),M22: fun(F2,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( type_definition(C,D,Rep2,Abs2,top_top(set(D)))
       => ( type_definition(E,F2,Rep3,Abs3,top_top(set(F2)))
         => ( ( aa(fun(E,F2),fun(E,C),aa(fun(F2,C),fun(fun(E,F2),fun(E,C)),comp(F2,C,E),aa(fun(F2,D),fun(F2,C),aa(fun(D,C),fun(fun(F2,D),fun(F2,C)),comp(D,C,F2),Abs2),M2)),Rep3) = aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),Abs2),M1)),Rep)),aa(fun(E,F2),fun(E,A),aa(fun(F2,A),fun(fun(E,F2),fun(E,A)),comp(F2,A,E),aa(fun(F2,B),fun(F2,A),aa(fun(B,A),fun(fun(F2,B),fun(F2,A)),comp(B,A,F2),Abs),M22)),Rep3)) )
           => ( aa(fun(F2,B),fun(F2,D),aa(fun(B,D),fun(fun(F2,B),fun(F2,D)),comp(B,D,F2),M1),M22) = M2 ) ) ) ) ) ).

% type_copy_map_comp0_undo
tff(fact_4968_type__copy__map__comp0,axiom,
    ! [F2: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(C,D),M1: fun(B,D),M22: fun(C,B),F: fun(D,F2),G: fun(E,C)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M2 = aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),M1),M22) )
       => ( aa(fun(E,C),fun(E,F2),aa(fun(C,F2),fun(fun(E,C),fun(E,F2)),comp(C,F2,E),aa(fun(C,D),fun(C,F2),aa(fun(D,F2),fun(fun(C,D),fun(C,F2)),comp(D,F2,C),F),M2)),G) = aa(fun(E,A),fun(E,F2),aa(fun(A,F2),fun(fun(E,A),fun(E,F2)),comp(A,F2,E),aa(fun(A,B),fun(A,F2),aa(fun(B,F2),fun(fun(A,B),fun(A,F2)),comp(B,F2,A),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),F),M1)),Rep)),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M22)),G)) ) ) ) ).

% type_copy_map_comp0
tff(fact_4969_type__copy__wit,axiom,
    ! [A: $tType,C: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),X: C,S2: fun(B,set(C)),Y3: B] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( pp(aa(set(C),bool,member(C,X),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),aa(fun(B,set(C)),fun(fun(A,B),fun(A,set(C))),comp(B,set(C),A),S2),Rep),aa(B,A,Abs,Y3))))
       => pp(aa(set(C),bool,member(C,X),aa(B,set(C),S2,Y3))) ) ) ).

% type_copy_wit
tff(fact_4970_congruentD,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B),Y3: A,Z: A] :
      ( equiv_congruent(A,B,R,F)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
       => ( aa(A,B,F,Y3) = aa(A,B,F,Z) ) ) ) ).

% congruentD
tff(fact_4971_congruentI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
      ( ! [Y4: A,Z4: 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),Z4)),R))
         => ( aa(A,B,F,Y4) = aa(A,B,F,Z4) ) )
     => equiv_congruent(A,B,R,F) ) ).

% congruentI
tff(fact_4972_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_afd(A,fun(A,bool))) ) ).

% irreflp_greater
tff(fact_4973_type__copy__map__id0,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M2: fun(B,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ( M2 = id(B) )
       => ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),aa(fun(B,B),fun(B,A),aa(fun(B,A),fun(fun(B,B),fun(B,A)),comp(B,A,B),Abs),M2)),Rep) = id(A) ) ) ) ).

% type_copy_map_id0
tff(fact_4974_type__copy__Rep__o__Abs,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Rep),Abs) = id(B) ) ) ).

% type_copy_Rep_o_Abs
tff(fact_4975_type__copy__Abs__o__Rep,axiom,
    ! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),Abs),Rep) = id(A) ) ) ).

% type_copy_Abs_o_Rep
tff(fact_4976_Abs__transfer,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: fun(A,B),Abs1: fun(B,A),Rep22: fun(C,D),Abs22: fun(D,C),R4: fun(B,fun(D,bool))] :
      ( type_definition(A,B,Rep1,Abs1,top_top(set(B)))
     => ( type_definition(C,D,Rep22,Abs22,top_top(set(D)))
       => pp(aa(fun(D,C),bool,aa(fun(B,A),fun(fun(D,C),bool),bNF_rel_fun(B,D,A,C,R4,aa(fun(B,fun(D,bool)),fun(A,fun(C,bool)),bNF_vimage2p(A,B,C,D,bool,Rep1,Rep22),R4)),Abs1),Abs22)) ) ) ).

% Abs_transfer
tff(fact_4977_typedef__rep__transfer,axiom,
    ! [A: $tType,B: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T4: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X4: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X4),Xa4))
          <=> ( X4 = 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,T4,fequal(A)),aTP_Lamp_ak(A,A)),Rep)) ) ) ).

% typedef_rep_transfer
tff(fact_4978_curr__in,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(A,B),C),A5: set(A),B4: set(B),C6: set(C)] :
      ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),F),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4)),C6)))
     => 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),F)),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C6)))) ) ).

% curr_in
tff(fact_4979_curr__surj,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,fun(B,C)),A5: set(A),B4: set(B),C6: set(C)] :
      ( pp(aa(set(fun(A,fun(B,C))),bool,member(fun(A,fun(B,C)),G),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C6))))
     => ? [X4: fun(product_prod(A,B),C)] :
          ( pp(aa(set(fun(product_prod(A,B),C)),bool,member(fun(product_prod(A,B),C),X4),bNF_Wellorder_Func(product_prod(A,B),C,product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4)),C6)))
          & ( aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),bNF_Wellorder_curr(A,B,C,A5),X4) = G ) ) ) ).

% curr_surj
tff(fact_4980_curr__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,A5: set(A),F: fun(product_prod(A,B),C),X2: A] :
      ( ( pp(aa(set(A),bool,member(A,X2),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),F),X2) = aa(A,fun(B,C),aTP_Lamp_bh(fun(product_prod(A,B),C),fun(A,fun(B,C)),F),X2) ) )
      & ( ~ pp(aa(set(A),bool,member(A,X2),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),F),X2) = undefined(fun(B,C)) ) ) ) ).

% curr_def
tff(fact_4981_curr__inj,axiom,
    ! [C: $tType,B: $tType,A: $tType,F1: fun(product_prod(A,B),C),A5: set(A),B4: set(B),C6: 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_xa(set(B),fun(A,set(B)),B4)),C6)))
     => ( 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_xa(set(B),fun(A,set(B)),B4)),C6)))
       => ( ( 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_4982_rel__pred__comp__def,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),P: fun(B,bool),X2: A] :
      ( rel_pred_comp(A,B,R4,P,X2)
    <=> ? [Y: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),R4,X2),Y))
          & pp(aa(B,bool,P,Y)) ) ) ).

% rel_pred_comp_def
tff(fact_4983_bij__betw__curr,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(A),B4: set(B),C6: 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_xa(set(B),fun(A,set(B)),B4)),C6),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C6))) ).

% bij_betw_curr
tff(fact_4984_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_afe(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).

% init_seg_of_def
tff(fact_4985_bij__betw__id,axiom,
    ! [A: $tType,A5: set(A)] : bij_betw(A,A,id(A),A5,A5) ).

% bij_betw_id
tff(fact_4986_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A5: set(A),B4: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A3),A5,B4)
        <=> ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),A3)),A5) = B4 ) ) ) ).

% bij_betw_add
tff(fact_4987_bij__betw__imp__surj__on,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => ( aa(set(A),set(B),image2(A,B,F),A5) = B4 ) ) ).

% bij_betw_imp_surj_on
tff(fact_4988_bij__betw__trans,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A5: set(A),B4: set(B),G: fun(B,C),C6: set(C)] :
      ( bij_betw(A,B,F,A5,B4)
     => ( bij_betw(B,C,G,B4,C6)
       => bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A5,C6) ) ) ).

% bij_betw_trans
tff(fact_4989_bij__betw__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A5: set(A),A10: set(B),F4: fun(B,C),A11: set(C)] :
      ( bij_betw(A,B,F,A5,A10)
     => ( bij_betw(B,C,F4,A10,A11)
      <=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F),A5,A11) ) ) ).

% bij_betw_comp_iff
tff(fact_4990_bij__betw__id__iff,axiom,
    ! [A: $tType,A5: set(A),B4: set(A)] :
      ( bij_betw(A,A,id(A),A5,B4)
    <=> ( A5 = B4 ) ) ).

% bij_betw_id_iff
tff(fact_4991_bij__betw__iff__bijections,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
    <=> ? [G6: fun(B,A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => ( pp(aa(set(B),bool,member(B,aa(A,B,F,X3)),B4))
                & ( aa(B,A,G6,aa(A,B,F,X3)) = X3 ) ) )
          & ! [X3: B] :
              ( pp(aa(set(B),bool,member(B,X3),B4))
             => ( pp(aa(set(A),bool,member(A,aa(B,A,G6,X3)),A5))
                & ( aa(A,B,F,aa(B,A,G6,X3)) = X3 ) ) ) ) ) ).

% bij_betw_iff_bijections
tff(fact_4992_bij__betw__apply,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B),A3: A] :
      ( bij_betw(A,B,F,A5,B4)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),B4)) ) ) ).

% bij_betw_apply
tff(fact_4993_bij__betw__cong,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),G: fun(A,B),A10: set(B)] :
      ( ! [A4: A] :
          ( pp(aa(set(A),bool,member(A,A4),A5))
         => ( aa(A,B,F,A4) = aa(A,B,G,A4) ) )
     => ( bij_betw(A,B,F,A5,A10)
      <=> bij_betw(A,B,G,A5,A10) ) ) ).

% bij_betw_cong
tff(fact_4994_bij__betw__inv,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => ? [G3: fun(B,A)] : bij_betw(B,A,G3,B4,A5) ) ).

% bij_betw_inv
tff(fact_4995_bij__betwE,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => ! [X2: A] :
          ( pp(aa(set(A),bool,member(A,X2),A5))
         => pp(aa(set(B),bool,member(B,aa(A,B,F,X2)),B4)) ) ) ).

% bij_betwE
tff(fact_4996_bij__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
    <=> ! [X3: B] :
        ? [Xa2: A] :
          ( ( aa(A,B,F,Xa2) = X3 )
          & ! [Y: A] :
              ( ( aa(A,B,F,Y) = X3 )
             => ( Y = Xa2 ) ) ) ) ).

% bij_iff
tff(fact_4997_bij__pointE,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y3: B] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ~ ! [X4: A] :
            ( ( Y3 = aa(A,B,F,X4) )
           => ~ ! [X9: A] :
                  ( ( Y3 = aa(A,B,F,X9) )
                 => ( X9 = X4 ) ) ) ) ).

% bij_pointE
tff(fact_4998_involuntory__imp__bij,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( ! [X4: A] : aa(A,A,F,aa(A,A,F,X4)) = X4
     => bij_betw(A,A,F,top_top(set(A)),top_top(set(A))) ) ).

% involuntory_imp_bij
tff(fact_4999_bij__betw__funpow,axiom,
    ! [A: $tType,F: fun(A,A),S2: set(A),N: nat] :
      ( bij_betw(A,A,F,S2,S2)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),S2,S2) ) ).

% bij_betw_funpow
tff(fact_5000_bij__betw__the__inv__into,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => bij_betw(B,A,the_inv_into(A,B,A5,F),B4,A5) ) ).

% bij_betw_the_inv_into
tff(fact_5001_f__the__inv__into__f__bij__betw,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B),X: B] :
      ( bij_betw(A,B,F,A5,B4)
     => ( ( bij_betw(A,B,F,A5,B4)
         => pp(aa(set(B),bool,member(B,X),B4)) )
       => ( aa(A,B,F,aa(B,A,the_inv_into(A,B,A5,F),X)) = X ) ) ) ).

% f_the_inv_into_f_bij_betw
tff(fact_5002_bij__betw__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => inj_on(A,B,F,A5) ) ).

% bij_betw_imp_inj_on
tff(fact_5003_sum_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
          ( bij_betw(B,C,H,S2,T4)
         => ( 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_aff(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T4) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_5004_prod_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
          ( bij_betw(B,C,H,S2,T4)
         => ( 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_afg(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T4) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_5005_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( bij_betw(A,B,F,A5,bot_bot(set(B)))
     => ( A5 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_5006_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(B)] :
      ( bij_betw(A,B,F,bot_bot(set(A)),A5)
     => ( A5 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_5007_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),A10: set(B),B4: set(A),B12: set(B)] :
      ( bij_betw(A,B,F,A5,A10)
     => ( 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,F),B4) = B12 )
         => bij_betw(A,B,F,B4,B12) ) ) ) ).

% bij_betw_subset
tff(fact_5008_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F4: fun(B,A),F: fun(A,B),A10: set(B)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),A5))
         => ( aa(B,A,F4,aa(A,B,F,X4)) = X4 ) )
     => ( ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),A10))
           => ( aa(A,B,F,aa(B,A,F4,X4)) = X4 ) )
       => ( 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,F),A5)),A10))
         => ( 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,F4),A10)),A5))
           => bij_betw(A,B,F,A5,A10) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_5009_bij__betw__imp__surj,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A)] :
      ( bij_betw(A,B,F,A5,top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).

% bij_betw_imp_surj
tff(fact_5010_bij__is__surj,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ).

% bij_is_surj
tff(fact_5011_bij__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => bij_betw(A,A,uminus_uminus(A),top_top(set(A)),top_top(set(A))) ) ).

% bij_uminus
tff(fact_5012_bij__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(B,C,G,top_top(set(B)),top_top(set(C)))
       => bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),top_top(set(A)),top_top(set(C))) ) ) ).

% bij_comp
tff(fact_5013_bij__betw__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
    <=> ( inj_on(A,B,F,A5)
        & ( aa(set(A),set(B),image2(A,B,F),A5) = B4 ) ) ) ).

% bij_betw_def
tff(fact_5014_bij__betw__imageI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( inj_on(A,B,F,A5)
     => ( ( aa(set(A),set(B),image2(A,B,F),A5) = B4 )
       => bij_betw(A,B,F,A5,B4) ) ) ).

% bij_betw_imageI
tff(fact_5015_inj__on__imp__bij__betw,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
     => bij_betw(A,B,F,A5,aa(set(A),set(B),image2(A,B,F),A5)) ) ).

% inj_on_imp_bij_betw
tff(fact_5016_bij__is__inj,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => inj_on(A,B,F,top_top(set(A))) ) ).

% bij_is_inj
tff(fact_5017_bij__id,axiom,
    ! [A: $tType] : bij_betw(A,A,id(A),top_top(set(A)),top_top(set(A))) ).

% bij_id
tff(fact_5018_bij__fn,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5019_trans__init__seg__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),T2: 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))),R),S)),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))),S),T2)),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))),R),T2)),init_seg_of(A))) ) ) ).

% trans_init_seg_of
tff(fact_5020_antisym__init__seg__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: 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))),R),S)),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))),S),R)),init_seg_of(A)))
       => ( R = S ) ) ) ).

% antisym_init_seg_of
tff(fact_5021_refl__on__init__seg__of,axiom,
    ! [A: $tType,R: 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))),R),R)),init_seg_of(A))) ).

% refl_on_init_seg_of
tff(fact_5022_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F4: fun(A,B),A10: set(A),A11: set(B),F: fun(C,A),A5: set(C)] :
      ( bij_betw(A,B,F4,A10,A11)
     => ( 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,F),A5)),A10))
       => ( bij_betw(C,A,F,A5,A10)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F4),F),A5,A11) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_5023_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F: fun(A,B),A10: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A5))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F,B2)),A10))
       => ( bij_betw(A,B,F,A5,A10)
        <=> bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A10),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_5024_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B2: A,A5: set(A),F: fun(A,B),A10: set(B)] :
      ( ~ pp(aa(set(A),bool,member(A,B2),A5))
     => ( ~ pp(aa(set(B),bool,member(B,aa(A,B,F,B2)),A10))
       => ( bij_betw(A,B,F,A5,A10)
         => bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A10),aa(set(B),set(B),insert2(B,aa(A,B,F,B2)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_5025_bijI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
       => bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).

% bijI
tff(fact_5026_bij__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
    <=> ( inj_on(A,B,F,top_top(set(A)))
        & ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) ) ) ) ).

% bij_def
tff(fact_5027_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),C6: set(A),B4: set(B),D5: set(B)] :
      ( bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),D5))
     => ( bij_betw(A,B,F,C6,D5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C6) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),D5) = bot_bot(set(B)) )
           => bij_betw(A,B,F,A5,B4) ) ) ) ) ).

% bij_betw_partition
tff(fact_5028_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),B4: set(B),C6: set(A),D5: set(B)] :
      ( bij_betw(A,B,F,A5,B4)
     => ( bij_betw(A,B,F,C6,D5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B4),D5) = bot_bot(set(B)) )
         => bij_betw(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B4),D5)) ) ) ) ).

% bij_betw_combine
tff(fact_5029_o__bij,axiom,
    ! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),G),F) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),F),G) = id(B) )
       => bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).

% o_bij
tff(fact_5030_bij__image__Compl__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F),A5)) ) ) ).

% bij_image_Compl_eq
tff(fact_5031_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A5: set(A),C6: set(B),G: fun(A,B),B4: set(A),D5: set(B)] :
      ( bij_betw(A,B,F,A5,C6)
     => ( bij_betw(A,B,G,B4,D5)
       => ( ( 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)),C6),D5) = 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_wc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A5),G),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)),C6),D5)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_5032_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),F: fun(B,C),A10: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( pp(aa(set(A),bool,member(A,I2),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,I2)),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,I2))) ) ) )
     => ( ! [I2: A] :
            ( pp(aa(set(A),bool,member(A,I2),I5))
           => bij_betw(B,C,F,aa(A,set(B),A5,I2),aa(A,set(C),A10,I2)) )
       => bij_betw(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A5),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A10),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_5033_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B4: set(B),A5: set(A)] :
      ( bij_betw(A,B,F,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),F),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,F),A5))) ) ) ).

% vimage_subset_eq
tff(fact_5034_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [S5: set(B),T3: set(C),H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = zero_zero(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,member(C,B3),T3))
                     => ( aa(C,A,G,B3) = 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_aff(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),T4) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5035_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S5: set(B),T3: set(C),H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),S5))
         => ( pp(aa(set(C),bool,finite_finite2(C),T3))
           => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
             => ( ! [A4: B] :
                    ( pp(aa(set(B),bool,member(B,A4),S5))
                   => ( aa(C,A,G,aa(B,C,H,A4)) = one_one(A) ) )
               => ( ! [B3: C] :
                      ( pp(aa(set(C),bool,member(C,B3),T3))
                     => ( aa(C,A,G,B3) = 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_afg(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T4) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5036_initial__segment__of__Diff,axiom,
    ! [A: $tType,P2: set(product_prod(A,A)),Q2: set(product_prod(A,A)),S: 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))),P2),Q2)),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))),P2),S)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Q2),S))),init_seg_of(A))) ) ).

% initial_segment_of_Diff
tff(fact_5037_bij__image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),B4: fun(C,set(A)),A5: set(C)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),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_wd(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B4)),A5)) ) ) ).

% bij_image_INT
tff(fact_5038_Chains__init__seg__of__Union,axiom,
    ! [A: $tType,R4: set(set(product_prod(A,A))),R: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),R4),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)),R),R4))
       => 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))),R),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R4))),init_seg_of(A))) ) ) ).

% Chains_init_seg_of_Union
tff(fact_5039_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),filter2(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5),Xs)) ).

% minus_coset_filter
tff(fact_5040_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),insert2(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),insert2(A,B2),bot_bot(set(A)))))
       => ( partial_flat_lub(A,B2,A5) = the(A,aa(A,fun(A,bool),aTP_Lamp_afh(set(A),fun(A,fun(A,bool)),A5),B2)) ) ) ) ).

% flat_lub_def
tff(fact_5041_bij__betw__Suc,axiom,
    ! [M2: set(nat),N5: set(nat)] :
      ( bij_betw(nat,nat,suc,M2,N5)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M2) = N5 ) ) ).

% bij_betw_Suc
tff(fact_5042_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_5043_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M2))
     => ? [H5: fun(A,nat)] : bij_betw(A,nat,H5,M2,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M2))) ) ).

% ex_bij_betw_finite_nat
tff(fact_5044_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M2))
     => ? [H5: fun(nat,A)] : bij_betw(nat,A,H5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M2)),M2) ) ).

% ex_bij_betw_nat_finite
tff(fact_5045_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M2: set(A)] :
      ( pp(aa(set(A),bool,finite_finite2(A),M2))
     => ? [H5: fun(nat,A)] : bij_betw(nat,A,H5,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M2)),M2) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_5046_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,filter2(A,aTP_Lamp_lw(set(A),fun(A,bool),A5),Xs)) ).

% union_coset_filter
tff(fact_5047_Chains__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_afi(set(product_prod(A,A)),fun(set(A),bool),R)) ).

% Chains_def
tff(fact_5048_sum_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_5049_sum_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_5050_prod_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_5051_prod_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_5052_Chains__inits__DiffI,axiom,
    ! [A: $tType,R4: set(set(product_prod(A,A))),S: set(product_prod(A,A))] :
      ( pp(aa(set(set(set(product_prod(A,A)))),bool,member(set(set(product_prod(A,A))),R4),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_afj(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),R4),S))),chains(set(product_prod(A,A)),init_seg_of(A)))) ) ).

% Chains_inits_DiffI
tff(fact_5053_Chains__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R))))) ).

% Chains_subset
tff(fact_5054_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F: fun(A,B),P: fun(A,bool),A3: A] :
          ( inj_on(A,B,F,aa(fun(A,bool),set(A),collect(A),P))
         => ( pp(aa(A,bool,P,A3))
           => ( ! [Y4: A] :
                  ( pp(aa(A,bool,P,Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,A3)),aa(A,B,F,Y4))) )
             => ( lattices_ord_arg_min(A,B,F,P) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_5055_in__measures_I2_J,axiom,
    ! [A: $tType,X: A,Y3: A,F: 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),Y3)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,Y3)))
        | ( ( aa(A,nat,F,X) = aa(A,nat,F,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),X),Y3)),measures(A,Fs))) ) ) ) ).

% in_measures(2)
tff(fact_5056_in__measures_I1_J,axiom,
    ! [A: $tType,X: 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),X),Y3)),measures(A,nil(fun(A,nat))))) ).

% in_measures(1)
tff(fact_5057_measures__less,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y3: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X)),aa(A,nat,F,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),X),Y3)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ).

% measures_less
tff(fact_5058_measures__lesseq,axiom,
    ! [A: $tType,F: fun(A,nat),X: A,Y3: A,Fs: list(fun(A,nat))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X)),aa(A,nat,F,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),X),Y3)),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),Y3)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ) ).

% measures_lesseq
tff(fact_5059_Chains__subset_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R)))),chains(A,R))) ) ).

% Chains_subset'
tff(fact_5060_refl__on__singleton,axiom,
    ! [A: $tType,X: A] : refl_on(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_5061_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,A5,R)
     => ( equiv_congruent(A,set(B),R,F)
       => ( 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),F),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) = aa(A,set(B),F,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_5062_refl__onD,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A5,R)
     => ( 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)),R)) ) ) ).

% refl_onD
tff(fact_5063_refl__onD1,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( refl_on(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
       => pp(aa(set(A),bool,member(A,X),A5)) ) ) ).

% refl_onD1
tff(fact_5064_refl__onD2,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( refl_on(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
       => pp(aa(set(A),bool,member(A,Y3),A5)) ) ) ).

% refl_onD2
tff(fact_5065_refl__on__domain,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( refl_on(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(A),bool,member(A,A3),A5))
          & pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ).

% refl_on_domain
tff(fact_5066_quotient__eqI,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A),Y7: set(A),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y7),equiv_quotient(A,A5,R)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y3),Y7))
             => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
               => ( X6 = Y7 ) ) ) ) ) ) ) ).

% quotient_eqI
tff(fact_5067_quotient__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A),Y7: set(A),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => ( pp(aa(set(set(A)),bool,member(set(A),Y7),equiv_quotient(A,A5,R)))
         => ( pp(aa(set(A),bool,member(A,X),X6))
           => ( pp(aa(set(A),bool,member(A,Y3),Y7))
             => ( ( X6 = Y7 )
              <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R)) ) ) ) ) ) ) ).

% quotient_eq_iff
tff(fact_5068_in__quotient__imp__closed,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => ( 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),Y3)),R))
           => pp(aa(set(A),bool,member(A,Y3),X6)) ) ) ) ) ).

% in_quotient_imp_closed
tff(fact_5069_congruent2__commuteI,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: set(product_prod(A,A)),F: fun(A,fun(A,B))] :
      ( equiv_equiv(A,A5,R)
     => ( ! [Y4: A,Z4: A] :
            ( pp(aa(set(A),bool,member(A,Y4),A5))
           => ( pp(aa(set(A),bool,member(A,Z4),A5))
             => ( aa(A,B,aa(A,fun(A,B),F,Y4),Z4) = aa(A,B,aa(A,fun(A,B),F,Z4),Y4) ) ) )
       => ( ! [Y4: A,Z4: A,W2: A] :
              ( pp(aa(set(A),bool,member(A,W2),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),Y4),Z4)),R))
               => ( aa(A,B,aa(A,fun(A,B),F,W2),Y4) = aa(A,B,aa(A,fun(A,B),F,W2),Z4) ) ) )
         => equiv_congruent2(A,A,B,R,R,F) ) ) ) ).

% congruent2_commuteI
tff(fact_5070_congruent2I,axiom,
    ! [C: $tType,B: $tType,A: $tType,A13: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
      ( equiv_equiv(A,A13,R1)
     => ( equiv_equiv(B,A24,R22)
       => ( ! [Y4: A,Z4: A,W2: B] :
              ( pp(aa(set(B),bool,member(B,W2),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),Y4),Z4)),R1))
               => ( aa(B,C,aa(A,fun(B,C),F,Y4),W2) = aa(B,C,aa(A,fun(B,C),F,Z4),W2) ) ) )
         => ( ! [Y4: B,Z4: B,W2: A] :
                ( pp(aa(set(A),bool,member(A,W2),A13))
               => ( 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),Y4),Z4)),R22))
                 => ( aa(B,C,aa(A,fun(B,C),F,W2),Y4) = aa(B,C,aa(A,fun(B,C),F,W2),Z4) ) ) )
           => equiv_congruent2(A,B,C,R1,R22,F) ) ) ) ) ).

% congruent2I
tff(fact_5071_equiv__type,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( equiv_equiv(A,A5,R)
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% equiv_type
tff(fact_5072_UN__equiv__class__type2,axiom,
    ! [A: $tType,B: $tType,C: $tType,A13: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),X13: set(A),X24: set(B),B4: set(set(C))] :
      ( equiv_equiv(A,A13,R1)
     => ( equiv_equiv(B,A24,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F)
         => ( pp(aa(set(set(A)),bool,member(set(A),X13),equiv_quotient(A,A13,R1)))
           => ( pp(aa(set(set(B)),bool,member(set(B),X24),equiv_quotient(B,A24,R22)))
             => ( ! [X12: A,X23: B] :
                    ( pp(aa(set(A),bool,member(A,X12),A13))
                   => ( pp(aa(set(B),bool,member(B,X23),A24))
                     => pp(aa(set(set(C)),bool,member(set(C),aa(B,set(C),aa(A,fun(B,set(C)),F,X12),X23)),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_afk(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),F),X24)),X13))),B4)) ) ) ) ) ) ) ).

% UN_equiv_class_type2
tff(fact_5073_UN__equiv__class__type,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),X6: set(A),B4: set(set(B))] :
      ( equiv_equiv(A,A5,R)
     => ( equiv_congruent(A,set(B),R,F)
       => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),A5))
               => pp(aa(set(set(B)),bool,member(set(B),aa(A,set(B),F,X4)),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),F),X6))),B4)) ) ) ) ) ).

% UN_equiv_class_type
tff(fact_5074_refl__on__def,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A5,R)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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)),R)) ) ) ) ).

% refl_on_def
tff(fact_5075_refl__onI,axiom,
    ! [A: $tType,R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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)),R)) )
       => refl_on(A,A5,R) ) ) ).

% refl_onI
tff(fact_5076_Refl__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,field2(A,R),R)
     => refl_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Refl_Restr
tff(fact_5077_refl__on__def_H,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( refl_on(A,A5,R)
    <=> ( ! [X3: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),R))
           => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_afl(set(A),fun(A,fun(A,bool)),A5)),X3)) )
        & ! [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)),R)) ) ) ) ).

% refl_on_def'
tff(fact_5078_equiv__class__eq__iff,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
      <=> ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,Y3),bot_bot(set(A)))) )
          & pp(aa(set(A),bool,member(A,X),A5))
          & pp(aa(set(A),bool,member(A,Y3),A5)) ) ) ) ).

% equiv_class_eq_iff
tff(fact_5079_eq__equiv__class__iff,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y3),A5))
         => ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,Y3),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),Y3)),R)) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_5080_equiv__class__eq,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_5081_eq__equiv__class,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A5,R)
       => ( 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)),R)) ) ) ) ).

% eq_equiv_class
tff(fact_5082_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(A),bool,member(A,X),A5))
       => ( pp(aa(set(A),bool,member(A,Y3),A5))
         => ( ( equiv_quotient(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A))),R) = equiv_quotient(A,aa(set(A),set(A),insert2(A,Y3),bot_bot(set(A))),R) )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R)) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_5083_UN__equiv__class__inject,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),X6: set(A),Y7: set(A)] :
      ( equiv_equiv(A,A5,R)
     => ( equiv_congruent(A,set(B),R,F)
       => ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F),Y7)) )
         => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
           => ( pp(aa(set(set(A)),bool,member(set(A),Y7),equiv_quotient(A,A5,R)))
             => ( ! [X4: A,Y4: A] :
                    ( pp(aa(set(A),bool,member(A,X4),A5))
                   => ( pp(aa(set(A),bool,member(A,Y4),A5))
                     => ( ( aa(A,set(B),F,X4) = aa(A,set(B),F,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),X4),Y4)),R)) ) ) )
               => ( X6 = Y7 ) ) ) ) ) ) ) ).

% UN_equiv_class_inject
tff(fact_5084_Refl__Field__Restr2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),field2(A,R)))
       => ( 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) = A5 ) ) ) ).

% Refl_Field_Restr2
tff(fact_5085_Refl__Field__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( refl_on(A,field2(A,R),R)
     => ( 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),field2(A,R)),A5) ) ) ).

% Refl_Field_Restr
tff(fact_5086_refines__equiv__image__eq,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),S2: 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))),R4),S2))
     => ( equiv_equiv(A,A5,R4)
       => ( equiv_equiv(A,A5,S2)
         => ( aa(set(set(A)),set(set(A)),image2(set(A),set(A),image(A,A,S2)),equiv_quotient(A,A5,R4)) = equiv_quotient(A,A5,S2) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_5087_subset__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),B2: A,A3: A] :
      ( equiv_equiv(A,A5,R)
     => ( 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,R),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(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)),R)) ) ) ) ).

% subset_equiv_class
tff(fact_5088_equiv__class__subset,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))) ) ) ).

% equiv_class_subset
tff(fact_5089_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,A3: A,B2: A] :
      ( equiv_equiv(A,A5,R)
     => ( 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,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(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)),R)) ) ) ).

% equiv_class_nondisjoint
tff(fact_5090_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),insert2(A,Y3),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),Y3)),R)) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_5091_Chains__alt__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R)
     => ( chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R))) ) ) ).

% Chains_alt_def
tff(fact_5092_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A13: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A1: A,A22: B] :
      ( equiv_equiv(A,A13,R1)
     => ( equiv_equiv(B,A24,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F)
         => ( pp(aa(set(A),bool,member(A,A1),A13))
           => ( 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_afm(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F),A22)),aa(set(A),set(A),image(A,A,R1),aa(set(A),set(A),insert2(A,A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F,A1),A22) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_5093_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A13: set(A),R1: set(product_prod(A,A)),A24: set(B),R22: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A3: B] :
      ( equiv_equiv(A,A13,R1)
     => ( equiv_equiv(B,A24,R22)
       => ( equiv_congruent2(A,B,set(C),R1,R22,F)
         => ( pp(aa(set(B),bool,member(B,A3),A24))
           => equiv_congruent(A,set(C),R1,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_afm(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R22),F),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_5094_proj__iff,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X: A,Y3: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,X),aa(set(A),set(A),insert2(A,Y3),bot_bot(set(A))))),A5))
       => ( ( aa(A,set(A),equiv_proj(A,A,R),X) = aa(A,set(A),equiv_proj(A,A,R),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),X),Y3)),R)) ) ) ) ).

% proj_iff
tff(fact_5095_disjnt__equiv__class,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert2(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)),R)) ) ) ).

% disjnt_equiv_class
tff(fact_5096_equiv__proj,axiom,
    ! [A: $tType,A5: set(A),R4: set(product_prod(A,A)),Z: product_prod(A,A)] :
      ( equiv_equiv(A,A5,R4)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),Z),R4))
       => ( aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),aa(fun(A,set(A)),fun(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A))),comp(A,set(A),product_prod(A,A)),equiv_proj(A,A,R4)),product_fst(A,A)),Z) = aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),aa(fun(A,set(A)),fun(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A))),comp(A,set(A),product_prod(A,A)),equiv_proj(A,A,R4)),product_snd(A,A)),Z) ) ) ) ).

% equiv_proj
tff(fact_5097_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C6: 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_xa(set(B),fun(A,set(B)),C6))),product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),C6))))
    <=> ( ( C6 = 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_5098_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C6: 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,C6,aTP_Lamp_xa(set(B),fun(A,set(B)),A5))),product_Sigma(A,B,C6,aTP_Lamp_xa(set(B),fun(A,set(B)),B4))))
    <=> ( ( C6 = 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_5099_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C6: 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,C6)),product_Sigma(A,B,B4,C6)))
    <=> ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B4)))
           => ( aa(A,set(B),C6,X3) = 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_5100_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))))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A5),aa(B,set(A),B4,X3))) ) ) ).

% disjoint_UN_iff
tff(fact_5101_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_5102_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( pairwise(A,aTP_Lamp_afn(fun(A,set(B)),fun(A,fun(A,bool)),F),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),F),A5)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_ox(fun(A,set(B)),fun(A,nat),F)),A5) ) ) ) ).

% sum_card_image
tff(fact_5103_pair__lessI2,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
       => 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),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ) ).

% pair_lessI2
tff(fact_5104_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [N: int,X: A] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(int,nat,nat2,N)) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( power_int(A,X,N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),X)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),N))) ) ) ) ) ).

% power_int_def
tff(fact_5105_power__int__1__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,one_one(A),N) = one_one(A) ) ).

% power_int_1_left
tff(fact_5106_power__int__mult__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : power_int(A,power_int(A,X,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% power_int_mult_numeral
tff(fact_5107_power__int__1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y3: A] : power_int(A,Y3,one_one(int)) = Y3 ) ).

% power_int_1_right
tff(fact_5108_power__int__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,sgn_sgn(A),power_int(A,A3,N)) = power_int(A,aa(A,A,sgn_sgn(A),A3),N) ) ).

% power_int_sgn
tff(fact_5109_power__int__mult__distrib__numeral1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [W: num,Y3: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y3),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(num,A,numeral_numeral(A),W),M)),power_int(A,Y3,M)) ) ).

% power_int_mult_distrib_numeral1
tff(fact_5110_power__int__mult__distrib__numeral2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,W: num,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(num,A,numeral_numeral(A),W)),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),W),M)) ) ).

% power_int_mult_distrib_numeral2
tff(fact_5111_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( power_int(A,X,N) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( N != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_5112_power__int__0__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          ( ( M != zero_zero(int) )
         => ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ).

% power_int_0_left
tff(fact_5113_power__int__0__right,axiom,
    ! [B: $tType] :
      ( ( inverse(B)
        & power(B) )
     => ! [X: B] : power_int(B,X,zero_zero(int)) = one_one(B) ) ).

% power_int_0_right
tff(fact_5114_abs__power__int__minus,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,aa(A,A,uminus_uminus(A),A3),N)) = aa(A,A,abs_abs(A),power_int(A,A3,N)) ) ).

% abs_power_int_minus
tff(fact_5115_power__int__of__nat,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A,N: nat] : power_int(A,X,aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),N) ) ).

% power_int_of_nat
tff(fact_5116_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_5117_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_5118_power__int__numeral,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [X: A,N: num] : power_int(A,X,aa(num,int,numeral_numeral(int),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),N)) ) ).

% power_int_numeral
tff(fact_5119_power__int__add__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),power_int(A,X,aa(num,int,numeral_numeral(int),N))) = power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_int_add_numeral
tff(fact_5120_power__int__add__numeral2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),M))),aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(num,int,numeral_numeral(int),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_int_add_numeral2
tff(fact_5121_power__int__minus1__right,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & monoid_mult(A) )
     => ! [Y3: A] : power_int(A,Y3,aa(int,int,uminus_uminus(int),one_one(int))) = aa(A,A,inverse_inverse(A),Y3) ) ).

% power_int_minus1_right
tff(fact_5122_pair__less__iff1,axiom,
    ! [X: nat,Y3: nat,Z: 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),Y3)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X),Z))),fun_pair_less))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y3),Z)) ) ).

% pair_less_iff1
tff(fact_5123_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,B2,N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2)) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5124_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_less_power_int
tff(fact_5125_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),power_int(A,X,N))) ) ) ).

% zero_le_power_int
tff(fact_5126_pairwise__trivial,axiom,
    ! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_abd(A,fun(A,bool)),I5) ).

% pairwise_trivial
tff(fact_5127_power__int__abs,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: int] : aa(A,A,abs_abs(A),power_int(A,A3,N)) = power_int(A,aa(A,A,abs_abs(A),A3),N) ) ).

% power_int_abs
tff(fact_5128_power__int__mult,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] : power_int(A,X,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = power_int(A,power_int(A,X,M),N) ) ).

% power_int_mult
tff(fact_5129_power__int__divide__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y3: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),X),Y3),M) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,Y3,M)) ) ).

% power_int_divide_distrib
tff(fact_5130_power__int__commutes,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,N)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),power_int(A,X,N)) ) ).

% power_int_commutes
tff(fact_5131_power__int__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y3: A,M: int] : power_int(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3),M) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,Y3,M)) ) ).

% power_int_mult_distrib
tff(fact_5132_power__int__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,aa(A,A,inverse_inverse(A),X),N) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).

% power_int_inverse
tff(fact_5133_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N = zero_zero(int) ) )
         => ( power_int(A,X,N) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_5134_power__int__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),X),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),power_int(A,X,N)) ) ).

% power_int_one_over
tff(fact_5135_power__int__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,N: int] : power_int(A,X,aa(int,int,uminus_uminus(int),N)) = aa(A,A,inverse_inverse(A),power_int(A,X,N)) ) ).

% power_int_minus
tff(fact_5136_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F: fun(B,A),S: set(B)] :
      ( pairwise(A,R,aa(set(B),set(A),image2(B,A,F),S))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_afo(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F),S) ) ).

% pairwise_image
tff(fact_5137_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          ( ( ( M = zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = one_one(A) ) )
          & ( ( M != zero_zero(int) )
           => ( power_int(A,zero_zero(A),M) = zero_zero(A) ) ) ) ) ).

% power_int_0_left_If
tff(fact_5138_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N)),power_int(A,A3,N5))) ) ) ) ).

% power_int_increasing
tff(fact_5139_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
         => ( 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,N)),power_int(A,A3,N5))) ) ) ) ).

% power_int_strict_increasing
tff(fact_5140_power__int__minus__one__minus,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [N: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,uminus_uminus(int),N)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N) ) ).

% power_int_minus_one_minus
tff(fact_5141_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( M != N ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_diff
tff(fact_5142_power__int__minus__one__diff__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: int,B2: int] : power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),A3),B2)) = power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),A3)) ) ).

% power_int_minus_one_diff_commute
tff(fact_5143_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),N5))
         => ( 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,N5)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5144_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y3: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),power_int(A,Y3,N))) ) ) ) ) ).

% power_int_mono
tff(fact_5145_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_5146_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),power_int(A,X,N))) ) ) ) ).

% one_le_power_int
tff(fact_5147_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,N: 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)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),power_int(A,A3,N))) ) ) ) ).

% one_less_power_int
tff(fact_5148_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: int,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N) != zero_zero(int) ) )
         => ( power_int(A,X,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,M)),power_int(A,X,N)) ) ) ) ).

% power_int_add
tff(fact_5149_power__int__minus__left__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( division_ring(A)
        & one(B)
        & uminus(B) )
     => ! [X: C,A3: A,N: int] :
          ( nO_MATCH(B,C,aa(B,B,uminus_uminus(B),one_one(B)),X)
         => ( power_int(A,aa(A,A,uminus_uminus(A),A3),N) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,aa(A,A,uminus_uminus(A),one_one(A)),N)),power_int(A,A3,N)) ) ) ) ).

% power_int_minus_left_distrib
tff(fact_5150_pair__lessI1,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,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),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ).

% pair_lessI1
tff(fact_5151_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,A3,N)),power_int(A,B2,N))) ) ) ) ) ).

% power_int_strict_mono
tff(fact_5152_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B2: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),zero_zero(int)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,B2,N)),power_int(A,A3,N))) ) ) ) ) ).

% power_int_antimono
tff(fact_5153_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: int,N5: int,A3: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),N5))
         => ( 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) )
                  | ( N5 != zero_zero(int) )
                  | ( N = zero_zero(int) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,A3,N5)),power_int(A,A3,N))) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5154_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,N)),one_one(A))) ) ) ) ) ).

% power_int_le_one
tff(fact_5155_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N)) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5156_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,M: int,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),power_int(A,X,M)),power_int(A,X,N)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N)) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5157_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: int] :
          ( ( ( X != zero_zero(A) )
            | ( N != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,X,aa(int,int,aa(int,fun(int,int),minus_minus(int),N),one_one(int)))),X) = power_int(A,X,N) ) ) ) ).

% power_int_minus_mult
tff(fact_5158_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_5159_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_5160_power__int__numeral__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: num,N: num] : power_int(A,aa(num,A,numeral_numeral(A),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),pow(M,N))) ) ).

% power_int_numeral_neg_numeral
tff(fact_5161_pair__leqI2,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S),T2))
       => 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),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ) ).

% pair_leqI2
tff(fact_5162_pair__leqI1,axiom,
    ! [A3: nat,B2: nat,S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
     => pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,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),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ).

% pair_leqI1
tff(fact_5163_wmax__insertI,axiom,
    ! [Y3: 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),Y3),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),Y3)),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)),insert2(product_prod(nat,nat),X),XS)),YS)),fun_max_weak)) ) ) ) ).

% wmax_insertI
tff(fact_5164_wmin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y3: 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),Y3)),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)),insert2(product_prod(nat,nat),Y3),YS))),fun_min_weak)) ) ) ) ).

% wmin_insertI
tff(fact_5165_smin__insertI,axiom,
    ! [X: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y3: 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),Y3)),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)),insert2(product_prod(nat,nat),Y3),YS))),fun_min_strict)) ) ) ) ).

% smin_insertI
tff(fact_5166_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_5167_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_5168_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_5169_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_5170_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)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% min_weak_def
tff(fact_5171_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)))),insert2(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).

% max_weak_def
tff(fact_5172_smax__insertI,axiom,
    ! [Y3: product_prod(nat,nat),Y7: 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),Y3),Y7))
     => ( 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),Y3)),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),Y7)),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)),insert2(product_prod(nat,nat),X),X6)),Y7)),fun_max_strict)) ) ) ) ).

% smax_insertI
tff(fact_5173_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)))
       => ( ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))))
              | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( ( A3 = B2 )
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_5174_assert__def,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ( pp(aa(A,bool,P,X))
       => ( heap_Time_assert(A,P,X) = aa(A,heap_Time_Heap(A),heap_Time_return(A),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( heap_Time_assert(A,P,X) = heap_Time_raise(A,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,fFalse)),aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,fFalse)),nil(char)))))))) ) ) ) ).

% assert_def
tff(fact_5175_wo__rel_Omax2__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 ) )
        & ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 ) ) ) ) ).

% wo_rel.max2_def
tff(fact_5176_wo__rel_OTOTALS,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ! [X2: A] :
          ( pp(aa(set(A),bool,member(A,X2),field2(A,R)))
         => ! [Xa3: A] :
              ( pp(aa(set(A),bool,member(A,Xa3),field2(A,R)))
             => ( pp(aa(set(product_prod(A,A)),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),Xa3)),R))
                | pp(aa(set(product_prod(A,A)),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),X2)),R)) ) ) ) ) ).

% wo_rel.TOTALS
tff(fact_5177_wo__rel_Owell__order__induct,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X4: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != 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),Y5),X4)),R)) )
               => pp(aa(A,bool,P,Y5)) )
           => pp(aa(A,bool,P,X4)) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% wo_rel.well_order_induct
tff(fact_5178_well__order__induct__imp,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),P: fun(A,bool),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [X4: A] :
            ( ! [Y5: A] :
                ( ( ( Y5 != 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),Y5),X4)),R)) )
               => ( pp(aa(set(A),bool,member(A,Y5),field2(A,R)))
                 => pp(aa(A,bool,P,Y5)) ) )
           => ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
             => pp(aa(A,bool,P,X4)) ) )
       => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
         => pp(aa(A,bool,P,A3)) ) ) ) ).

% well_order_induct_imp
tff(fact_5179_wo__rel_Omax2__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R)) ) ) ) ) ).

% wo_rel.max2_greater
tff(fact_5180_wo__rel_Omax2__equals2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).

% wo_rel.max2_equals2
tff(fact_5181_wo__rel_Omax2__equals1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 )
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R)) ) ) ) ) ).

% wo_rel.max2_equals1
tff(fact_5182_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))),field2(A,R)))
       => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
         => ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
             => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_5183_smax__emptyI,axiom,
    ! [Y7: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,finite_finite2(product_prod(nat,nat)),Y7))
     => ( ( Y7 != 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)))),Y7)),fun_max_strict)) ) ) ).

% smax_emptyI
tff(fact_5184_natLeq__on__wo__rel,axiom,
    ! [N: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_wo_rel
tff(fact_5185_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
            & pp(aa(set(A),bool,member(A,bNF_We1388413361240627857o_max2(A,R,A3,B2)),aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A)))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_5186_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_5187_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = char2(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),one_one(nat)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_5188_char_Osize_I2_J,axiom,
    ! [X1: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : aa(char,nat,size_size(char),char2(X1,X22,X32,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size(2)
tff(fact_5189_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,A3),B4))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),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),B3)),R)) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R,B4) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_5190_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_5191_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image2(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_5192_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),field2(A,R)))
       => ( 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,R,B4)),B2)),R)) ) ) ) ).

% wo_rel.minim_least
tff(fact_5193_char_Osize__gen,axiom,
    ! [X1: bool,X22: bool,X32: bool,X42: bool,X52: bool,X62: bool,X72: bool,X82: bool] : size_char(char2(X1,X22,X32,X42,X52,X62,X72,X82)) = zero_zero(nat) ).

% char.size_gen
tff(fact_5194_wo__rel_Ominim__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( bNF_We6954850376910717587_minim(A,R,A5) = the(A,bNF_We4791949203932849705sMinim(A,R,A5)) ) ) ).

% wo_rel.minim_def
tff(fact_5195_range__nat__of__char,axiom,
    aa(set(char),set(nat),image2(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_5196_char__of__comp__of__char,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ( aa(fun(char,A),fun(char,char),aa(fun(A,char),fun(fun(char,A),fun(char,char)),comp(A,char,char),unique5772411509450598832har_of(A)),comm_s6883823935334413003f_char(A)) = id(char) ) ) ).

% char_of_comp_of_char
tff(fact_5197_wo__rel_OisMinim__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(A,bool,bNF_We4791949203932849705sMinim(A,R,A5),B2))
      <=> ( pp(aa(set(A),bool,member(A,B2),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),B2),X3)),R)) ) ) ) ) ).

% wo_rel.isMinim_def
tff(fact_5198_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_afp(A,fun(A,bool)),aTP_Lamp_afq(A,fun(A,bool)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_5199_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F5: fun(A,filter(B)),P: fun(B,bool)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,member(A,A4),B4))
           => ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),B4))
               => ? [X2: A] :
                    ( pp(aa(set(A),bool,member(A,X2),B4))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F5,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F5,A4)),aa(A,filter(B),F5,B3)))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F5),B4)))
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),B4))
              & eventually(B,P,aa(A,filter(B),F5,X3)) ) ) ) ) ).

% eventually_INF_base
tff(fact_5200_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),aa(A,nat,euclid6346220572633701492n_size(A),B2)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_5201_eventually__const,axiom,
    ! [A: $tType,F5: filter(A),P: bool] :
      ( ( F5 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_pl(bool,fun(A,bool),P),F5)
      <=> pp(P) ) ) ).

% eventually_const
tff(fact_5202_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A] :
          ( ( aa(A,nat,euclid6346220572633701492n_size(A),B2) = zero_zero(nat) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_5203_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_5204_euclidean__size__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) = one_one(nat) ) ) ).

% euclidean_size_1
tff(fact_5205_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)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
        <=> ( B2 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_5206_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),Net: filter(A)] :
      ( ! [X4: A] : ~ pp(aa(A,bool,P,X4))
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_5207_eventually__const__iff,axiom,
    ! [A: $tType,P: bool,F5: filter(A)] :
      ( eventually(A,aTP_Lamp_pl(bool,fun(A,bool),P),F5)
    <=> ( pp(P)
        | ( F5 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_5208_trivial__limit__def,axiom,
    ! [A: $tType,F5: filter(A)] :
      ( ( F5 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_ap(A,bool),F5) ) ).

% trivial_limit_def
tff(fact_5209_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_um(fun(A,fun(B,bool)),fun(A,bool),P),F5)
    <=> ? [Y8: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_afr(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P),Y8),F5) ) ).

% eventually_ex
tff(fact_5210_eventually__all__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(B)
     => ! [P: fun(A,fun(B,bool)),Net: filter(A)] :
          ( ! [Y4: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_afs(fun(A,fun(B,bool)),fun(B,fun(A,bool)),P),Y4),Net)
         => eventually(A,aTP_Lamp_aft(fun(A,fun(B,bool)),fun(A,bool),P),Net) ) ) ).

% eventually_all_finite
tff(fact_5211_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F5: filter(A),F: fun(B,A),G5: filter(B)] :
      ( eventually(A,P,F5)
     => ( filterlim(B,A,F,F5,G5)
       => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afu(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F),G5) ) ) ).

% eventually_compose_filterlim
tff(fact_5212_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F12: filter(A),F13: filter(A),F23: filter(B),F24: filter(B),F: fun(B,A),G: fun(B,A)] :
      ( ( F12 = F13 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afv(fun(B,A),fun(fun(B,A),fun(B,bool)),F),G),F23)
         => ( filterlim(B,A,F,F12,F23)
          <=> filterlim(B,A,G,F13,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_5213_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F,F23,F12)
    <=> ! [P5: fun(B,bool)] :
          ( eventually(B,P5,F23)
         => eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_yp(fun(A,B),fun(fun(B,bool),fun(A,bool)),F),P5),F12) ) ) ).

% filterlim_iff
tff(fact_5214_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C6: bool,P: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afw(bool,fun(fun(A,bool),fun(A,bool)),C6),P),F5)
    <=> ( pp(C6)
       => eventually(A,P,F5) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_5215_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C6: bool,P: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afx(bool,fun(fun(A,bool),fun(A,bool)),C6),P),F5)
    <=> ( pp(C6)
        | eventually(A,P,F5) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_5216_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,bool),C6: bool,F5: filter(A)] :
      ( eventually(A,aa(bool,fun(A,bool),aTP_Lamp_afy(fun(A,bool),fun(bool,fun(A,bool)),P),C6),F5)
    <=> ( eventually(A,P,F5)
        | pp(C6) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_5217_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
     => ( eventually(A,P,F5)
       => eventually(A,Q,F5) ) ) ).

% eventually_mp
tff(fact_5218_eventually__True,axiom,
    ! [A: $tType,F5: filter(A)] : eventually(A,aTP_Lamp_aq(A,bool),F5) ).

% eventually_True
tff(fact_5219_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F5)
     => ( eventually(A,Q,F5)
       => eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_av(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5) ) ) ).

% eventually_conj
tff(fact_5220_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool),R4: fun(A,bool)] :
      ( eventually(A,P,F5)
     => ( eventually(A,Q,F5)
       => ( ! [I2: A] :
              ( pp(aa(A,bool,P,I2))
             => ( pp(aa(A,bool,Q,I2))
               => pp(aa(A,bool,R4,I2)) ) )
         => eventually(A,R4,F5) ) ) ) ).

% eventually_elim2
tff(fact_5221_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afz(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
     => ( eventually(A,P,F5)
      <=> eventually(A,Q,F5) ) ) ).

% eventually_subst
tff(fact_5222_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F5)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
       => eventually(A,Q,F5) ) ) ).

% eventually_rev_mp
tff(fact_5223_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_av(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
    <=> ( eventually(A,P,F5)
        & eventually(A,Q,F5) ) ) ).

% eventually_conj_iff
tff(fact_5224_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F5)
     => ( ~ eventually(A,Q,F5)
       => ~ eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5) ) ) ).

% not_eventually_impI
tff(fact_5225_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_aga(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_5226_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))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X4),Net) )
       => eventually(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_agb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_5227_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_agb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),Net)
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X3),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5228_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),B2) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_5229_euclidean__size__unit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) ) ) ) ).

% euclidean_size_unit
tff(fact_5230_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_lb(A,fun(A,bool)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5231_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_agc(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_5232_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F5: filter(B),G5: filter(A),F8: filter(B),G7: filter(A),F4: fun(A,B)] :
      ( filterlim(A,B,F,F5,G5)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F5),F8))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G7),G5))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),F4),G7)
           => filterlim(A,B,F4,F8,G7) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_5233_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),S2: set(B),F5: filter(A)] :
      ( filterlim(A,B,F,principal(B,S2),F5)
    <=> eventually(A,aa(set(B),fun(A,bool),aTP_Lamp_yr(fun(A,B),fun(set(B),fun(A,bool)),F),S2),F5) ) ).

% filterlim_principal
tff(fact_5234_le__principal,axiom,
    ! [A: $tType,F5: filter(A),A5: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),principal(A,A5)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5),F5) ) ).

% le_principal
tff(fact_5235_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I: A,I5: set(A),P: fun(B,bool),F5: fun(A,filter(B))] :
      ( pp(aa(set(A),bool,member(A,I),I5))
     => ( eventually(B,P,aa(A,filter(B),F5,I))
       => eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F5),I5))) ) ) ).

% eventually_INF1
tff(fact_5236_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),S: set(A)] :
      ( eventually(A,P,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),principal(A,S)))
    <=> eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_age(fun(A,bool),fun(set(A),fun(A,bool)),P),S),F5) ) ).

% eventually_inf_principal
tff(fact_5237_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
        <=> ( ( aa(A,nat,euclid6346220572633701492n_size(A),A3) = aa(A,nat,euclid6346220572633701492n_size(A),one_one(A)) )
            & ( A3 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_5238_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,bool),F5: 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),F5),A5)))
      <=> ? [Q7: fun(A,fun(B,bool))] :
            ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => eventually(B,aa(A,fun(B,bool),Q7,X3),aa(A,filter(B),F5,X3)) )
            & ! [Y: B] :
                ( ! [X3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),A5))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q7,X3),Y)) )
               => pp(aa(B,bool,P,Y)) ) ) ) ) ).

% eventually_INF_finite
tff(fact_5239_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),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)))) ) ) ).

% size_mult_mono
tff(fact_5240_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),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)))) ) ) ).

% size_mult_mono'
tff(fact_5241_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)))
         => ( aa(A,nat,euclid6346220572633701492n_size(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,nat,euclid6346220572633701492n_size(A),B2) ) ) ) ).

% euclidean_size_times_unit
tff(fact_5242_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),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_5243_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),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ) ).

% dvd_imp_size_le
tff(fact_5244_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),aa(A,nat,euclid6346220572633701492n_size(A),modulo_modulo(A,A3,B2))),aa(A,nat,euclid6346220572633701492n_size(A),B2))) ) ) ).

% mod_size_less
tff(fact_5245_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F,at_bot(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agf(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5246_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F,at_bot(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_agf(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ).

% filterlim_at_bot
tff(fact_5247_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F,at_bot(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_agg(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ).

% filterlim_at_bot_dense
tff(fact_5248_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_jg(nat,fun(nat,bool)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_5249_euclidean__size__int__def,axiom,
    euclid6346220572633701492n_size(int) = aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int)) ).

% euclidean_size_int_def
tff(fact_5250_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F5: 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),F5),B4)))
    <=> ? [X10: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X10),B4))
          & pp(aa(set(B),bool,finite_finite2(B),X10))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F5),X10))) ) ) ).

% eventually_INF
tff(fact_5251_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F,at_bot(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agh(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5252_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ie(nat,fun(nat,bool)),aTP_Lamp_cq(nat,fun(nat,bool)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_5253_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y7: set(B),X6: set(A),F5: filter(B),F: 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,G),Y7)),X6))
     => ( eventually(B,aTP_Lamp_agi(set(B),fun(B,bool),Y7),F5)
       => ( map_filter_on(A,C,X6,F,map_filter_on(B,A,Y7,G,F5)) = map_filter_on(B,C,Y7,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),F),G),F5) ) ) ) ).

% map_filter_on_comp
tff(fact_5254_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) )
             => ! [Q3: A,R2: A] :
                  ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B2) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R2)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
                   => ( ( R2 != zero_zero(A) )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q3 )
                       => ( ( modulo_modulo(A,A3,B2) = R2 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B2)),R2) ) ) ) ) ) ) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_5255_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( modulo_modulo(A,A3,B2) = R ) ) ) ) ) ) ).

% mod_eqI
tff(fact_5256_division__segment__1,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( euclid7384307370059645450egment(A,one_one(A)) = one_one(A) ) ) ).

% division_segment_1
tff(fact_5257_division__segment__numeral,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [K: num] : euclid7384307370059645450egment(A,aa(num,A,numeral_numeral(A),K)) = one_one(A) ) ).

% division_segment_numeral
tff(fact_5258_division__segment__of__nat,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : euclid7384307370059645450egment(A,aa(nat,A,semiring_1_of_nat(A),N)) = one_one(A) ) ).

% division_segment_of_nat
tff(fact_5259_division__segment__nat__def,axiom,
    ! [N: nat] : euclid7384307370059645450egment(nat,N) = one_one(nat) ).

% division_segment_nat_def
tff(fact_5260_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : euclid7384307370059645450egment(A,A3) != zero_zero(A) ) ).

% division_segment_not_0
tff(fact_5261_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_5262_is__unit__division__segment,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),euclid7384307370059645450egment(A,A3)),one_one(A))) ) ).

% is_unit_division_segment
tff(fact_5263_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F5: filter(A),P: fun(B,bool),F: fun(A,B)] :
      ( eventually(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),X6),F5)
     => ( eventually(B,P,map_filter_on(A,B,X6,F,F5))
      <=> eventually(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),aTP_Lamp_agj(set(A),fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),X6),P),F),F5) ) ) ).

% eventually_map_filter_on
tff(fact_5264_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A3,B2)) = euclid7384307370059645450egment(A,B2) ) ) ) ) ).

% division_segment_mod
tff(fact_5265_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),aa(A,nat,euclid6346220572633701492n_size(A),A3)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
              | ( B2 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_5266_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,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),Q2),B2)),R)),B2) = Q2 ) ) ) ) ) ).

% div_bounded
tff(fact_5267_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B2: A,R: A,Q2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R) = euclid7384307370059645450egment(A,B2) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,euclid6346220572633701492n_size(A),R)),aa(A,nat,euclid6346220572633701492n_size(A),B2)))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q2),B2)),R) = A3 )
               => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2) = Q2 ) ) ) ) ) ) ).

% div_eqI
tff(fact_5268_eventually__cofinite,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( eventually(A,P,cofinite(A))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_lv(fun(A,bool),fun(A,bool),P)))) ) ).

% eventually_cofinite
tff(fact_5269_Sup__filter__def,axiom,
    ! [A: $tType,S2: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S2) = abs_filter(A,aTP_Lamp_agk(set(filter(A)),fun(fun(A,bool),bool),S2)) ).

% Sup_filter_def
tff(fact_5270_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F: fun(A,set(B)),A5: set(B),F5: filter(A)] :
      ( filterlim(A,set(B),F,finite5375528669736107172at_top(B,A5),F5)
    <=> ! [X10: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite2(B),X10))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X10),A5)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_agl(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F),A5),X10),F5) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_5271_cofinite__def,axiom,
    ! [A: $tType] : cofinite(A) = abs_filter(A,aTP_Lamp_agm(fun(A,bool),bool)) ).

% cofinite_def
tff(fact_5272_sup__filter__def,axiom,
    ! [A: $tType,F5: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F5),F8) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_agn(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F5),F8)) ).

% sup_filter_def
tff(fact_5273_bot__filter__def,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_ago(fun(A,bool),bool)) ).

% bot_filter_def
tff(fact_5274_top__filter__def,axiom,
    ! [A: $tType] : top_top(filter(A)) = abs_filter(A,fAll(A)) ).

% top_filter_def
tff(fact_5275_principal__def,axiom,
    ! [A: $tType,S2: set(A)] : principal(A,S2) = abs_filter(A,aa(set(A),fun(fun(A,bool),bool),ball(A),S2)) ).

% principal_def
tff(fact_5276_map__filter__on__def,axiom,
    ! [B: $tType,A: $tType,X6: set(A),F: fun(A,B),F5: filter(A)] : map_filter_on(A,B,X6,F,F5) = 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_agq(set(A),fun(fun(A,B),fun(filter(A),fun(fun(B,bool),bool))),X6),F),F5)) ).

% map_filter_on_def
tff(fact_5277_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_ags(set(A),fun(set(A),filter(set(A))),A5)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_agt(set(A),fun(set(A),bool),A5)))) ).

% finite_subsets_at_top_def
tff(fact_5278_inf__filter__def,axiom,
    ! [A: $tType,F5: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),F8) = abs_filter(A,aa(filter(A),fun(fun(A,bool),bool),aTP_Lamp_agu(filter(A),fun(filter(A),fun(fun(A,bool),bool)),F5),F8)) ).

% inf_filter_def
tff(fact_5279_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F,at_top(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_agv(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5280_coinduct3,axiom,
    ! [A: $tType,F: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( order_mono(set(A),set(A),F)
     => ( 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),F,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_agw(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),F),X6)))))
         => pp(aa(set(A),bool,member(A,A3),complete_lattice_gfp(set(A),F))) ) ) ) ).

% coinduct3
tff(fact_5281_def__coinduct3,axiom,
    ! [A: $tType,A5: set(A),F: fun(set(A),set(A)),A3: A,X6: set(A)] :
      ( ( A5 = complete_lattice_gfp(set(A),F) )
     => ( order_mono(set(A),set(A),F)
       => ( 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),F,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_agx(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),A5),F),X6)))))
           => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% def_coinduct3
tff(fact_5282_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_agy(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_5283_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_agz(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5284_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),F5: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_zy(fun(nat,A),fun(nat,A),F),F5,at_top(nat))
    <=> filterlim(nat,A,F,F5,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_5285_gfp__rolling,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [G: fun(A,B),F: fun(B,A)] :
          ( order_mono(A,B,G)
         => ( order_mono(B,A,F)
           => ( aa(A,B,G,complete_lattice_gfp(A,aa(fun(B,A),fun(A,A),aTP_Lamp_wg(fun(A,B),fun(fun(B,A),fun(A,A)),G),F))) = complete_lattice_gfp(B,aa(fun(B,A),fun(B,B),aTP_Lamp_wh(fun(A,B),fun(fun(B,A),fun(B,B)),G),F)) ) ) ) ) ).

% gfp_rolling
tff(fact_5286_le__sequentially,axiom,
    ! [F5: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F5),at_top(nat)))
    <=> ! [N10: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N10),F5) ) ).

% le_sequentially
tff(fact_5287_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,fun(A,A))] :
          ( ! [X4: A,Y4: A,W2: A,Z4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F,X4),W2)),aa(A,A,aa(A,fun(A,A),F,Y4),Z4))) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_aha(fun(A,fun(A,A)),fun(A,A),F)) = complete_lattice_gfp(A,aTP_Lamp_wj(fun(A,fun(A,A)),fun(A,A),F)) ) ) ) ).

% gfp_gfp
tff(fact_5288_gfp__const,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [T2: A] : complete_lattice_gfp(A,aTP_Lamp_wq(A,fun(A,A),T2)) = T2 ) ).

% gfp_const
tff(fact_5289_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_ahb(A,fun(A,bool),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_5290_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A)] : complete_lattice_gfp(A,F) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ahc(fun(A,A),fun(A,bool),F))) ) ).

% gfp_def
tff(fact_5291_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_ahd(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P)) )
     => ( order_mono(set(A),set(A),aTP_Lamp_ahd(fun(set(A),fun(A,bool)),fun(set(A),set(A)),P))
       => ( pp(aa(set(A),bool,member(A,A3),X6))
         => ( ! [Z4: A] :
                ( pp(aa(set(A),bool,member(A,Z4),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)),Z4)) )
           => pp(aa(set(A),bool,member(A,A3),A5)) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_5292_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_5293_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_5294_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_5295_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_5296_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),N: nat] :
          ( order_mono(A,A,F)
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F)) = complete_lattice_gfp(A,F) ) ) ) ).

% gfp_funpow
tff(fact_5297_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_ahe(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5298_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F: fun(A,A),K: nat] :
          ( order_mono(A,A,F)
         => ( ( 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)),F),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),F),top_top(A)) )
           => ( complete_lattice_gfp(A,F) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_5299_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahf(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ).

% filterlim_at_top
tff(fact_5300_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F,at_top(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahf(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5301_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F: fun(B,A),F5: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F,at_top(A),F5)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ahg(fun(B,A),fun(fun(B,A),fun(B,bool)),F),G),F5)
           => filterlim(B,A,G,at_top(A),F5) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5302_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_ahh(fun(A,B),fun(B,fun(A,bool)),F),Z7),F5) ) ) ).

% filterlim_at_top_dense
tff(fact_5303_coinduct3__lemma,axiom,
    ! [A: $tType,X6: set(A),F: 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),F,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ahi(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F)))))
     => ( order_mono(set(A),set(A),F)
       => 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_ahi(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F))),aa(set(A),set(A),F,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ahi(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X6),F))))) ) ) ).

% coinduct3_lemma
tff(fact_5304_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_ahj(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_5305_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),F5: filter(B)] : filtercomap(A,B,F,F5) = abs_filter(A,aa(filter(B),fun(fun(A,bool),bool),aTP_Lamp_ahk(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),F),F5)) ).

% filtercomap_def
tff(fact_5306_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_ahl(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C2))) ) ).

% at_top_sub
tff(fact_5307_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5308_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F5: filter(A),F: fun(B,A)] :
      ( eventually(A,P,F5)
     => eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afu(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F),filtercomap(B,A,F,F5)) ) ).

% eventually_filtercomapI
tff(fact_5309_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),I)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).

% image_add_atLeast
tff(fact_5310_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_co(nat,bool),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_5311_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_5312_filtercomap__ident,axiom,
    ! [A: $tType,F5: filter(A)] : filtercomap(A,A,aTP_Lamp_ak(A,A),F5) = F5 ).

% filtercomap_ident
tff(fact_5313_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),G: fun(B,C),F5: filter(C)] : filtercomap(A,B,F,filtercomap(B,C,G,F5)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ahm(fun(A,B),fun(fun(B,C),fun(A,C)),F),G),F5) ).

% filtercomap_filtercomap
tff(fact_5314_filterlim__filtercomap__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(B,C),G5: filter(C),F5: filter(A)] :
      ( filterlim(A,B,F,filtercomap(B,C,G,G5),F5)
    <=> filterlim(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),G5,F5) ) ).

% filterlim_filtercomap_iff
tff(fact_5315_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),F5: fun(C,filter(B)),B4: set(C)] : filtercomap(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F5),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_ahn(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F),F5)),B4)) ).

% filtercomap_INF
tff(fact_5316_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,C),F5: 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_aho(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F),F5)),B4))),filtercomap(A,C,F,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F5),B4))))) ).

% filtercomap_SUP
tff(fact_5317_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_5318_AboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] : order_AboveS(A,R,A5) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_ahp(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A5)) ).

% AboveS_def
tff(fact_5319_prod__decode__triangle__add,axiom,
    ! [K: nat,M: nat] : aa(nat,product_prod(nat,nat),nat_prod_decode,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M)) = aa(nat,product_prod(nat,nat),nat_prod_decode_aux(K),M) ).

% prod_decode_triangle_add
tff(fact_5320_cSUP__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(C),B4: fun(C,set(D)),F: fun(D,B)] :
          ( ( A5 != bot_bot(set(C)) )
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,member(C,X4),A5))
               => ( aa(C,set(D),B4,X4) != bot_bot(set(D)) ) )
           => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ahq(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B4),F)),A5)))
             => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B4),A5)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_ahr(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B4),F)),A5)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5321_bdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ahs(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5))
        <=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A5))
            & condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A5)) ) ) ) ).

% bdd_above_image_sup
tff(fact_5322_bdd__above__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I5: set(B),A5: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( condit941137186595557371_above(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5)))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => condit941137186595557371_above(A,aa(B,set(A),A5,X3)) ) ) ) ) ).

% bdd_above_UN
tff(fact_5323_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F: fun(B,A),A5: set(B),Y3: A,I: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A5))
         => ( 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,F),A5))),Y3))
           => ( pp(aa(set(B),bool,member(B,I),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y3)) ) ) ) ) ).

% cSUP_lessD
tff(fact_5324_prod__decode__def,axiom,
    nat_prod_decode = nat_prod_decode_aux(zero_zero(nat)) ).

% prod_decode_def
tff(fact_5325_less__cSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A5: set(B),F: fun(B,A),A3: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A5))
           => ( 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,F),A5))))
            <=> ? [X3: B] :
                  ( pp(aa(set(B),bool,member(B,X3),A5))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X3))) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5326_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ( A5 != bot_bot(set(B)) )
         => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F),A5))
           => ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A5))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),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_aht(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_5327_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S2)
           => ( aa(set(A),A,complete_Sup_Sup(A),S2) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ahu(set(A),fun(A,bool),S2))) ) ) ) ) ).

% cSup_cInf
tff(fact_5328_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,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,A),fun(C,B),aTP_Lamp_ahv(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5)),I5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A5),I5))))) ) ) ) ) ).

% mono_cSUP
tff(fact_5329_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F)
         => ( condit941137186595557371_above(A,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F),A5))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A5)))) ) ) ) ) ).

% mono_cSup
tff(fact_5330_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),field2(A,R)))
       => ( ( order_AboveS(A,R,B4) != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,member(A,B2),B4))
           => ( ( bNF_Wellorder_wo_suc(A,R,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,R,B4))),R)) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_5331_list__decode_Oelims,axiom,
    ! [X: nat,Y3: list(nat)] :
      ( ( nat_list_decode(X) = Y3 )
     => ( ( ( X = zero_zero(nat) )
         => ( Y3 != nil(nat) ) )
       => ~ ! [N2: nat] :
              ( ( X = aa(nat,nat,suc,N2) )
             => ( Y3 != aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ahw(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_5332_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),B4: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B4),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R,B4)))
         => ( ! [A14: A] :
                ( pp(aa(set(A),bool,member(A,A14),order_AboveS(A,R,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),A14)),R)) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R,B4) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_5333_list__decode_Osimps_I1_J,axiom,
    nat_list_decode(zero_zero(nat)) = nil(nat) ).

% list_decode.simps(1)
tff(fact_5334_wo__rel_Osuc__least__AboveS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R,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,R,B4)),A3)),R)) ) ) ).

% wo_rel.suc_least_AboveS
tff(fact_5335_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] : nat_list_decode(aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ahw(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% list_decode.simps(2)
tff(fact_5336_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B2: A] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( order_ofilter(A,R,A5)
       => ( ( order_AboveS(A,R,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,R,A5))),R))
           => ( ( B2 != bNF_Wellorder_wo_suc(A,R,A5) )
             => pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_5337_list__decode_Opelims,axiom,
    ! [X: nat,Y3: list(nat)] :
      ( ( nat_list_decode(X) = Y3 )
     => ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y3 = nil(nat) )
             => ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat))) ) )
         => ~ ! [N2: nat] :
                ( ( X = aa(nat,nat,suc,N2) )
               => ( ( Y3 = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ahw(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) )
                 => ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N2))) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_5338_cINF__UNION,axiom,
    ! [D: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(C),B4: fun(C,set(D)),F: fun(D,B)] :
          ( ( A5 != bot_bot(set(C)) )
         => ( ! [X4: C] :
                ( pp(aa(set(C),bool,member(C,X4),A5))
               => ( aa(C,set(D),B4,X4) != bot_bot(set(D)) ) )
           => ( pp(aa(set(B),bool,condit1013018076250108175_below(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ahq(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B4),F)),A5))))
             => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),image2(C,set(D),B4),A5)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_ahx(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B4),F)),A5)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5339_bdd__below__image__inf,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F: fun(B,A),G: fun(B,A),A5: set(B)] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ahy(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5)))
        <=> ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A5)))
            & pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,G),A5))) ) ) ) ).

% bdd_below_image_inf
tff(fact_5340_bdd__below__UN,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [I5: set(B),A5: fun(B,set(A))] :
          ( pp(aa(set(B),bool,finite_finite2(B),I5))
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))))
          <=> ! [X3: B] :
                ( pp(aa(set(B),bool,member(B,X3),I5))
               => pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(B,set(A),A5,X3))) ) ) ) ) ).

% bdd_below_UN
tff(fact_5341_list__decode_Opsimps_I1_J,axiom,
    ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat)))
   => ( nat_list_decode(zero_zero(nat)) = nil(nat) ) ) ).

% list_decode.psimps(1)
tff(fact_5342_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),A0))
     => ( ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat)))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N2)))
             => ( ! [X2: nat,Y5: nat] :
                    ( ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y5) = aa(nat,product_prod(nat,nat),nat_prod_decode,N2) )
                   => pp(aa(nat,bool,P,Y5)) )
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,A0)) ) ) ) ).

% list_decode.pinduct
tff(fact_5343_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F: fun(B,A),A5: set(B),Y3: A,I: B] :
          ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A5)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5))))
           => ( pp(aa(set(B),bool,member(B,I),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),aa(B,A,F,I))) ) ) ) ) ).

% less_cINF_D
tff(fact_5344_wo__rel_Oofilter__UNION,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(A))] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( ! [I2: B] :
            ( pp(aa(set(B),bool,member(B,I2),I5))
           => order_ofilter(A,R,aa(B,set(A),A5,I2)) )
       => order_ofilter(A,R,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_5345_cINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A5: set(B),F: fun(B,A),A3: A] :
          ( ( A5 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A5)))
           => ( 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,F),A5))),A3))
            <=> ? [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,F,X3)),A3)) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5346_cINF__inf__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ( A5 != bot_bot(set(B)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,F),A5)))
           => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(B),set(A),image2(B,A,G),A5)))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F),A5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,G),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_ahz(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A5)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5347_cSUP__eq__cINF__D,axiom,
    ! [B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [F: fun(C,B),A5: set(C),A3: C] :
          ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,F),A5)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A5)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,F),A5))
           => ( pp(aa(set(B),bool,condit1013018076250108175_below(B),aa(set(C),set(B),image2(C,B,F),A5)))
             => ( pp(aa(set(C),bool,member(C,A3),A5))
               => ( aa(C,B,F,A3) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F),A5)) ) ) ) ) ) ) ).

% cSUP_eq_cINF_D
tff(fact_5348_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N)))
     => ( nat_list_decode(aa(nat,nat,suc,N)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ahw(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ) ) ).

% list_decode.psimps(2)
tff(fact_5349_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S2: set(A)] :
          ( ( S2 != bot_bot(set(A)) )
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),S2))
           => ( aa(set(A),A,complete_Inf_Inf(A),S2) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aia(set(A),fun(A,bool),S2))) ) ) ) ) ).

% cInf_cSup
tff(fact_5350_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F: fun(A,B),A5: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F)
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),aa(set(C),set(A),image2(C,A,A5),I5)))
           => ( ( I5 != bot_bot(set(C)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,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_ahv(fun(A,B),fun(fun(C,A),fun(C,B)),F),A5)),I5)))) ) ) ) ) ).

% mono_cINF
tff(fact_5351_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F: fun(A,B),A5: set(A)] :
          ( order_mono(A,B,F)
         => ( pp(aa(set(A),bool,condit1013018076250108175_below(A),A5))
           => ( ( A5 != bot_bot(set(A)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,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,F),A5)))) ) ) ) ) ).

% mono_cInf
tff(fact_5352_ofilterIncl__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),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_aib(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R))) ).

% ofilterIncl_def
tff(fact_5353_wo__rel_Oofilter__under__UNION,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R)
     => ( order_ofilter(A,R,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,R)),A5)) ) ) ) ).

% wo_rel.ofilter_under_UNION
tff(fact_5354_bsqr__max2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A1: A,A22: A,B1: A,B22: A] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(product_prod(A,A),product_prod(A,A))),bool,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,R)))
       => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R,A1,A22)),bNF_We1388413361240627857o_max2(A,R,B1,B22))),R)) ) ) ).

% bsqr_max2
tff(fact_5355_under__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : aa(A,set(A),order_under(A,R),A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aic(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% under_def
tff(fact_5356_well__order__on__domain,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_well_order_on(A,A5,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(A),bool,member(A,A3),A5))
          & pp(aa(set(A),bool,member(A,B2),A5)) ) ) ) ).

% well_order_on_domain
tff(fact_5357_ofilter__Restr__under,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),A3: A] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))),A3) = aa(A,set(A),order_under(A,R),A3) ) ) ) ) ).

% ofilter_Restr_under
tff(fact_5358_Well__order__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => order_well_order_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Well_order_Restr
tff(fact_5359_natLeq__on__well__order__on,axiom,
    ! [N: nat] : order_well_order_on(nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_well_order_on
tff(fact_5360_bsqr__ofilter,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),D5: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R),D5)
       => ( 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))),D5),product_Sigma(A,A,field2(A,R),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),R))))
         => ( ~ ? [A4: A] : field2(A,R) = aa(A,set(A),order_under(A,R),A4)
           => ? [A15: set(A)] :
                ( order_ofilter(A,R,A15)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A15),field2(A,R)))
                & pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),D5),product_Sigma(A,A,A15,aTP_Lamp_wx(set(A),fun(A,set(A)),A15)))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_5361_well__order__on__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),field2(A,R)))
       => 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ) ).

% well_order_on_Restr
tff(fact_5362_Field__Restr__ofilter,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) = A5 ) ) ) ).

% Field_Restr_ofilter
tff(fact_5363_natLeq__on__Well__order,axiom,
    ! [N: nat] : order_well_order_on(nat,field2(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N)))),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N)))) ).

% natLeq_on_Well_order
tff(fact_5364_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( order_well_order_on(A,field2(A,R),R)
      <=> ! [A9: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A9),field2(A,R)))
           => ( ( A9 != bot_bot(set(A)) )
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,member(A,X3),A9))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,member(A,Xa2),A9))
                     => pp(aa(set(product_prod(A,A)),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),Xa2)),R)) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_5365_ofilter__Restr__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))),A5) ) ) ) ).

% ofilter_Restr_subset
tff(fact_5366_ofilter__Restr__Int,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(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_5367_UNION__inj__on__ofilter,axiom,
    ! [C: $tType,A: $tType,B: $tType,R: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(A)),F: fun(A,C)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( ! [I2: B] :
            ( pp(aa(set(B),bool,member(B,I2),I5))
           => order_ofilter(A,R,aa(B,set(A),A5,I2)) )
       => ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),I5))
             => inj_on(A,C,F,aa(B,set(A),A5,I2)) )
         => inj_on(A,C,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A5),I5))) ) ) ) ).

% UNION_inj_on_ofilter
tff(fact_5368_ofilter__subset__embedS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_5369_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_5370_ordLess__irreflexive,axiom,
    ! [A: $tType,R: 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))),R),R)),bNF_We4044943003108391690rdLess(A,A))) ).

% ordLess_irreflexive
tff(fact_5371_ordLess__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_transitive
tff(fact_5372_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_aid(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLess_def
tff(fact_5373_finite__ordLess__infinite,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
         => ( ~ pp(aa(set(B),bool,finite_finite2(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))),R),R3)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).

% finite_ordLess_infinite
tff(fact_5374_ofilter__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A5),field2(A,R)))
        <=> 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))),R)),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% ofilter_ordLess
tff(fact_5375_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_5376_underS__Restr__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( ( field2(A,R) != 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))),R),product_Sigma(A,A,order_underS(A,R,A3),aa(A,fun(A,set(A)),aTP_Lamp_aie(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),A3)))),R)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% underS_Restr_ordLess
tff(fact_5377_ofilter__embed,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
      <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),field2(A,R)))
          & pp(aa(fun(A,A),bool,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))),R),id(A))) ) ) ) ).

% ofilter_embed
tff(fact_5378_iso__comp__embed,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( bNF_Wellorder_iso(A,B,R,R3,F)
     => ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R3,R7),F4))
       => pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).

% iso_comp_embed
tff(fact_5379_embed__comp__iso,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R3),F))
     => ( bNF_Wellorder_iso(B,C,R3,R7,F4)
       => pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).

% embed_comp_iso
tff(fact_5380_comp__embed,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R3),F))
     => ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R3,R7),F4))
       => pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).

% comp_embed
tff(fact_5381_comp__iso,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( bNF_Wellorder_iso(A,B,R,R3,F)
     => ( bNF_Wellorder_iso(B,C,R3,R7,F4)
       => bNF_Wellorder_iso(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ).

% comp_iso
tff(fact_5382_iso__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X2: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( bNF_Wellorder_iso(A,A2,X2,Xa3,Xb)
    <=> ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X2,Xa3),Xb))
        & bij_betw(A,A2,Xb,field2(A,X2),field2(A2,Xa3)) ) ) ).

% iso_defs(2)
tff(fact_5383_iso__forward,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: A,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: 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),Y3)),R))
     => ( bNF_Wellorder_iso(A,B,R,R3,F)
       => 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,F,X)),aa(A,B,F,Y3))),R3)) ) ) ).

% iso_forward
tff(fact_5384_underS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_underS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aif(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% underS_def
tff(fact_5385_underS__I,axiom,
    ! [A: $tType,I: A,J: A,R4: set(product_prod(A,A))] :
      ( ( I != 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),I),J)),R4))
       => pp(aa(set(A),bool,member(A,I),order_underS(A,R4,J))) ) ) ).

% underS_I
tff(fact_5386_underS__E,axiom,
    ! [A: $tType,I: A,R4: set(product_prod(A,A)),J: A] :
      ( pp(aa(set(A),bool,member(A,I),order_underS(A,R4,J)))
     => ( ( I != 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),I),J)),R4)) ) ) ).

% underS_E
tff(fact_5387_ordLess__not__embed,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),bNF_We4044943003108391690rdLess(A,B)))
     => ~ ? [X_12: fun(B,A)] : pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R3,R),X_12)) ) ).

% ordLess_not_embed
tff(fact_5388_embed__implies__iso__Restr,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(B,A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R3,R),F))
         => bNF_Wellorder_iso(B,A,R3,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),field2(B,R3)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_aig(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R3),F))),F) ) ) ) ).

% embed_implies_iso_Restr
tff(fact_5389_embedS__comp__iso,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( bNF_Wellorder_embedS(A,B,R,R3,F)
     => ( bNF_Wellorder_iso(B,C,R3,R7,F4)
       => bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ).

% embedS_comp_iso
tff(fact_5390_iso__iff2,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B)] :
      ( bNF_Wellorder_iso(A,B,R,R3,F)
    <=> ( bij_betw(A,B,F,field2(A,R),field2(B,R3))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),field2(A,R)))
           => ! [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),field2(A,R)))
               => ( pp(aa(set(product_prod(A,A)),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),Xa2)),R))
                <=> 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,F,X3)),aa(A,B,F,Xa2))),R3)) ) ) ) ) ) ).

% iso_iff2
tff(fact_5391_BNF__Wellorder__Constructions_OordLess__Field,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),F: 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))),R1),R22)),bNF_We4044943003108391690rdLess(A,B)))
     => ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R1,R22),F))
       => ( aa(set(A),set(B),image2(A,B,F),field2(A,R1)) != field2(B,R22) ) ) ) ).

% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_5392_embedS__defs_I2_J,axiom,
    ! [A: $tType,A2: $tType,X2: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( bNF_Wellorder_embedS(A,A2,X2,Xa3,Xb)
    <=> ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X2,Xa3),Xb))
        & ~ bij_betw(A,A2,Xb,field2(A,X2),field2(A2,Xa3)) ) ) ).

% embedS_defs(2)
tff(fact_5393_ordLess__iff,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),bNF_We4044943003108391690rdLess(A,B)))
    <=> ( order_well_order_on(A,field2(A,R),R)
        & order_well_order_on(B,field2(B,R3),R3)
        & ~ ? [X_1: fun(B,A)] : pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R3,R),X_1)) ) ) ).

% ordLess_iff
tff(fact_5394_embedS__comp__embed,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( bNF_Wellorder_embedS(A,B,R,R3,F)
         => ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R3,R7),F4))
           => bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).

% embedS_comp_embed
tff(fact_5395_embed__comp__embedS,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R3),F))
         => ( bNF_Wellorder_embedS(B,C,R3,R7,F4)
           => bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).

% embed_comp_embedS
tff(fact_5396_underS__incl__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( order_679001287576687338der_on(A,field2(A,R),R)
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,B2),field2(A,R)))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2)))
          <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).

% underS_incl_iff
tff(fact_5397_embed__defs_I2_J,axiom,
    ! [A2: $tType,A: $tType,X2: set(product_prod(A,A)),Xa3: set(product_prod(A2,A2)),Xb: fun(A,A2)] :
      ( pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,X2,Xa3),Xb))
    <=> ! [Xc2: A] :
          ( pp(aa(set(A),bool,member(A,Xc2),field2(A,X2)))
         => bij_betw(A,A2,Xb,aa(A,set(A),order_under(A,X2),Xc2),aa(A2,set(A2),order_under(A2,Xa3),aa(A,A2,Xb,Xc2))) ) ) ).

% embed_defs(2)
tff(fact_5398_iso__comp__embedS,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,R3),R3)
       => ( bNF_Wellorder_iso(A,B,R,R3,F)
         => ( bNF_Wellorder_embedS(B,C,R3,R7,F4)
           => bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).

% iso_comp_embedS
tff(fact_5399_embed__ordLess__ofilterIncl,axiom,
    ! [B: $tType,A: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: 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))),R1),R22)),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))),R22),R33)),bNF_We4044943003108391690rdLess(B,C)))
       => ( pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R1,R33),F132))
         => ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R22,R33),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),field2(A,R1))),aa(set(B),set(C),image2(B,C,F232),field2(B,R22)))),bNF_We413866401316099525erIncl(C,R33))) ) ) ) ) ).

% embed_ordLess_ofilterIncl
tff(fact_5400_ofilter__subset__embed,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( order_ofilter(A,R,B4)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),B4))
          <=> pp(aa(fun(A,A),bool,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4)))),id(A))) ) ) ) ) ).

% ofilter_subset_embed
tff(fact_5401_ordLess__iff__ordIso__Restr,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,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))),R3),R)),bNF_We4044943003108391690rdLess(B,A)))
        <=> ? [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),field2(A,R)))
              & 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),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,order_underS(A,R,X3),aa(A,fun(A,set(A)),aTP_Lamp_aie(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X3))))),bNF_Wellorder_ordIso(B,A))) ) ) ) ) ).

% ordLess_iff_ordIso_Restr
tff(fact_5402_ordLeq__iff__ordLess__Restr,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(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))),R),R3)),bNF_Wellorder_ordLeq(A,B)))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),field2(A,R)))
             => 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))),R),product_Sigma(A,A,order_underS(A,R,X3),aa(A,fun(A,set(A)),aTP_Lamp_aie(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X3)))),R3)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ) ).

% ordLeq_iff_ordLess_Restr
tff(fact_5403_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A),B4: set(A)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_ofilter(A,R,A5)
       => ( order_ofilter(A,R,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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))))),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_5404_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R: 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),R)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [P6: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),field2(B,P6)),field2(B,R)))
          & 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),P6)),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))),P6),R)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_ordLeq
tff(fact_5405_ordLeq__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordLeq_ordIso_trans
tff(fact_5406_ordIso__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordIso_ordLeq_trans
tff(fact_5407_ordLeq__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_Wellorder_ordLeq(A,C))) ) ) ).

% ordLeq_transitive
tff(fact_5408_ordIso__transitive,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_Wellorder_ordIso(A,C))) ) ) ).

% ordIso_transitive
tff(fact_5409_ordIso__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),R3)),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))),R),R3)),bNF_Wellorder_ordLeq(A,B))) ) ).

% ordIso_imp_ordLeq
tff(fact_5410_ordIso__iff__ordLeq,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),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))),R),R3)),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))),R3),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% ordIso_iff_ordLeq
tff(fact_5411_ordIso__symmetric,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),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))),R3),R)),bNF_Wellorder_ordIso(B,A))) ) ).

% ordIso_symmetric
tff(fact_5412_ordLeq__iff__ordLess__or__ordIso,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),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))),R),R3)),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))),R),R3)),bNF_Wellorder_ordIso(A,B))) ) ) ).

% ordLeq_iff_ordLess_or_ordIso
tff(fact_5413_ordLess__ordLeq__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_ordLeq_trans
tff(fact_5414_ordLeq__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLeq_ordLess_trans
tff(fact_5415_ordLess__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),R3)),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))),R),R3)),bNF_Wellorder_ordLeq(A,B))) ) ).

% ordLess_imp_ordLeq
tff(fact_5416_not__ordLess__ordLeq,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),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))),R3),R)),bNF_Wellorder_ordLeq(B,A))) ) ).

% not_ordLess_ordLeq
tff(fact_5417_ordLess__ordIso__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordLess_ordIso_trans
tff(fact_5418_ordIso__ordLess__trans,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),R7: 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))),R),R3)),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))),R3),R7)),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))),R),R7)),bNF_We4044943003108391690rdLess(A,C))) ) ) ).

% ordIso_ordLess_trans
tff(fact_5419_not__ordLess__ordIso,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),R3)),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))),R),R3)),bNF_Wellorder_ordIso(A,B))) ) ).

% not_ordLess_ordIso
tff(fact_5420_finite__well__order__on__ordIso,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( order_well_order_on(A,A5,R)
       => ( 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))),R),R3)),bNF_Wellorder_ordIso(A,A))) ) ) ) ).

% finite_well_order_on_ordIso
tff(fact_5421_exists__minim__Well__order,axiom,
    ! [A: $tType,R4: set(set(product_prod(A,A)))] :
      ( ( R4 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X4: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X4),R4))
           => order_well_order_on(A,field2(A,X4),X4) )
       => ? [X4: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X4),R4))
            & ! [Xa3: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),Xa3),R4))
               => 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))),X4),Xa3)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Well_order
tff(fact_5422_ordLeq__total,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(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))),R),R3)),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))),R3),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% ordLeq_total
tff(fact_5423_ordLeq__reflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% ordLeq_reflexive
tff(fact_5424_ordLeq__Well__order__simp,axiom,
    ! [A: $tType,B: $tType,R: 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))),R),R3)),bNF_Wellorder_ordLeq(A,B)))
     => ( order_well_order_on(A,field2(A,R),R)
        & order_well_order_on(B,field2(B,R3),R3) ) ) ).

% ordLeq_Well_order_simp
tff(fact_5425_ordIso__reflexive,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R)),bNF_Wellorder_ordIso(A,A))) ) ).

% ordIso_reflexive
tff(fact_5426_Well__order__iso__copy,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: set(product_prod(A,A)),F: fun(A,B),A10: set(B)] :
      ( order_well_order_on(A,A5,R)
     => ( bij_betw(A,B,F,A5,A10)
       => ? [R8: set(product_prod(B,B))] :
            ( order_well_order_on(B,A10,R8)
            & 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))),R),R8)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% Well_order_iso_copy
tff(fact_5427_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R3: set(product_prod(A,A)),R: 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),R)),bNF_We4044943003108391690rdLess(A,B)))
    <=> ? [P6: set(product_prod(B,B))] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),field2(B,P6)),field2(B,R)))
          & 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),P6)),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))),P6),R)),bNF_We4044943003108391690rdLess(B,B))) ) ) ).

% internalize_ordLess
tff(fact_5428_not__ordLess__iff__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,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))),R3),R)),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))),R),R3)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% not_ordLess_iff_ordLeq
tff(fact_5429_not__ordLeq__iff__ordLess,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(B,field2(B,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))),R3),R)),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))),R),R3)),bNF_We4044943003108391690rdLess(A,B))) ) ) ) ).

% not_ordLeq_iff_ordLess
tff(fact_5430_ordLess__or__ordLeq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( order_well_order_on(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))),R),R3)),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))),R3),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% ordLess_or_ordLeq
tff(fact_5431_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_aih(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordLeq_def
tff(fact_5432_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_aii(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).

% ordIso_def
tff(fact_5433_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_aij(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)),aa(fun(set(set(A)),set(A)),fun(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_5434_dir__image__ordIso,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( inj_on(A,B,F,field2(A,R))
       => 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))),R),bNF_We2720479622203943262_image(A,B,R,F))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% dir_image_ordIso
tff(fact_5435_map__comp__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,option(C)),G: fun(A,option(B)),X2: A] : map_comp(B,C,A,F,G,X2) = aa(option(B),option(C),aa(fun(B,option(C)),fun(option(B),option(C)),aa(option(C),fun(fun(B,option(C)),fun(option(B),option(C))),case_option(option(C),B),none(C)),F),aa(A,option(B),G,X2)) ).

% map_comp_def
tff(fact_5436_map__comp__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,C: $tType,M23: fun(B,option(A)),K: B,M12: fun(A,option(C))] :
      ( ( aa(B,option(A),M23,K) = none(A) )
     => ( map_comp(A,C,B,M12,M23,K) = none(C) ) ) ).

% map_comp_simps(1)
tff(fact_5437_map__comp__simps_I2_J,axiom,
    ! [B: $tType,C: $tType,A: $tType,M23: fun(B,option(A)),K: B,K7: A,M12: fun(A,option(C))] :
      ( ( aa(B,option(A),M23,K) = aa(A,option(A),some(A),K7) )
     => ( map_comp(A,C,B,M12,M23,K) = aa(A,option(C),M12,K7) ) ) ).

% map_comp_simps(2)
tff(fact_5438_map__comp__empty_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,M: fun(C,option(B)),X2: A] : map_comp(C,B,A,M,aTP_Lamp_aik(A,option(C)),X2) = none(B) ).

% map_comp_empty(1)
tff(fact_5439_map__comp__empty_I2_J,axiom,
    ! [C: $tType,B: $tType,D: $tType,M: fun(C,option(B)),X2: C] : map_comp(B,D,C,aTP_Lamp_ail(B,option(D)),M,X2) = none(D) ).

% map_comp_empty(2)
tff(fact_5440_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_xa(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ).

% card_of_Times2
tff(fact_5441_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_nh(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordLeq(B,product_prod(B,A)))) ) ).

% card_of_Times1
tff(fact_5442_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_wx(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_5443_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_wx(set(A),fun(A,set(A)),A5))))),bNF_Wellorder_ordLeq(A,product_prod(A,A)))) ).

% card_of_Times3
tff(fact_5444_card__of__Sigma__mono1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: fun(A,set(C))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),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,X4))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),B4,X4)))),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_5445_card__of__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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_xn(set(C),fun(A,set(C)),C6)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,B4,aTP_Lamp_xm(set(C),fun(B,set(C)),C6))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% card_of_Times_mono1
tff(fact_5446_card__of__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C6: 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,C6,aTP_Lamp_aim(set(A),fun(C,set(A)),A5)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,C6,aTP_Lamp_ain(set(B),fun(C,set(B)),B4))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% card_of_Times_mono2
tff(fact_5447_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_xa(set(B),fun(A,set(B)),B4)))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B4,aTP_Lamp_nh(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% card_of_Times_commute
tff(fact_5448_Func__Times__Range,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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_xm(set(C),fun(B,set(C)),C6))))),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_aio(set(A),fun(set(C),fun(fun(A,B),set(fun(A,C)))),A5),C6))))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% Func_Times_Range
tff(fact_5449_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),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))))),bNF_Wellorder_ordIso(bool,A))) ) ).

% card_of_bool
tff(fact_5450_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_5451_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_5452_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),pow2(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_5453_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),pow2(A,A5)))),bNF_We4044943003108391690rdLess(A,set(A)))) ).

% card_of_Pow
tff(fact_5454_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)))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),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,X4))),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_5455_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_5456_map__comp__Some__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M12: fun(B,option(A)),M23: fun(C,option(B)),K: C,V: A] :
      ( ( map_comp(B,A,C,M12,M23,K) = aa(A,option(A),some(A),V) )
    <=> ? [K8: B] :
          ( ( aa(C,option(B),M23,K) = aa(B,option(B),some(B),K8) )
          & ( aa(B,option(A),M12,K8) = aa(A,option(A),some(A),V) ) ) ) ).

% map_comp_Some_iff
tff(fact_5457_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_xa(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_nh(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_5458_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_xa(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_5459_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_xa(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordIso(A,product_prod(A,B)))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_5460_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_nh(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_5461_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_nh(set(A),fun(B,set(A)),A5))))),bNF_Wellorder_ordIso(A,product_prod(B,A)))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_5462_card__of__Func__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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_xa(set(B),fun(A,set(B)),B4)),C6))),bNF_Ca6860139660246222851ard_of(fun(A,fun(B,C)),bNF_Wellorder_Func(A,fun(B,C),A5,bNF_Wellorder_Func(B,C,B4,C6))))),bNF_Wellorder_ordIso(fun(product_prod(A,B),C),fun(A,fun(B,C))))) ).

% card_of_Func_Times
tff(fact_5463_card__of__image,axiom,
    ! [B: $tType,A: $tType,F: 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,F),A5))),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordLeq(A,B))) ).

% card_of_image
tff(fact_5464_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_5465_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_5466_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_5467_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_5468_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_5469_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_5470_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_5471_card__of__mono2,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),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,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3)))),bNF_Wellorder_ordLeq(A,B))) ) ).

% card_of_mono2
tff(fact_5472_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_5473_card__of__cong,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),R3)),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,field2(A,R))),bNF_Ca6860139660246222851ard_of(B,field2(B,R3)))),bNF_Wellorder_ordIso(A,B))) ) ).

% card_of_cong
tff(fact_5474_card__of__ordLeqI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( inj_on(A,B,F,A5)
     => ( ! [A4: A] :
            ( pp(aa(set(A),bool,member(A,A4),A5))
           => pp(aa(set(B),bool,member(B,aa(A,B,F,A4)),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_5475_dir__image__def,axiom,
    ! [A2: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,A2)] : bNF_We2720479622203943262_image(A,A2,R,F) = 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_aip(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),R),F)) ).

% dir_image_def
tff(fact_5476_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ? [F3: fun(B,A),B9: set(B)] : bij_betw(B,A,F3,B9,A5) ) ).

% ex_bij_betw
tff(fact_5477_card__of__least,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( order_well_order_on(A,A5,R)
     => 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)),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_least
tff(fact_5478_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),R3)),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,field2(A,R))),R3)),bNF_We4044943003108391690rdLess(A,B))) ) ).

% BNF_Cardinal_Order_Relation.ordLess_Field
tff(fact_5479_card__of__ordIso,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ? [F6: fun(A,B)] : bij_betw(A,B,F6,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_5480_card__of__ordIsoI,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A),B4: set(B)] :
      ( bij_betw(A,B,F,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_5481_type__copy__set__bd,axiom,
    ! [A: $tType,D: $tType,C: $tType,B: $tType,S2: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X: D] :
      ( ! [Y4: 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),S2,Y4))),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)),aa(fun(A,set(B)),fun(fun(D,A),fun(D,set(B))),comp(A,set(B),D),S2),Rep),X))),Bd)),bNF_Wellorder_ordLeq(B,C))) ) ).

% type_copy_set_bd
tff(fact_5482_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ? [G6: fun(B,A)] : aa(set(B),set(A),image2(B,A,G6),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_5483_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B4: set(A),F: 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,F),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_5484_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),insert2(B,B2),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_singl_ordLeq
tff(fact_5485_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B4: set(A),A5: set(B)] :
      ( ( B4 != bot_bot(set(A)) )
     => ( ~ ? [F6: fun(B,A)] : aa(set(B),set(A),image2(B,A,F6),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_5486_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C6: 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,C6))),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B8),C6))
          & 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,B8))),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,B8)),bNF_Ca6860139660246222851ard_of(B,C6))),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_5487_card__of__Field__ordLess,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% card_of_Field_ordLess
tff(fact_5488_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_aiq(set(B),fun(fun(A,B),bool),B4))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).

% card_of_Func_UNIV
tff(fact_5489_map__comp__None__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,M12: fun(B,option(A)),M23: fun(C,option(B)),K: C] :
      ( ( map_comp(B,A,C,M12,M23,K) = none(A) )
    <=> ( ( aa(C,option(B),M23,K) = none(B) )
        | ? [K8: B] :
            ( ( aa(C,option(B),M23,K) = aa(B,option(B),some(B),K8) )
            & ( aa(B,option(A),M12,K8) = none(A) ) ) ) ) ).

% map_comp_None_iff
tff(fact_5490_ordLeq__Times__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C6: 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))),R),R3)),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,field2(A,R),aTP_Lamp_xn(set(C),fun(A,set(C)),C6)))),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,field2(B,R3),aTP_Lamp_xm(set(C),fun(B,set(C)),C6))))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% ordLeq_Times_mono1
tff(fact_5491_ordLeq__Times__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: 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))),R),R3)),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_air(set(product_prod(A,A)),fun(C,set(A)),R)))),bNF_Ca6860139660246222851ard_of(product_prod(C,B),product_Sigma(C,B,A5,aTP_Lamp_ais(set(product_prod(B,B)),fun(C,set(B)),R3))))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% ordLeq_Times_mono2
tff(fact_5492_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B4: set(B)] :
      ( ? [F6: fun(A,B)] :
          ( inj_on(A,B,F6,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,F6),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_5493_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
    <=> ? [B8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B8),field2(B,R)))
          & pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),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,B8))),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,B8)),R)),bNF_Wellorder_ordLeq(B,B))) ) ) ).

% internalize_card_of_ordLeq
tff(fact_5494_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ~ ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,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,F6),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_5495_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)))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),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,X4))),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_5496_regularCard__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R)
    <=> ! [K9: set(A)] :
          ( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),K9),field2(A,R)))
            & bNF_Ca7293521722713021262ofinal(A,K9,R) )
         => 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)),R)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% regularCard_def
tff(fact_5497_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_5498_Card__order__iff__Restr__underS,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_well_order_on(A,field2(A,R),R)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
      <=> ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),field2(A,R)))
           => 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))),R),product_Sigma(A,A,order_underS(A,R,X3),aa(A,fun(A,set(A)),aTP_Lamp_aie(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),X3)))),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_We4044943003108391690rdLess(A,A))) ) ) ) ).

% Card_order_iff_Restr_underS
tff(fact_5499_Card__order__trans,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y3: A,Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( X != 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),X),Y3)),R))
         => ( ( Y3 != Z )
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
             => ( ( X != Z )
                & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R)) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_5500_ordLeq__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R)),bNF_Wellorder_ordLeq(A,A))) ) ).

% ordLeq_refl
tff(fact_5501_ordIso__refl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),R)),bNF_Wellorder_ordIso(A,A))) ) ).

% ordIso_refl
tff(fact_5502_card__order__on__ordIso,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A5),R))
     => ( pp(aa(set(product_prod(A,A)),bool,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))),R),R3)),bNF_Wellorder_ordIso(A,A))) ) ) ).

% card_order_on_ordIso
tff(fact_5503_infinite__Card__order__limit,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
              & ( A3 != 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),A3),X4)),R)) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_5504_dir__image,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R: set(product_prod(A,A))] :
      ( ! [X4: A,Y4: A] :
          ( ( aa(A,B,F,X4) = aa(A,B,F,Y4) )
        <=> ( X4 = Y4 ) )
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_We2720479622203943262_image(A,B,R,F))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% dir_image
tff(fact_5505_Card__order__ordIso2,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),R3)),bNF_Wellorder_ordIso(A,B)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)) ) ) ).

% Card_order_ordIso2
tff(fact_5506_Card__order__ordIso,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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),R)),bNF_Wellorder_ordIso(B,A)))
       => pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3)) ) ) ).

% Card_order_ordIso
tff(fact_5507_card__of__unique,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A5),R))
     => 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))),R),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_unique
tff(fact_5508_card__order__on__def,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,A5),R))
    <=> ( order_well_order_on(A,A5,R)
        & ! [R9: set(product_prod(A,A))] :
            ( order_well_order_on(A,A5,R9)
           => 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))),R),R9)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ).

% card_order_on_def
tff(fact_5509_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( 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)))),R),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A5,aTP_Lamp_ait(set(product_prod(A,A)),fun(B,set(A)),R))))),bNF_Wellorder_ordLeq(A,product_prod(B,A)))) ) ) ).

% Card_order_Times2
tff(fact_5510_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),B4: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( 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)))),R),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_xa(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(A,product_prod(A,B)))) ) ) ).

% Card_order_Times1
tff(fact_5511_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => pp(aa(set(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,A),product_prod(A,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,field2(A,R),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),R)))),R)),bNF_Wellorder_ordLeq(product_prod(A,A),A))) ) ) ).

% Card_order_Times_same_infinite
tff(fact_5512_Card__order__Pow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca6860139660246222851ard_of(set(A),pow2(A,field2(A,R))))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% Card_order_Pow
tff(fact_5513_Card__order__iff__ordLeq__card__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_Wellorder_ordLeq(A,A))) ) ).

% Card_order_iff_ordLeq_card_of
tff(fact_5514_card__of__Field__ordIso,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(A,field2(A,R))),R)),bNF_Wellorder_ordIso(A,A))) ) ).

% card_of_Field_ordIso
tff(fact_5515_Card__order__iff__ordIso__card__of,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
    <=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Ca6860139660246222851ard_of(A,field2(A,R)))),bNF_Wellorder_ordIso(A,A))) ) ).

% Card_order_iff_ordIso_card_of
tff(fact_5516_ordIso__card__of__imp__Card__order,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordIso(A,B)))
     => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) ) ).

% ordIso_card_of_imp_Card_order
tff(fact_5517_exists__minim__Card__order,axiom,
    ! [A: $tType,R4: set(set(product_prod(A,A)))] :
      ( ( R4 != bot_bot(set(set(product_prod(A,A)))) )
     => ( ! [X4: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X4),R4))
           => pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,X4)),X4)) )
       => ? [X4: set(product_prod(A,A))] :
            ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),X4),R4))
            & ! [Xa3: set(product_prod(A,A))] :
                ( pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),Xa3),R4))
               => 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))),X4),Xa3)),bNF_Wellorder_ordLeq(A,A))) ) ) ) ) ).

% exists_minim_Card_order
tff(fact_5518_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R)),bNF_Wellorder_ordLeq(B,A))) ) ).

% Card_order_empty
tff(fact_5519_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),A5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Ca6860139660246222851ard_of(B,A5))),bNF_Wellorder_ordIso(A,B)))
       => ( pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
        <=> pp(aa(set(B),bool,finite_finite2(B),A5)) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_5520_card__of__underS,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(A),bool,member(A,A3),field2(A,R)))
       => 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,R,A3))),R)),bNF_We4044943003108391690rdLess(A,A))) ) ) ).

% card_of_underS
tff(fact_5521_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(A,set(B)),B4: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( bNF_Ca7133664381575040944arCard(A,R)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R,As)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As),field2(A,R)))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),bNF_We4044943003108391690rdLess(B,A)))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(A,set(B),As,X4))) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_5522_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),B2: B] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( field2(A,R) != bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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),insert2(B,B2),bot_bot(set(B))))),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ).

% Card_order_singl_ordLeq
tff(fact_5523_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),A5: set(B),B4: set(B)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),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)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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))),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_5524_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,field2(A,R),R)
        | pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R)),bNF_Wellorder_ordLeq(B,A))) ) ).

% card_of_empty1
tff(fact_5525_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( ( field2(B,P2) != 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))),P2),R)),bNF_Wellorder_ordLeq(B,A)))
           => ( pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_aiu(set(product_prod(B,B)),fun(A,set(B)),P2)))),R)),bNF_Wellorder_ordIso(product_prod(A,B),A)))
              & pp(aa(set(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,field2(B,P2),aTP_Lamp_ait(set(product_prod(A,A)),fun(B,set(A)),R)))),R)),bNF_Wellorder_ordIso(product_prod(B,A),A))) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_5526_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),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,X4))),R)),bNF_Wellorder_ordLeq(C,A))) )
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,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))),R)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_5527_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),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)),R)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)))),bool,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_xm(set(C),fun(B,set(C)),B4)))),R)),bNF_Wellorder_ordLeq(product_prod(B,C),A))) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_5528_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I5)),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),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,X4))),R)),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)))),R)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_5529_ex__toCard__pred,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R))
       => ? [X_13: fun(A,B)] : pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A5,R),X_13)) ) ) ).

% ex_toCard_pred
tff(fact_5530_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(set(A),set(B)),B4: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))))
           => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),bNF_Wellorder_ordLeq(B,A)))
             => ? [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X4),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
                  & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),As,X4))) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_5531_isCardSuc__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca6246979054910435723ardSuc(A,R),R3))
    <=> ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(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)))),R),R3)),bNF_We4044943003108391690rdLess(A,set(A))))
        & ! [R10: set(product_prod(set(A),set(A)))] :
            ( ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),R10)),R10))
              & 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)))),R),R10)),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)))),R3),R10)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% isCardSuc_def
tff(fact_5532_cardSuc__ordLeq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_Wellorder_ordLeq(A,set(A)))) ) ).

% cardSuc_ordLeq
tff(fact_5533_cardSuc__greater,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),aa(set(product_prod(A,A)),fun(set(product_prod(set(A),set(A))),product_prod(set(product_prod(A,A)),set(product_prod(set(A),set(A))))),product_Pair(set(product_prod(A,A)),set(product_prod(set(A),set(A)))),R),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_We4044943003108391690rdLess(A,set(A)))) ) ).

% cardSuc_greater
tff(fact_5534_cardSuc__ordLess__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,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))),R),R3)),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,R)),R3)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_ordLess_ordLeq
tff(fact_5535_cardSuc__least,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,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))),R),R3)),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,R)),R3)),bNF_Wellorder_ordLeq(set(A),B))) ) ) ) ).

% cardSuc_least
tff(fact_5536_cardSuc__mono__ordLeq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ( 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,R)),bNF_Ca8387033319878233205ardSuc(B,R3))),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))),R),R3)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% cardSuc_mono_ordLeq
tff(fact_5537_cardSuc__invar__ordIso,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ( 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,R)),bNF_Ca8387033319878233205ardSuc(B,R3))),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))),R),R3)),bNF_Wellorder_ordIso(A,B))) ) ) ) ).

% cardSuc_invar_ordIso
tff(fact_5538_cardSuc__least__aux,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(set(A),set(A)))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(set(A),set(A))),bool,bNF_Ca8970107618336181345der_on(set(A),field2(set(A),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)))),R),R3)),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,R)),R3)),bNF_Wellorder_ordLeq(set(A),set(A)))) ) ) ) ).

% cardSuc_least_aux
tff(fact_5539_cardSuc__ordLeq__ordLess,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R3)),R3))
       => ( 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)))),R3),bNF_Ca8387033319878233205ardSuc(A,R))),bNF_We4044943003108391690rdLess(B,set(A))))
        <=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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),R)),bNF_Wellorder_ordLeq(B,A))) ) ) ) ).

% cardSuc_ordLeq_ordLess
tff(fact_5540_toCard__pred__toCard,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R))
       => pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A5,R),bNF_Greatest_toCard(A,B,A5,R))) ) ) ).

% toCard_pred_toCard
tff(fact_5541_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(set(A),set(B)),B4: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As)
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))))
         => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),bNF_Wellorder_ordLeq(B,A)))
           => ? [X4: set(A)] :
                ( pp(aa(set(set(A)),bool,member(set(A),X4),field2(set(A),bNF_Ca8387033319878233205ardSuc(A,R))))
                & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B4),aa(set(A),set(B),As,X4))) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_5542_toCard__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: set(product_prod(B,B)),X: A,Y3: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R))
       => ( pp(aa(set(A),bool,member(A,X),A5))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => ( ( aa(A,B,bNF_Greatest_toCard(A,B,A5,R),X) = aa(A,B,bNF_Greatest_toCard(A,B,A5,R),Y3) )
            <=> ( X = Y3 ) ) ) ) ) ) ).

% toCard_inj
tff(fact_5543_cinfinite__mono,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: 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))),R1),R22)),bNF_Wellorder_ordLeq(A,B)))
     => ( bNF_Ca4139267488887388095finite(A,R1)
       => bNF_Ca4139267488887388095finite(B,R22) ) ) ).

% cinfinite_mono
tff(fact_5544_Cinfinite__limit2,axiom,
    ! [A: $tType,X1: A,R: set(product_prod(A,A)),X22: A] :
      ( pp(aa(set(A),bool,member(A,X1),field2(A,R)))
     => ( pp(aa(set(A),bool,member(A,X22),field2(A,R)))
       => ( ( bNF_Ca4139267488887388095finite(A,R)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
              & ( X1 != 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),X1),X4)),R))
              & ( X22 != 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),X22),X4)),R)) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_5545_Cinfinite__limit,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,X),field2(A,R)))
     => ( ( bNF_Ca4139267488887388095finite(A,R)
          & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
       => ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
            & ( X != 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),X),X4)),R)) ) ) ) ).

% Cinfinite_limit
tff(fact_5546_Cinfinite__cong,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: 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))),R1),R22)),bNF_Wellorder_ordIso(A,B)))
     => ( ( bNF_Ca4139267488887388095finite(A,R1)
          & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1)) )
       => ( bNF_Ca4139267488887388095finite(B,R22)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R22)),R22)) ) ) ) ).

% Cinfinite_cong
tff(fact_5547_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X6: set(A),R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,finite_finite2(A),X6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),field2(A,R)))
       => ( ( bNF_Ca4139267488887388095finite(A,R)
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
         => ? [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
              & ! [Xa3: A] :
                  ( pp(aa(set(A),bool,member(A,Xa3),X6))
                 => ( ( Xa3 != 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),Xa3),X4)),R)) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_5548_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: 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)),R)),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)),R)),bNF_Wellorder_ordLeq(A,B)))
       => ( ( bNF_Ca4139267488887388095finite(B,R)
            & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
         => 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))),R)),bNF_Wellorder_ordLeq(A,B))) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_5549_UNION__Cinfinite__bound,axiom,
    ! [A: $tType,B: $tType,C: $tType,I5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),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,X4))),R)),bNF_Wellorder_ordLeq(C,B))) )
       => ( ( bNF_Ca4139267488887388095finite(B,R)
            & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
         => 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)))),R)),bNF_Wellorder_ordLeq(C,B))) ) ) ) ).

% UNION_Cinfinite_bound
tff(fact_5550_fromCard__toCard,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R: 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)),R)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R))
       => ( pp(aa(set(A),bool,member(A,B2),A5))
         => ( bNF_Gr5436034075474128252omCard(A,B,A5,R,aa(A,B,bNF_Greatest_toCard(A,B,A5,R),B2)) = B2 ) ) ) ) ).

% fromCard_toCard
tff(fact_5551_Cfinite__ordLess__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( ( bNF_Ca4139267488887388095finite(B,S)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,S)),S)) )
       => 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))),R),S)),bNF_We4044943003108391690rdLess(A,B))) ) ) ).

% Cfinite_ordLess_Cinfinite
tff(fact_5552_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] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,Fbd)),Fbd))
     => ( ! [X4: 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,X4))),Fbd)),bNF_Wellorder_ordLeq(C,A)))
       => ( ! [X4: 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,X4))),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_5553_cprod__com,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: 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,P1,P22)),bNF_Cardinal_cprod(B,A,P22,P1))),bNF_Wellorder_ordIso(product_prod(A,B),product_prod(B,A)))) ).

% cprod_com
tff(fact_5554_Cfinite__cprod__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( ( bNF_Ca4139267488887388095finite(B,S)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,S)),S)) )
       => 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,R,S)),S)),bNF_Wellorder_ordLeq(product_prod(A,B),B))) ) ) ).

% Cfinite_cprod_Cinfinite
tff(fact_5555_cprod__def,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] : bNF_Cardinal_cprod(A,B,R1,R22) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R1),aTP_Lamp_aiu(set(product_prod(B,B)),fun(A,set(B)),R22))) ).

% cprod_def
tff(fact_5556_cprod__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),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,P1,P22)),bNF_Cardinal_cprod(B,D,R1,R22))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,D)))) ) ) ).

% cprod_mono
tff(fact_5557_cprod__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q2: 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))),P1),R1)),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,P1,Q2)),bNF_Cardinal_cprod(B,C,R1,Q2))),bNF_Wellorder_ordLeq(product_prod(A,C),product_prod(B,C)))) ) ).

% cprod_mono1
tff(fact_5558_cprod__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),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,Q2,P22)),bNF_Cardinal_cprod(C,B,Q2,R22))),bNF_Wellorder_ordLeq(product_prod(C,A),product_prod(C,B)))) ) ).

% cprod_mono2
tff(fact_5559_cprod__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),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,P1,P22)),bNF_Cardinal_cprod(B,D,R1,R22))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,D)))) ) ) ).

% cprod_cong
tff(fact_5560_cprod__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: 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))),P1),R1)),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,P1,P22)),bNF_Cardinal_cprod(B,C,R1,P22))),bNF_Wellorder_ordIso(product_prod(A,C),product_prod(B,C)))) ) ).

% cprod_cong1
tff(fact_5561_cprod__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),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,Q2,P22)),bNF_Cardinal_cprod(C,B,Q2,R22))),bNF_Wellorder_ordIso(product_prod(C,A),product_prod(C,B)))) ) ).

% cprod_cong2
tff(fact_5562_cprod__infinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(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,R,R)),R)),bNF_Wellorder_ordIso(product_prod(A,A),A))) ) ).

% cprod_infinite
tff(fact_5563_cprod__cinfinite__bound,axiom,
    ! [B: $tType,C: $tType,A: $tType,P2: set(product_prod(A,A)),R: set(product_prod(B,B)),Q2: 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))),P2),R)),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))),Q2),R)),bNF_Wellorder_ordLeq(C,B)))
       => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P2)),P2))
         => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
           => ( ( bNF_Ca4139267488887388095finite(B,R)
                & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
             => 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,P2,Q2)),R)),bNF_Wellorder_ordLeq(product_prod(A,C),B))) ) ) ) ) ) ).

% cprod_cinfinite_bound
tff(fact_5564_cprod__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),P2: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(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,P2,P7)),bNF_Cardinal_cprod(A,A,R,R))),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,P2,P7)),R)),bNF_Wellorder_ordIso(product_prod(B,C),A))) ) ) ) ).

% cprod_dup
tff(fact_5565_card__of__Csum__Times_H,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),I5: set(B),A5: fun(B,set(C))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ! [X4: B] :
            ( pp(aa(set(B),bool,member(B,X4),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,X4))),R)),bNF_Wellorder_ordLeq(C,A))) )
       => pp(aa(set(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(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_aiv(fun(B,set(C)),fun(B,set(product_prod(C,C))),A5))),bNF_Cardinal_cprod(B,A,bNF_Ca6860139660246222851ard_of(B,I5),R))),bNF_Wellorder_ordLeq(product_prod(B,C),product_prod(B,A)))) ) ) ).

% card_of_Csum_Times'
tff(fact_5566_card__of__Csum__Times,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),B4: set(C)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),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,X4))),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_aiw(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_5567_cprod__infinite1_H,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( ( ~ pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(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))),P2),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B)))
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,P2)),P2)) )
       => ( 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))),P2),R)),bNF_Wellorder_ordLeq(B,A)))
         => pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)))),bool,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,R,P2)),R)),bNF_Wellorder_ordIso(product_prod(A,B),A))) ) ) ) ).

% cprod_infinite1'
tff(fact_5568_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_5569_cinfinite__not__czero,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,B))] :
      ( bNF_Ca4139267488887388095finite(B,R)
     => ~ 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))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(B,A))) ) ).

% cinfinite_not_czero
tff(fact_5570_czeroE,axiom,
    ! [B: $tType,A: $tType,R: 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))),R),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B)))
     => ( field2(A,R) = bot_bot(set(A)) ) ) ).

% czeroE
tff(fact_5571_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_5572_czeroI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( field2(A,R) = bot_bot(set(A)) )
       => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(A,B))) ) ) ).

% czeroI
tff(fact_5573_Cnotzero__imp__not__empty,axiom,
    ! [A: $tType,R: 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))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( field2(A,R) != bot_bot(set(A)) ) ) ).

% Cnotzero_imp_not_empty
tff(fact_5574_Cnotzero__mono,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),Q2: 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))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,Q2)),Q2))
       => ( 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))),R),Q2)),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))),Q2),bNF_Cardinal_czero(B))),bNF_Wellorder_ordIso(B,B)))
            & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,Q2)),Q2)) ) ) ) ) ).

% Cnotzero_mono
tff(fact_5575_Cinfinite__Cnotzero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( ~ pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) ) ) ).

% Cinfinite_Cnotzero
tff(fact_5576_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)))
      & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,bNF_Ca6860139660246222851ard_of(A,top_top(set(A))))),bNF_Ca6860139660246222851ard_of(A,top_top(set(A))))) ) ).

% Cnotzero_UNIV
tff(fact_5577_cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: 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))),R1),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1)) )
     => ( ( bNF_Ca4139267488887388095finite(B,R22)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R22)),R22)) )
       => bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22)) ) ) ).

% cinfinite_cprod2
tff(fact_5578_Csum__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),Rs: fun(A,set(product_prod(B,B)))] : bNF_Cardinal_Csum(A,B,R,Rs) = bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,field2(A,R),aTP_Lamp_aix(fun(A,set(product_prod(B,B))),fun(A,set(B)),Rs))) ).

% Csum_def
tff(fact_5579_SIGMA__CSUM,axiom,
    ! [B: $tType,A: $tType,I5: set(A),As: fun(A,set(B))] : bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,I5,As)) = bNF_Cardinal_Csum(A,B,bNF_Ca6860139660246222851ard_of(A,I5),aTP_Lamp_aiw(fun(A,set(B)),fun(A,set(product_prod(B,B))),As)) ).

% SIGMA_CSUM
tff(fact_5580_ordLeq__cprod2,axiom,
    ! [A: $tType,B: $tType,P1: set(product_prod(A,A)),P22: 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))),P1),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P1)),P1)) )
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,P22)),P22))
       => 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)))),P22),bNF_Cardinal_cprod(A,B,P1,P22))),bNF_Wellorder_ordLeq(B,product_prod(A,B)))) ) ) ).

% ordLeq_cprod2
tff(fact_5581_Cinfinite__cprod2,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: 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))),R1),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1)) )
     => ( ( bNF_Ca4139267488887388095finite(B,R22)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R22)),R22)) )
       => ( bNF_Ca4139267488887388095finite(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22))
          & pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,bNF_Ca8970107618336181345der_on(product_prod(A,B),field2(product_prod(A,B),bNF_Cardinal_cprod(A,B,R1,R22))),bNF_Cardinal_cprod(A,B,R1,R22))) ) ) ) ).

% Cinfinite_cprod2
tff(fact_5582_cone__ordLeq__Cnotzero,axiom,
    ! [A: $tType,R: 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))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(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),R)),bNF_Wellorder_ordLeq(product_unit,A))) ) ).

% cone_ordLeq_Cnotzero
tff(fact_5583_cexp__mono,axiom,
    ! [E: $tType,F2: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),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))),P22),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(C,E)))
           => pp(aa(set(product_prod(set(product_prod(D,D)),set(product_prod(F2,F2)))),bool,member(product_prod(set(product_prod(D,D)),set(product_prod(F2,F2))),aa(set(product_prod(F2,F2)),product_prod(set(product_prod(D,D)),set(product_prod(F2,F2))),aa(set(product_prod(D,D)),fun(set(product_prod(F2,F2)),product_prod(set(product_prod(D,D)),set(product_prod(F2,F2)))),product_Pair(set(product_prod(D,D)),set(product_prod(F2,F2))),R22),bNF_Cardinal_czero(F2))),bNF_Wellorder_ordIso(D,F2))) )
         => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,P22)),P22))
           => 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,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ) ).

% cexp_mono
tff(fact_5584_cexp__mono2,axiom,
    ! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => ( ( 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))),P22),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))),R22),bNF_Cardinal_czero(E))),bNF_Wellorder_ordIso(B,E))) )
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22))
           => 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,Q2,P22)),bNF_Cardinal_cexp(C,B,Q2,R22))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ) ).

% cexp_mono2
tff(fact_5585_cexp__cone,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(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,R,bNF_Cardinal_cone)),R)),bNF_Wellorder_ordIso(fun(product_unit,A),A))) ) ).

% cexp_cone
tff(fact_5586_cexp__cprod,axiom,
    ! [A: $tType,C: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(C,C)),R33: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1))
     => 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,R1,R22),R33)),bNF_Cardinal_cexp(A,product_prod(C,B),R1,bNF_Cardinal_cprod(C,B,R22,R33)))),bNF_Wellorder_ordIso(fun(B,fun(C,A)),fun(product_prod(C,B),A)))) ) ).

% cexp_cprod
tff(fact_5587_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_5588_cprod__cexp,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(B,B)),S: set(product_prod(C,C)),T2: 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,R,S),T2)),bNF_Cardinal_cprod(fun(A,B),fun(A,C),bNF_Cardinal_cexp(B,A,R,T2),bNF_Cardinal_cexp(C,A,S,T2)))),bNF_Wellorder_ordIso(fun(A,product_prod(B,C)),product_prod(fun(A,B),fun(A,C))))) ).

% cprod_cexp
tff(fact_5589_cexp__cprod__ordLeq,axiom,
    ! [A: $tType,B: $tType,C: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1))
     => ( ( bNF_Ca4139267488887388095finite(B,R22)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R22)),R22)) )
       => ( ( ~ 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))),R33),bNF_Cardinal_czero(C))),bNF_Wellorder_ordIso(C,C)))
            & pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,R33)),R33)) )
         => ( 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))),R33),R22)),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,R1,R22),R33)),bNF_Cardinal_cexp(A,B,R1,R22))),bNF_Wellorder_ordIso(fun(C,fun(B,A)),fun(B,A)))) ) ) ) ) ).

% cexp_cprod_ordLeq
tff(fact_5590_cexp__mono_H,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),bNF_Wellorder_ordLeq(C,D)))
       => ( ( ( field2(C,P22) = bot_bot(set(C)) )
           => ( field2(D,R22) = 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,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22))),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B)))) ) ) ) ).

% cexp_mono'
tff(fact_5591_cexp__mono1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q2: 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))),P1),R1)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => 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,P1,Q2)),bNF_Cardinal_cexp(B,C,R1,Q2))),bNF_Wellorder_ordLeq(fun(C,A),fun(C,B)))) ) ) ).

% cexp_mono1
tff(fact_5592_cexp__mono2_H,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => ( ( ( field2(A,P22) = bot_bot(set(A)) )
           => ( field2(B,R22) = 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,Q2,P22)),bNF_Cardinal_cexp(C,B,Q2,R22))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_mono2'
tff(fact_5593_ordLeq__cexp1,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),Q2: 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))),R),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,Q2)),Q2))
       => 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)))),Q2),bNF_Cardinal_cexp(B,A,Q2,R))),bNF_Wellorder_ordLeq(B,fun(A,B)))) ) ) ).

% ordLeq_cexp1
tff(fact_5594_cexp__cong2,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22))
         => 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,Q2,P22)),bNF_Cardinal_cexp(C,B,Q2,R22))),bNF_Wellorder_ordIso(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_cong2
tff(fact_5595_cexp__cong1,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q2: 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))),P1),R1)),bNF_Wellorder_ordIso(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => 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,P1,Q2)),bNF_Cardinal_cexp(B,C,R1,Q2))),bNF_Wellorder_ordIso(fun(C,A),fun(C,B)))) ) ) ).

% cexp_cong1
tff(fact_5596_cexp__cong,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),bNF_Wellorder_ordIso(C,D)))
       => ( pp(aa(set(product_prod(D,D)),bool,bNF_Ca8970107618336181345der_on(D,field2(D,R22)),R22))
         => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,P22)),P22))
           => 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,P1,P22)),bNF_Cardinal_cexp(B,D,R1,R22))),bNF_Wellorder_ordIso(fun(C,A),fun(D,B)))) ) ) ) ) ).

% cexp_cong
tff(fact_5597_cexp__mono2__Cnotzero,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),bNF_Wellorder_ordLeq(A,B)))
     => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
       => ( ( ~ 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))),P22),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
            & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22)) )
         => 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,Q2,P22)),bNF_Cardinal_cexp(C,B,Q2,R22))),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C)))) ) ) ) ).

% cexp_mono2_Cnotzero
tff(fact_5598_ordLeq__cexp2,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R: 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),Q2)),bNF_Wellorder_ordLeq(bool,A)))
     => ( pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R))
       => 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)))),R),bNF_Cardinal_cexp(A,B,Q2,R))),bNF_Wellorder_ordLeq(B,fun(B,A)))) ) ) ).

% ordLeq_cexp2
tff(fact_5599_Cfinite__cexp__Cinfinite,axiom,
    ! [A: $tType,B: $tType,S: set(product_prod(A,A)),T2: set(product_prod(B,B))] :
      ( ( bNF_Cardinal_cfinite(A,S)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,S)),S)) )
     => ( ( bNF_Ca4139267488887388095finite(B,T2)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,T2)),T2)) )
       => 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,S,T2)),bNF_Cardinal_cexp(bool,B,bNF_Cardinal_ctwo,T2))),bNF_Wellorder_ordLeq(fun(B,A),fun(B,bool)))) ) ) ).

% Cfinite_cexp_Cinfinite
tff(fact_5600_cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R: 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),Q2)),bNF_Wellorder_ordLeq(bool,A)))
     => ( ( bNF_Ca4139267488887388095finite(B,R)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
       => bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q2,R)) ) ) ).

% cinfinite_cexp
tff(fact_5601_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)))
    & pp(aa(set(product_prod(bool,bool)),bool,bNF_Ca8970107618336181345der_on(bool,field2(bool,bNF_Cardinal_ctwo)),bNF_Cardinal_ctwo)) ) ).

% ctwo_Cnotzero
tff(fact_5602_ordLess__ctwo__cexp,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(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)))),R),bNF_Cardinal_cexp(bool,A,bNF_Cardinal_ctwo,R))),bNF_We4044943003108391690rdLess(A,fun(A,bool)))) ) ).

% ordLess_ctwo_cexp
tff(fact_5603_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_5604_ctwo__ordLess__Cinfinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(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),R)),bNF_We4044943003108391690rdLess(bool,A))) ) ).

% ctwo_ordLess_Cinfinite
tff(fact_5605_ctwo__ordLeq__Cinfinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(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),R)),bNF_Wellorder_ordLeq(bool,A))) ) ).

% ctwo_ordLeq_Cinfinite
tff(fact_5606_Cinfinite__cexp,axiom,
    ! [A: $tType,B: $tType,Q2: set(product_prod(A,A)),R: 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),Q2)),bNF_Wellorder_ordLeq(bool,A)))
     => ( ( bNF_Ca4139267488887388095finite(B,R)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
       => ( bNF_Ca4139267488887388095finite(fun(B,A),bNF_Cardinal_cexp(A,B,Q2,R))
          & pp(aa(set(product_prod(fun(B,A),fun(B,A))),bool,bNF_Ca8970107618336181345der_on(fun(B,A),field2(fun(B,A),bNF_Cardinal_cexp(A,B,Q2,R))),bNF_Cardinal_cexp(A,B,Q2,R))) ) ) ) ).

% Cinfinite_cexp
tff(fact_5607_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),insert2(A,A1),aa(set(A),set(A),insert2(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_xa(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_5608_natLeq__ordLeq__cinfinite,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ( bNF_Ca4139267488887388095finite(A,R)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R)) )
     => pp(aa(set(product_prod(set(product_prod(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),R)),bNF_Wellorder_ordLeq(nat,A))) ) ).

% natLeq_ordLeq_cinfinite
tff(fact_5609_cprod__cexp__csum__cexp__Cinfinite,axiom,
    ! [C: $tType,B: $tType,A: $tType,T2: set(product_prod(A,A)),R: set(product_prod(B,B)),S: set(product_prod(C,C))] :
      ( ( bNF_Ca4139267488887388095finite(A,T2)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,T2)),T2)) )
     => pp(aa(set(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),bool,member(product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),aa(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),fun(set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))),product_prod(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C)))))),product_Pair(set(product_prod(fun(A,product_prod(B,C)),fun(A,product_prod(B,C)))),set(product_prod(fun(A,sum_sum(B,C)),fun(A,sum_sum(B,C))))),bNF_Cardinal_cexp(product_prod(B,C),A,bNF_Cardinal_cprod(B,C,R,S),T2)),bNF_Cardinal_cexp(sum_sum(B,C),A,bNF_Cardinal_csum(B,C,R,S),T2))),bNF_Wellorder_ordLeq(fun(A,product_prod(B,C)),fun(A,sum_sum(B,C))))) ) ).

% cprod_cexp_csum_cexp_Cinfinite
tff(fact_5610_Plus__csum,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(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A5,B4))),bNF_Cardinal_csum(A,B,bNF_Ca6860139660246222851ard_of(A,A5),bNF_Ca6860139660246222851ard_of(B,B4)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B)))) ).

% Plus_csum
tff(fact_5611_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_5612_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_5613_card__of__Plus__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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),C6))),bNF_Ca6860139660246222851ard_of(sum_sum(A,sum_sum(B,C)),sum_Plus(A,sum_sum(B,C),A5,sum_Plus(B,C,B4,C6))))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).

% card_of_Plus_assoc
tff(fact_5614_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_5615_card__of__Times__Plus__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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_aiy(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),B4),C6)))),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_xa(set(B),fun(A,set(B)),B4)),product_Sigma(A,C,A5,aTP_Lamp_xn(set(C),fun(A,set(C)),C6)))))),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_5616_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_aiz(A,set(bool)))))),bNF_Wellorder_ordIso(sum_sum(A,A),product_prod(A,bool)))) ).

% card_of_Plus_Times_bool
tff(fact_5617_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_5618_csum__assoc,axiom,
    ! [C: $tType,B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B)),P32: set(product_prod(C,C))] : 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_Cardinal_csum(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,P1,P22),P32)),bNF_Cardinal_csum(A,sum_sum(B,C),P1,bNF_Cardinal_csum(B,C,P22,P32)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),C),sum_sum(A,sum_sum(B,C))))) ).

% csum_assoc
tff(fact_5619_csum__csum,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C)),R42: set(product_prod(D,D))] : pp(aa(set(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),bool,member(product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),aa(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),fun(set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))),product_prod(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D)))))),product_Pair(set(product_prod(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,B),sum_sum(C,D)))),set(product_prod(sum_sum(sum_sum(A,C),sum_sum(B,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))),bNF_Cardinal_csum(sum_sum(A,B),sum_sum(C,D),bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_csum(C,D,R33,R42))),bNF_Cardinal_csum(sum_sum(A,C),sum_sum(B,D),bNF_Cardinal_csum(A,C,R1,R33),bNF_Cardinal_csum(B,D,R22,R42)))),bNF_Wellorder_ordIso(sum_sum(sum_sum(A,B),sum_sum(C,D)),sum_sum(sum_sum(A,C),sum_sum(B,D))))) ).

% csum_csum
tff(fact_5620_csum__com,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,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_Cardinal_csum(A,B,P1,P22)),bNF_Cardinal_csum(B,A,P22,P1))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(B,A)))) ).

% csum_com
tff(fact_5621_cprod__csum__distrib1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B)),R33: set(product_prod(C,C))] : pp(aa(set(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),bool,member(product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),aa(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),fun(set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))),product_prod(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C)))))),product_Pair(set(product_prod(sum_sum(product_prod(A,B),product_prod(A,C)),sum_sum(product_prod(A,B),product_prod(A,C)))),set(product_prod(product_prod(A,sum_sum(B,C)),product_prod(A,sum_sum(B,C))))),bNF_Cardinal_csum(product_prod(A,B),product_prod(A,C),bNF_Cardinal_cprod(A,B,R1,R22),bNF_Cardinal_cprod(A,C,R1,R33))),bNF_Cardinal_cprod(A,sum_sum(B,C),R1,bNF_Cardinal_csum(B,C,R22,R33)))),bNF_Wellorder_ordIso(sum_sum(product_prod(A,B),product_prod(A,C)),product_prod(A,sum_sum(B,C))))) ).

% cprod_csum_distrib1
tff(fact_5622_natLeq__underS__less,axiom,
    ! [N: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,N) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N)) ).

% natLeq_underS_less
tff(fact_5623_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_5624_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_5625_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_5626_Card__order__Plus1,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),B4: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),B4)))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).

% Card_order_Plus1
tff(fact_5627_Card__order__Plus2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A5: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),R),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,A5,field2(A,R))))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).

% Card_order_Plus2
tff(fact_5628_Un__csum,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_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,A5),bNF_Ca6860139660246222851ard_of(A,B4)))),bNF_Wellorder_ordLeq(A,sum_sum(A,A)))) ).

% Un_csum
tff(fact_5629_ordLeq__csum2,axiom,
    ! [B: $tType,A: $tType,P22: set(product_prod(A,A)),P1: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P22)),P22))
     => 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)))),P22),bNF_Cardinal_csum(B,A,P1,P22))),bNF_Wellorder_ordLeq(A,sum_sum(B,A)))) ) ).

% ordLeq_csum2
tff(fact_5630_ordLeq__csum1,axiom,
    ! [B: $tType,A: $tType,P1: set(product_prod(A,A)),P22: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P1)),P1))
     => 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)))),P1),bNF_Cardinal_csum(A,B,P1,P22))),bNF_Wellorder_ordLeq(A,sum_sum(A,B)))) ) ).

% ordLeq_csum1
tff(fact_5631_csum__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),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_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R1,R22))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% csum_mono
tff(fact_5632_csum__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q2: 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))),P1),R1)),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_Cardinal_csum(A,C,P1,Q2)),bNF_Cardinal_csum(B,C,R1,Q2))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% csum_mono1
tff(fact_5633_csum__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),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_Cardinal_csum(C,A,Q2,P22)),bNF_Cardinal_csum(C,B,Q2,R22))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% csum_mono2
tff(fact_5634_csum__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),P22: set(product_prod(C,C)),R22: 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))),P1),R1)),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))),P22),R22)),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_Cardinal_csum(A,C,P1,P22)),bNF_Cardinal_csum(B,D,R1,R22))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% csum_cong
tff(fact_5635_csum__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,P1: set(product_prod(A,A)),R1: set(product_prod(B,B)),Q2: 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))),P1),R1)),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_Cardinal_csum(A,C,P1,Q2)),bNF_Cardinal_csum(B,C,R1,Q2))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% csum_cong1
tff(fact_5636_csum__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,P22: set(product_prod(A,A)),R22: set(product_prod(B,B)),Q2: 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))),P22),R22)),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_Cardinal_csum(C,A,Q2,P22)),bNF_Cardinal_csum(C,B,Q2,R22))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% csum_cong2
tff(fact_5637_cprod__csum__cexp,axiom,
    ! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(B,B))] : pp(aa(set(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))))),bool,member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),aa(set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),fun(set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B)))))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(fun(bool,sum_sum(A,B)),fun(bool,sum_sum(A,B))))),bNF_Cardinal_cprod(A,B,R1,R22)),bNF_Cardinal_cexp(sum_sum(A,B),bool,bNF_Cardinal_csum(A,B,R1,R22),bNF_Cardinal_ctwo))),bNF_Wellorder_ordLeq(product_prod(A,B),fun(bool,sum_sum(A,B))))) ).

% cprod_csum_cexp
tff(fact_5638_csum__Cfinite__cexp__Cinfinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B)),T2: set(product_prod(C,C))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
     => ( ( bNF_Cardinal_cfinite(B,S)
          & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,S)),S)) )
       => ( ( bNF_Ca4139267488887388095finite(C,T2)
            & pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,T2)),T2)) )
         => pp(aa(set(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))))),bool,member(product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),aa(set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),aa(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),fun(set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))),product_prod(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool)))))),product_Pair(set(product_prod(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,B)))),set(product_prod(fun(C,sum_sum(A,bool)),fun(C,sum_sum(A,bool))))),bNF_Cardinal_cexp(sum_sum(A,B),C,bNF_Cardinal_csum(A,B,R,S),T2)),bNF_Cardinal_cexp(sum_sum(A,bool),C,bNF_Cardinal_csum(A,bool,R,bNF_Cardinal_ctwo),T2))),bNF_Wellorder_ordLeq(fun(C,sum_sum(A,B)),fun(C,sum_sum(A,bool))))) ) ) ) ).

% csum_Cfinite_cexp_Cinfinite
tff(fact_5639_card__Plus,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))
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A5,B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4)) ) ) ) ).

% card_Plus
tff(fact_5640_ordLeq__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: 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))),R),R3)),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,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A5,field2(B,R3))))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordLeq_Plus_mono2
tff(fact_5641_ordLeq__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C6: 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))),R),R3)),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,field2(A,R),C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C6)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordLeq_Plus_mono1
tff(fact_5642_ordLeq__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),P2: set(product_prod(C,C)),P7: 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))),R),R3)),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))),P2),P7)),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,field2(A,R),field2(C,P2)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7))))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordLeq_Plus_mono
tff(fact_5643_ordIso__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(A,A)),R3: 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))),R),R3)),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,field2(A,R)))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,A5,field2(B,R3))))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% ordIso_Plus_cong2
tff(fact_5644_ordIso__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),C6: 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))),R),R3)),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,field2(A,R),C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,field2(B,R3),C6)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% ordIso_Plus_cong1
tff(fact_5645_ordIso__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set(product_prod(A,A)),R3: set(product_prod(B,B)),P2: set(product_prod(C,C)),P7: 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))),R),R3)),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))),P2),P7)),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,field2(A,R),field2(C,P2)))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,field2(B,R3),field2(D,P7))))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% ordIso_Plus_cong
tff(fact_5646_card__of__Plus__mono,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C6: set(C),D5: 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,C6)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordLeq(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,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,C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B4,D5)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_mono
tff(fact_5647_card__of__Plus__mono1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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,C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B4,C6)))),bNF_Wellorder_ordLeq(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_mono1
tff(fact_5648_card__of__Plus__mono2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C6: 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,C6,A5))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C6,B4)))),bNF_Wellorder_ordLeq(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_mono2
tff(fact_5649_card__of__Plus__cong,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C6: set(C),D5: 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,C6)),bNF_Ca6860139660246222851ard_of(D,D5))),bNF_Wellorder_ordIso(C,D)))
       => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(sum_sum(B,D),sum_sum(B,D))))),bool,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,C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,D),sum_Plus(B,D,B4,D5)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,D)))) ) ) ).

% card_of_Plus_cong
tff(fact_5650_card__of__Plus__cong1,axiom,
    ! [B: $tType,C: $tType,A: $tType,A5: set(A),B4: set(B),C6: 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,C6))),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,B4,C6)))),bNF_Wellorder_ordIso(sum_sum(A,C),sum_sum(B,C)))) ) ).

% card_of_Plus_cong1
tff(fact_5651_card__of__Plus__cong2,axiom,
    ! [B: $tType,A: $tType,C: $tType,A5: set(A),B4: set(B),C6: 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,C6,A5))),bNF_Ca6860139660246222851ard_of(sum_sum(C,B),sum_Plus(C,B,C6,B4)))),bNF_Wellorder_ordIso(sum_sum(C,A),sum_sum(C,B)))) ) ).

% card_of_Plus_cong2
tff(fact_5652_csum__dup,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),P2: set(product_prod(B,B)),P7: set(product_prod(C,C))] :
      ( bNF_Ca4139267488887388095finite(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),bool,member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(B,C),sum_sum(B,C))),fun(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A))))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),bNF_Cardinal_csum(B,C,P2,P7)),bNF_Cardinal_csum(A,A,R,R))),bNF_Wellorder_ordIso(sum_sum(B,C),sum_sum(A,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_Cardinal_csum(B,C,P2,P7)),R)),bNF_Wellorder_ordIso(sum_sum(B,C),A))) ) ) ) ).

% csum_dup
tff(fact_5653_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_5654_card__Plus__conv__if,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)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A5,B4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B4)) ) )
      & ( ~ ( pp(aa(set(A),bool,finite_finite2(A),A5))
            & pp(aa(set(B),bool,finite_finite2(B),B4)) )
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A5,B4)) = zero_zero(nat) ) ) ) ).

% card_Plus_conv_if
tff(fact_5655_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_5656_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_5657_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_5658_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C6: set(A),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),C6))
     => ( 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,C6))),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,C6))),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,C6))),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_5659_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_5660_Restr__natLeq2,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N),aTP_Lamp_aja(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq2
tff(fact_5661_Restr__natLeq,axiom,
    ! [N: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),N)),aTP_Lamp_ajb(nat,fun(nat,set(nat)),N))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_sp(nat,fun(nat,fun(nat,bool)),N))) ).

% Restr_natLeq
tff(fact_5662_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),P2: set(product_prod(B,B))] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),P2),R)),bNF_Wellorder_ordLeq(B,A)))
         => ( pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,field2(A,R),field2(B,P2)))),R)),bNF_Wellorder_ordIso(sum_sum(A,B),A)))
            & pp(aa(set(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),fun(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,field2(B,P2),field2(A,R)))),R)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_5663_csum__cinfinite__bound,axiom,
    ! [B: $tType,C: $tType,A: $tType,P2: set(product_prod(A,A)),R: set(product_prod(B,B)),Q2: 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))),P2),R)),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))),Q2),R)),bNF_Wellorder_ordLeq(C,B)))
       => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,P2)),P2))
         => ( pp(aa(set(product_prod(C,C)),bool,bNF_Ca8970107618336181345der_on(C,field2(C,Q2)),Q2))
           => ( ( bNF_Ca4139267488887388095finite(B,R)
                & pp(aa(set(product_prod(B,B)),bool,bNF_Ca8970107618336181345der_on(B,field2(B,R)),R)) )
             => pp(aa(set(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),aa(set(product_prod(sum_sum(A,C),sum_sum(A,C))),fun(set(product_prod(B,B)),product_prod(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B)))),product_Pair(set(product_prod(sum_sum(A,C),sum_sum(A,C))),set(product_prod(B,B))),bNF_Cardinal_csum(A,C,P2,Q2)),R)),bNF_Wellorder_ordLeq(sum_sum(A,C),B))) ) ) ) ) ) ).

% csum_cinfinite_bound
tff(fact_5664_csum__absorb2_H,axiom,
    ! [A: $tType,B: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R22)),R22))
     => ( 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))),R1),R22)),bNF_Wellorder_ordLeq(B,A)))
       => ( ( bNF_Ca4139267488887388095finite(B,R1)
            | bNF_Ca4139267488887388095finite(A,R22) )
         => 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_Cardinal_csum(B,A,R1,R22)),R22)),bNF_Wellorder_ordIso(sum_sum(B,A),A))) ) ) ) ).

% csum_absorb2'
tff(fact_5665_csum__absorb1_H,axiom,
    ! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
      ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R22)),R22))
     => ( 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))),R1),R22)),bNF_Wellorder_ordLeq(B,A)))
       => ( ( bNF_Ca4139267488887388095finite(B,R1)
            | bNF_Ca4139267488887388095finite(A,R22) )
         => 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_Cardinal_csum(A,B,R22,R1)),R22)),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ) ).

% csum_absorb1'
tff(fact_5666_csum__absorb1,axiom,
    ! [B: $tType,A: $tType,R22: set(product_prod(A,A)),R1: set(product_prod(B,B))] :
      ( ( bNF_Ca4139267488887388095finite(A,R22)
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R22)),R22)) )
     => ( 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))),R1),R22)),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_Cardinal_csum(A,B,R22,R1)),R22)),bNF_Wellorder_ordIso(sum_sum(A,B),A))) ) ) ).

% csum_absorb1
tff(fact_5667_card__of__nat,axiom,
    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))),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_nat
tff(fact_5668_csum__Cnotzero1,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: 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))),R1),bNF_Cardinal_czero(A))),bNF_Wellorder_ordIso(A,A)))
        & pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R1)),R1)) )
     => ( ~ pp(aa(set(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),bool,member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),fun(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B))))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),bNF_Cardinal_csum(A,B,R1,R22)),bNF_Cardinal_czero(sum_sum(A,B)))),bNF_Wellorder_ordIso(sum_sum(A,B),sum_sum(A,B))))
        & pp(aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),bool,bNF_Ca8970107618336181345der_on(sum_sum(A,B),field2(sum_sum(A,B),bNF_Cardinal_csum(A,B,R1,R22))),bNF_Cardinal_csum(A,B,R1,R22))) ) ) ).

% csum_Cnotzero1
tff(fact_5669_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A1: A,A22: A,A5: set(A),B1: B,B22: B,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),insert2(A,A1),aa(set(A),set(A),insert2(A,A22),bot_bot(set(A))))),A5)) )
     => ( ( ( B1 != B22 )
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),insert2(B,B1),aa(set(B),set(B),insert2(B,B22),bot_bot(set(B))))),B4)) )
       => 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_xa(set(B),fun(A,set(B)),B4))))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B)))) ) ) ).

% card_of_Plus_Times
tff(fact_5670_card__of__Field__natLeq,axiom,
    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))),bNF_Ca6860139660246222851ard_of(nat,field2(nat,bNF_Ca8665028551170535155natLeq))),bNF_Ca8665028551170535155natLeq)),bNF_Wellorder_ordIso(nat,nat))) ).

% card_of_Field_natLeq
tff(fact_5671_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),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)),R)),bNF_Wellorder_ordLeq(C,A)))
         => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,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))),R)),bNF_Wellorder_ordLeq(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_5672_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,A)),A5: set(B),B4: set(C)] :
      ( ~ pp(aa(set(A),bool,finite_finite2(A),field2(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,bNF_Ca8970107618336181345der_on(A,field2(A,R)),R))
       => ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,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)),R)),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)),R)),bNF_We4044943003108391690rdLess(C,A)))
           => pp(aa(set(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)))),bool,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))),R)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A))) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_5673_infinite__iff__natLeq__ordLeq,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_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,A5))),bNF_Wellorder_ordLeq(nat,A))) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_5674_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J5: set(A),I5: set(B),F: fun(D,C),F5: fun(B,filter(D)),G5: fun(A,filter(C))] :
      ( ! [M4: A] :
          ( pp(aa(set(A),bool,member(A,M4),J5))
         => ? [X2: B] :
              ( pp(aa(set(B),bool,member(B,X2),I5))
              & pp(aa(filter(C),bool,aa(filter(C),fun(filter(C),bool),ord_less_eq(filter(C)),filtermap(D,C,F,aa(B,filter(D),F5,X2))),aa(A,filter(C),G5,M4))) ) )
     => filterlim(D,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),G5),J5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),F5),I5))) ) ).

% filterlim_INF_INF
tff(fact_5675_num__of__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,num),bool,aa(fun(int,num),fun(fun(code_integer,num),bool),bNF_rel_fun(int,code_integer,num,num,code_pcr_integer,fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),code_num_of_integer)) ).

% num_of_integer.transfer
tff(fact_5676_subset__singleton__iff__Uniq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ? [A7: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A5),aa(set(A),set(A),insert2(A,A7),bot_bot(set(A)))))
    <=> uniq(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5)) ) ).

% subset_singleton_iff_Uniq
tff(fact_5677_filtermap__id_H,axiom,
    ! [A: $tType,X2: filter(A)] : filtermap(A,A,aTP_Lamp_ak(A,A),X2) = X2 ).

% filtermap_id'
tff(fact_5678_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F: fun(B,A),F5: filter(B)] :
      ( eventually(A,P,filtermap(B,A,F,F5))
    <=> eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_afu(fun(A,bool),fun(fun(B,A),fun(B,bool)),P),F),F5) ) ).

% eventually_filtermap
tff(fact_5679_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,B),F12: filter(B),G: fun(C,A),F23: filter(C)] :
      ( filterlim(A,B,F,F12,filtermap(C,A,G,F23))
    <=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ud(fun(A,B),fun(fun(C,A),fun(C,B)),F),G),F12,F23) ) ).

% filterlim_filtermap
tff(fact_5680_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),F5: filter(C)] : filtermap(B,A,F,filtermap(C,B,G,F5)) = filtermap(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_mf(fun(B,A),fun(fun(C,B),fun(C,A)),F),G),F5) ).

% filtermap_filtermap
tff(fact_5681_filtermap__ident,axiom,
    ! [A: $tType,F5: filter(A)] : filtermap(A,A,aTP_Lamp_ak(A,A),F5) = F5 ).

% filtermap_ident
tff(fact_5682_alt__ex1E_H,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(A,bool,P,X2))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => ( Y4 = X2 ) ) )
     => ~ ( ? [X_13: A] : pp(aa(A,bool,P,X_13))
         => ~ uniq(A,P) ) ) ).

% alt_ex1E'
tff(fact_5683_ex1__iff__ex__Uniq,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y: A] :
              ( pp(aa(A,bool,P,Y))
             => ( Y = X3 ) ) )
    <=> ( ? [X_1: A] : pp(aa(A,bool,P,X_1))
        & uniq(A,P) ) ) ).

% ex1_iff_ex_Uniq
tff(fact_5684_integer_Oid__abs__transfer,axiom,
    pp(aa(fun(int,code_integer),bool,aa(fun(int,int),fun(fun(int,code_integer),bool),bNF_rel_fun(int,int,int,code_integer,fequal(int),code_pcr_integer),aTP_Lamp_gf(int,int)),code_integer_of_int)) ).

% integer.id_abs_transfer
tff(fact_5685_integer_Orep__transfer,axiom,
    pp(aa(fun(code_integer,int),bool,aa(fun(int,int),fun(fun(code_integer,int),bool),bNF_rel_fun(int,code_integer,int,int,code_pcr_integer,fequal(int)),aTP_Lamp_gf(int,int)),code_int_of_integer)) ).

% integer.rep_transfer
tff(fact_5686_inj__on__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F,A5)
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),A5))
         => uniq(A,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_ajc(fun(A,B),fun(set(A),fun(A,fun(A,bool))),F),A5),X3)) ) ) ).

% inj_on_iff_Uniq
tff(fact_5687_pairwise__disjnt__iff,axiom,
    ! [A: $tType,A16: set(set(A))] :
      ( pairwise(set(A),disjnt(A),A16)
    <=> ! [X3: A] : uniq(set(A),aa(A,fun(set(A),bool),aTP_Lamp_ajd(set(set(A)),fun(A,fun(set(A),bool)),A16),X3)) ) ).

% pairwise_disjnt_iff
tff(fact_5688_gcd__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),gcd_gcd(int)),gcd_gcd(code_integer))) ).

% gcd_integer.transfer
tff(fact_5689_the1__equality_H,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( uniq(A,P)
     => ( pp(aa(A,bool,P,A3))
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality'
tff(fact_5690_dup_Otransfer,axiom,
    pp(aa(fun(code_integer,code_integer),bool,aa(fun(int,int),fun(fun(code_integer,code_integer),bool),bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer),aTP_Lamp_kz(int,int)),code_dup)) ).

% dup.transfer
tff(fact_5691_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),F5: filter(B),G5: filter(A),F: fun(B,A)] :
      ( filterlim(A,B,G,F5,G5)
     => ( filterlim(B,A,F,G5,F5)
       => ( eventually(A,aa(fun(B,A),fun(A,bool),aTP_Lamp_aje(fun(A,B),fun(fun(B,A),fun(A,bool)),G),F),G5)
         => ( filtermap(B,A,F,F5) = G5 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_5692_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F5: fun(C,filter(B)),B4: set(C)] : filtermap(B,A,F,aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F5),B4))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ajf(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F5)),B4)) ).

% filtermap_SUP
tff(fact_5693_filtermap__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),F5: filter(A)] : filtermap(A,B,F,F5) = abs_filter(B,aa(filter(A),fun(fun(B,bool),bool),aTP_Lamp_ajg(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),F),F5)) ).

% filtermap_def
tff(fact_5694_sub_Otransfer,axiom,
    pp(aa(fun(num,fun(num,code_integer)),bool,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,code_integer)),bool),bNF_rel_fun(num,num,fun(num,int),fun(num,code_integer),fequal(num),bNF_rel_fun(num,num,int,code_integer,fequal(num),code_pcr_integer)),aTP_Lamp_ly(num,fun(num,int))),code_sub)) ).

% sub.transfer
tff(fact_5695_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F5: fun(C,filter(B)),B4: set(C)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F5),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_ajf(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F5)),B4)))) ).

% filtermap_INF
tff(fact_5696_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : uniq(list(A),aTP_Lamp_ajh(set(A),fun(list(A),bool),A5)) ) ).

% strict_sorted_equal_Uniq
tff(fact_5697_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F5: filter(A),X: B] : prod_filter(A,B,F5,principal(B,aa(set(B),set(B),insert2(B,X),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_fx(B,fun(A,product_prod(A,B))),X),F5) ).

% prod_filter_principal_singleton2
tff(fact_5698_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_lb(A,fun(A,bool)),aTP_Lamp_aji(A,fun(A,bool))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_5699_Gcd__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).

% Gcd_fin_def
tff(fact_5700_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F5: filter(A),G5: filter(B),H7: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F5,G5),H7) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_ajk(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F5,prod_filter(B,C,G5,H7))) ).

% prod_filter_assoc
tff(fact_5701_bounded__quasi__semilattice__set_OF_Ocong,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A] : bounde2362111253966948842tice_F(A,F,Top,Bot) = bounde2362111253966948842tice_F(A,F,Top,Bot) ).

% bounded_quasi_semilattice_set.F.cong
tff(fact_5702_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),bool),F5: filter(A)] :
      ( eventually(product_prod(A,A),P,prod_filter(A,A,F5,F5))
    <=> ? [Q7: fun(A,bool)] :
          ( eventually(A,Q7,F5)
          & ! [X3: A,Y: A] :
              ( pp(aa(A,bool,Q7,X3))
             => ( pp(aa(A,bool,Q7,Y))
               => pp(aa(product_prod(A,A),bool,P,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Y))) ) ) ) ) ).

% eventually_prod_same
tff(fact_5703_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),bool),F5: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),P,prod_filter(A,B,F5,G5))
    <=> ? [Pf: fun(A,bool),Pg: fun(B,bool)] :
          ( eventually(A,Pf,F5)
          & eventually(B,Pg,G5)
          & ! [X3: A,Y: B] :
              ( pp(aa(A,bool,Pf,X3))
             => ( pp(aa(B,bool,Pg,Y))
               => 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),X3),Y))) ) ) ) ) ).

% eventually_prod_filter
tff(fact_5704_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G5: filter(B),F5: filter(A),G: fun(A,C),H7: filter(C)] :
      ( filterlim(A,B,F,G5,F5)
     => ( filterlim(A,C,G,H7,F5)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ajl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F),G),prod_filter(B,C,G5,H7),F5) ) ) ).

% filterlim_Pair
tff(fact_5705_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,A),G: fun(C,B),F5: filter(C)] : pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_ym(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G),F5)),prod_filter(A,B,filtermap(C,A,F,F5),filtermap(C,B,G,F5)))) ).

% filtermap_Pair
tff(fact_5706_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] : prod_filter(A,B,principal(A,A5),principal(B,B4)) = principal(product_prod(A,B),product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4))) ).

% principal_prod_principal
tff(fact_5707_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,F5: filter(A),G5: filter(B)] : prod_filter(A,B,F5,G5) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,bool),fun(B,bool))),set(filter(product_prod(A,B))),image2(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B)),aa(fun(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),fun(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B))),product_case_prod(fun(A,bool),fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_ajm(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,bool),fun(B,bool)),bool),set(product_prod(fun(A,bool),fun(B,bool))),collect(product_prod(fun(A,bool),fun(B,bool))),aa(fun(fun(A,bool),fun(fun(B,bool),bool)),fun(product_prod(fun(A,bool),fun(B,bool)),bool),product_case_prod(fun(A,bool),fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_ajn(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),F5),G5))))) ).

% prod_filter_def
tff(fact_5708_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),bool)] :
      ( eventually(product_prod(nat,nat),P,prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N10: nat] :
        ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N10),M7))
         => ! [N6: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N10),N6))
             => pp(aa(product_prod(nat,nat),bool,P,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),N6),M7))) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5709_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F5: filter(A),Q: fun(B,bool),G5: filter(B)] :
      ( eventually(A,P,F5)
     => ( eventually(B,Q,G5)
       => eventually(product_prod(A,B),aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_ajo(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),P),Q),prod_filter(A,B,F5,G5)) ) ) ).

% eventually_prodI
tff(fact_5710_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A5: filter(A),P: fun(B,bool),B4: filter(B)] :
      ( ( A5 != bot_bot(filter(A)) )
     => ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ajp(fun(B,bool),fun(A,fun(B,bool)),P)),prod_filter(A,B,A5,B4))
      <=> eventually(B,P,B4) ) ) ).

% eventually_prod2
tff(fact_5711_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B4: filter(A),P: fun(B,bool),A5: filter(B)] :
      ( ( B4 != bot_bot(filter(A)) )
     => ( eventually(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),aTP_Lamp_ajq(fun(B,bool),fun(B,fun(A,bool)),P)),prod_filter(B,A,A5,B4))
      <=> eventually(B,P,A5) ) ) ).

% eventually_prod1
tff(fact_5712_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J5: set(B),A5: fun(A,filter(C)),B4: fun(B,filter(D))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ( J5 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A5),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B4),J5))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_ajs(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J5),A5),B4)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_5713_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,filter(B)),B4: filter(C)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A5),I5)),B4) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_ajt(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A5),B4)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_5714_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J5: set(A),A5: filter(B),B4: fun(A,filter(C))] :
      ( ( J5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A5,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B4),J5))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_aju(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A5),B4)),J5)) ) ) ).

% prod_filter_INF2
tff(fact_5715_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,X: A,F5: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),insert2(A,X),bot_bot(set(A)))),F5) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),F5) ).

% prod_filter_principal_singleton
tff(fact_5716_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_ajv(A,fun(A,bool)),aTP_Lamp_ajw(A,fun(A,bool))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_5717_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_5718_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_5719_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y3: A] :
          ( ( arg_min_list(A,B,X,Xa) = Y3 )
         => ( 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),Xa)))
           => ( ! [X4: A] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) )
                 => ( ( Y3 = X4 )
                   => ~ 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),X4),nil(A))))) ) )
             => ( ! [X4: A,Y4: A,Zs2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                   => ( ( Y3 = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X4)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))))),X4),arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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),X4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))))) ) )
               => ~ ( ( Xa = nil(A) )
                   => ( ( Y3 = 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_5720_bounded__quasi__semilattice__set_Oin__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( pp(aa(set(A),bool,member(A,A3),A5))
       => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) ) ) ) ).

% bounded_quasi_semilattice_set.in_idem
tff(fact_5721_bounded__quasi__semilattice__set_Onormalize,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) ) ) ).

% bounded_quasi_semilattice_set.normalize
tff(fact_5722_bounded__quasi__semilattice__set_Oempty,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),bot_bot(set(A))) = Top ) ) ).

% bounded_quasi_semilattice_set.empty
tff(fact_5723_bounded__quasi__semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) = Bot ) ) ) ).

% bounded_quasi_semilattice_set.infinite
tff(fact_5724_bounded__quasi__semilattice__set_Osubset,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),B4: set(A),A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( 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),F,aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),B4)),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) ) ) ) ).

% bounded_quasi_semilattice_set.subset
tff(fact_5725_bounded__quasi__semilattice__set_Oinsert,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5)) ) ) ).

% bounded_quasi_semilattice_set.insert
tff(fact_5726_bounded__quasi__semilattice__set_Ounion,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A5: set(A),B4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),F,aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5)),aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),B4)) ) ) ).

% bounded_quasi_semilattice_set.union
tff(fact_5727_bounded__quasi__semilattice__set_Oeq__fold,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A5: set(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) = finite_fold(A,A,F,Top,A5) ) )
        & ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),A5) = Bot ) ) ) ) ).

% bounded_quasi_semilattice_set.eq_fold
tff(fact_5728_bounded__quasi__semilattice__set_Oset__eq__fold,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),Xs: list(A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F,Top,Bot),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,F,Xs),Top) ) ) ).

% bounded_quasi_semilattice_set.set_eq_fold
tff(fact_5729_sorted__wrt_Opelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( sorted_wrt(A,X,Xa)
     => ( 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),Xa)))
       => ( ( ( Xa = 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)))) )
         => ~ ! [X4: A,Ys3: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),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),X4),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,X4),Xa3)) )
                      & sorted_wrt(A,X,Ys3) ) ) ) ) ) ) ).

% sorted_wrt.pelims(2)
tff(fact_5730_sorted__wrt_Opelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y3: bool] :
      ( ( sorted_wrt(A,X,Xa)
      <=> pp(Y3) )
     => ( 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),Xa)))
       => ( ( ( Xa = nil(A) )
           => ( pp(Y3)
             => ~ 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)))) ) )
         => ~ ! [X4: A,Ys3: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys3) )
               => ( ( pp(Y3)
                  <=> ( ! [Xa2: A] :
                          ( pp(aa(set(A),bool,member(A,Xa2),aa(list(A),set(A),set2(A),Ys3)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),X,X4),Xa2)) )
                      & 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),X4),Ys3)))) ) ) ) ) ) ).

% sorted_wrt.pelims(1)
tff(fact_5731_sorted__wrt_Opelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( ~ sorted_wrt(A,X,Xa)
     => ( 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),Xa)))
       => ~ ! [X4: A,Ys3: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),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),X4),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,X4),Xa4)) )
                  & sorted_wrt(A,X,Ys3) ) ) ) ) ) ).

% sorted_wrt.pelims(3)
tff(fact_5732_gen__length__def,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,gen_length(A,N),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)) ).

% gen_length_def
tff(fact_5733_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X_13: A] : pp(aa(A,bool,lattic501386751177426532rg_min(A,B,F,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),S2)),X_13)) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_5734_relImage__def,axiom,
    ! [A: $tType,B: $tType,R4: set(product_prod(B,B)),F: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R4,F) = 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_ajx(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R4),F)) ).

% relImage_def
tff(fact_5735_length__code,axiom,
    ! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).

% length_code
tff(fact_5736_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,A)),F: 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))),R4),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F),A5),aa(set(B),fun(A,set(A)),aTP_Lamp_ajy(fun(B,A),fun(set(B),fun(A,set(A))),F),A5))))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A5,R4,F),F) = R4 ) ) ).

% relImage_relInvImage
tff(fact_5737_relImage__Gr,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,A)),A5: set(A),F: 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))),R4),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5))))
     => ( bNF_Gr4221423524335903396lImage(A,B,R4,F) = relcomp(B,A,B,converse(A,B,bNF_Gr(A,B,A5,F)),relcomp(A,A,B,R4,bNF_Gr(A,B,A5,F))) ) ) ).

% relImage_Gr
tff(fact_5738_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_ajz(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_5739_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R)))
    <=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R)) ) ).

% converse_iff
tff(fact_5740_trancl__converseD,axiom,
    ! [A: $tType,X: A,Y3: A,R: 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),Y3)),transitive_trancl(A,converse(A,A,R))))
     => pp(aa(set(product_prod(A,A)),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)),converse(A,A,transitive_trancl(A,R)))) ) ).

% trancl_converseD
tff(fact_5741_trancl__converseI,axiom,
    ! [A: $tType,X: A,Y3: A,R: 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),Y3)),converse(A,A,transitive_trancl(A,R))))
     => pp(aa(set(product_prod(A,A)),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)),transitive_trancl(A,converse(A,A,R)))) ) ).

% trancl_converseI
tff(fact_5742_rtrancl__converseD,axiom,
    ! [A: $tType,X: A,Y3: A,R: 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),Y3)),transitive_rtrancl(A,converse(A,A,R))))
     => pp(aa(set(product_prod(A,A)),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),X)),transitive_rtrancl(A,R))) ) ).

% rtrancl_converseD
tff(fact_5743_rtrancl__converseI,axiom,
    ! [A: $tType,Y3: A,X: A,R: 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),Y3),X)),transitive_rtrancl(A,R)))
     => pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,converse(A,A,R)))) ) ).

% rtrancl_converseI
tff(fact_5744_converse_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R: 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,R)))
     => 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)),R)) ) ).

% converse.cases
tff(fact_5745_converse_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: B,A22: A,R: 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,R)))
    <=> ? [A7: A,B7: B] :
          ( ( A1 = B7 )
          & ( A22 = A7 )
          & 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),A7),B7)),R)) ) ) ).

% converse.simps
tff(fact_5746_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: B,R: 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,R)))
     => 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)),R)) ) ).

% converseD
tff(fact_5747_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R: set(product_prod(B,A))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Yx),converse(B,A,R)))
     => ~ ! [X4: B,Y4: A] :
            ( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y4),X4) )
           => ~ 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),Y4)),R)) ) ) ).

% converseE
tff(fact_5748_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: 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)),R))
     => 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,R))) ) ).

% converseI
tff(fact_5749_in__listrel1__converse,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: 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),Y3)),listrel1(A,converse(A,A,R))))
    <=> 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),Y3)),converse(list(A),list(A),listrel1(A,R)))) ) ).

% in_listrel1_converse
tff(fact_5750_converse__Times,axiom,
    ! [A: $tType,B: $tType,A5: set(B),B4: set(A)] : converse(B,A,product_Sigma(B,A,A5,aTP_Lamp_nh(set(A),fun(B,set(A)),B4))) = product_Sigma(A,B,B4,aTP_Lamp_xa(set(B),fun(A,set(B)),A5)) ).

% converse_Times
tff(fact_5751_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : converse(B,A,R) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_aka(set(product_prod(B,A)),fun(A,fun(B,bool)),R))) ).

% converse_unfold
tff(fact_5752_converse__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(C,set(product_prod(B,A))),S2: 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)),R),S2))) = 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_akb(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R)),S2)) ).

% converse_UNION
tff(fact_5753_converse__INTER,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(C,set(product_prod(B,A))),S2: 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)),R),S2))) = 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_akb(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R)),S2)) ).

% converse_INTER
tff(fact_5754_relInvImage__Gr,axiom,
    ! [A: $tType,B: $tType,R4: set(product_prod(A,A)),B4: set(A),A5: set(B),F: 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))),R4),product_Sigma(A,A,B4,aTP_Lamp_wx(set(A),fun(A,set(A)),B4))))
     => ( bNF_Gr7122648621184425601vImage(B,A,A5,R4,F) = relcomp(B,A,B,bNF_Gr(B,A,A5,F),relcomp(A,A,B,R4,converse(B,A,bNF_Gr(B,A,A5,F)))) ) ) ).

% relInvImage_Gr
tff(fact_5755_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),R4: set(product_prod(B,B)),F: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A5,R4,F) = 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_akc(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A5),R4),F)) ).

% relInvImage_def
tff(fact_5756_irrefl__tranclI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A] :
      ( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R)),transitive_rtrancl(A,R)) = bot_bot(set(product_prod(A,A))) )
     => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R))) ) ).

% irrefl_tranclI
tff(fact_5757_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: set(product_prod(B,A)),A5: set(C),B4: fun(C,set(B))] :
      ( single_valued(A,B,converse(B,A,R))
     => ( ( A5 != bot_bot(set(C)) )
       => ( aa(set(B),set(A),image(B,A,R),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_uj(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B4)),A5)) ) ) ) ).

% Image_INT_eq
tff(fact_5758_trans__wf__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( trans(A,R)
     => ( wf(A,R)
      <=> ! [A7: A] : wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(A),set(A),image(A,A,converse(A,A,R)),aa(set(A),set(A),insert2(A,A7),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_akd(set(product_prod(A,A)),fun(A,fun(A,set(A))),R),A7)))) ) ) ).

% trans_wf_iff
tff(fact_5759_ord__to__filter__compat,axiom,
    ! [A: $tType,R0: set(product_prod(A,A))] : bNF_Wellorder_compat(set(product_prod(A,A)),set(A),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),fun(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),inf_inf(set(product_prod(set(product_prod(A,A)),set(product_prod(A,A))))),bNF_We4044943003108391690rdLess(A,A)),product_Sigma(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),insert2(set(product_prod(A,A)),R0),bot_bot(set(set(product_prod(A,A)))))),aTP_Lamp_ake(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),R0))),bNF_We413866401316099525erIncl(A,R0),bNF_We8469521843155493636filter(A,R0)) ).

% ord_to_filter_compat
tff(fact_5760_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,B)),X: A,Y3: B,Z: B] :
      ( single_valued(A,B,R)
     => ( 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)),R))
       => ( 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),Z)),R))
         => ( Y3 = Z ) ) ) ) ).

% single_valuedD
tff(fact_5761_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( ! [X4: A,Y4: B,Z4: 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),Y4)),R))
         => ( 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),Z4)),R))
           => ( Y4 = Z4 ) ) )
     => single_valued(A,B,R) ) ).

% single_valuedI
tff(fact_5762_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,B))] :
      ( single_valued(A,B,R)
    <=> ! [X3: A,Y: 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),Y)),R))
         => ! [Z5: 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),Z5)),R))
             => ( Y = Z5 ) ) ) ) ).

% single_valued_def
tff(fact_5763_transD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y3: A,Z: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R)) ) ) ) ).

% transD
tff(fact_5764_transE,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y3: A,Z: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),R))
         => pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),R)) ) ) ) ).

% transE
tff(fact_5765_transI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X4: A,Y4: A,Z4: 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),Y4)),R))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R))
           => pp(aa(set(product_prod(A,A)),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),Z4)),R)) ) )
     => trans(A,R) ) ).

% transI
tff(fact_5766_trans__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( trans(A,R)
    <=> ! [X3: A,Y: 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),X3),Y)),R))
         => ( pp(aa(set(product_prod(A,A)),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),Z5)),R))
           => pp(aa(set(product_prod(A,A)),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),Z5)),R)) ) ) ) ).

% trans_def
tff(fact_5767_lexord__trans,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: set(product_prod(A,A)),Z: 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),Y3)),lexord(A,R)))
     => ( 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)),Y3),Z)),lexord(A,R)))
       => ( trans(A,R)
         => 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),Z)),lexord(A,R))) ) ) ) ).

% lexord_trans
tff(fact_5768_compat__def,axiom,
    ! [A: $tType,A2: $tType,R: set(product_prod(A,A)),R3: set(product_prod(A2,A2)),F: fun(A,A2)] :
      ( bNF_Wellorder_compat(A,A2,R,R3,F)
    <=> ! [A7: A,B7: 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),A7),B7)),R))
         => pp(aa(set(product_prod(A2,A2)),bool,member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,F,A7)),aa(A,A2,F,B7))),R3)) ) ) ).

% compat_def
tff(fact_5769_lenlex__trans,axiom,
    ! [A: $tType,X: list(A),Y3: list(A),R: set(product_prod(A,A)),Z: 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),Y3)),lenlex(A,R)))
     => ( 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)),Y3),Z)),lenlex(A,R)))
       => ( trans(A,R)
         => 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),Z)),lenlex(A,R))) ) ) ) ).

% lenlex_trans
tff(fact_5770_trans__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( trans(A,R)
     => 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% trans_Restr
tff(fact_5771_single__valued__confluent,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X: A,Y3: A,Z: A] :
      ( single_valued(A,A,R)
     => ( pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,R)))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z)),transitive_rtrancl(A,R)))
         => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z)),transitive_rtrancl(A,R)))
            | pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z),Y3)),transitive_rtrancl(A,R))) ) ) ) ) ).

% single_valued_confluent
tff(fact_5772_under__incr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R),A3)),aa(A,set(A),order_under(A,R),B2))) ) ) ).

% under_incr
tff(fact_5773_trans__singleton,axiom,
    ! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_5774_trans__join,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( trans(A,R)
    <=> ! [X3: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),R))
         => pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_akg(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X3)) ) ) ).

% trans_join
tff(fact_5775_wf__finite__segments,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => ( trans(A,R)
       => ( ! [X4: A] : pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_aic(set(product_prod(A,A)),fun(A,fun(A,bool)),R),X4))))
         => wf(A,R) ) ) ) ).

% wf_finite_segments
tff(fact_5776_ord__to__filter__def,axiom,
    ! [A: $tType,R0: set(product_prod(A,A)),R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),bNF_We8469521843155493636filter(A,R0),R) = aa(set(A),set(A),image2(A,A,fChoice(fun(A,A),bNF_Wellorder_embed(A,A,R,R0))),field2(A,R)) ).

% ord_to_filter_def
tff(fact_5777_underS__incr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( trans(A,R)
     => ( antisym(A,R)
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2))) ) ) ) ).

% underS_incr
tff(fact_5778_less__than__iff,axiom,
    ! [X: nat,Y3: 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),Y3)),less_than))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y3)) ) ).

% less_than_iff
tff(fact_5779_some__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => ( X4 = A3 ) )
       => ( fChoice(A,P) = A3 ) ) ) ).

% some_equality
tff(fact_5780_some__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aTP_Lamp_kn(A,fun(A,bool),X)) = X ).

% some_eq_trivial
tff(fact_5781_some__sym__eq__trivial,axiom,
    ! [A: $tType,X: A] : fChoice(A,aa(A,fun(A,bool),fequal(A),X)) = X ).

% some_sym_eq_trivial
tff(fact_5782_antisymD,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
      ( antisym(A,R)
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
         => ( A3 = B2 ) ) ) ) ).

% antisymD
tff(fact_5783_antisymI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [X4: 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),X4),Y4)),R))
         => ( pp(aa(set(product_prod(A,A)),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),X4)),R))
           => ( X4 = Y4 ) ) )
     => antisym(A,R) ) ).

% antisymI
tff(fact_5784_antisym__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( antisym(A,R)
    <=> ! [X3: 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),X3),Y)),R))
         => ( pp(aa(set(product_prod(A,A)),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),X3)),R))
           => ( X3 = Y ) ) ) ) ).

% antisym_def
tff(fact_5785_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,bool),A5: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,P)))
      <=> pp(A5) )
     => ( ? [X_1: A] : pp(aa(A,bool,P,X_1))
      <=> pp(A5) ) ) ).

% verit_sko_ex'
tff(fact_5786_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ! [X_1: A] : pp(aa(A,bool,P,X_1))
    <=> pp(aa(A,bool,P,fChoice(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P)))) ) ).

% verit_sko_forall
tff(fact_5787_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,bool),A5: bool] :
      ( ( pp(aa(A,bool,P,fChoice(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P))))
      <=> pp(A5) )
     => ( ! [X_1: A] : pp(aa(A,bool,P,X_1))
      <=> pp(A5) ) ) ).

% verit_sko_forall'
tff(fact_5788_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B4: A,A5: A,P: fun(A,bool)] :
      ( ( B4 = A5 )
     => ( ( fChoice(A,P) = A5 )
      <=> ( fChoice(A,P) = B4 ) ) ) ).

% verit_sko_forall''
tff(fact_5789_verit__sko__ex__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ? [X_1: A] : pp(aa(A,bool,P,X_1))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_ex_indirect
tff(fact_5790_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P8: fun(A,bool)] :
      ( ( X = fChoice(A,P) )
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
          <=> pp(aa(A,bool,P8,X4)) )
       => ( ? [X_1: A] : pp(aa(A,bool,P8,X_1))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5791_verit__sko__forall__indirect,axiom,
    ! [A: $tType,X: A,P: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X_1: A] : pp(aa(A,bool,P,X_1))
      <=> pp(aa(A,bool,P,X)) ) ) ).

% verit_sko_forall_indirect
tff(fact_5792_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),P8: fun(A,bool)] :
      ( ( X = fChoice(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P)) )
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
          <=> pp(aa(A,bool,P8,X4)) )
       => ( ! [X_1: A] : pp(aa(A,bool,P8,X_1))
        <=> pp(aa(A,bool,P,X)) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5793_someI2,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,A3))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => pp(aa(A,bool,Q,X4)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2
tff(fact_5794_someI__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => pp(aa(A,bool,P,fChoice(A,P))) ) ).

% someI_ex
tff(fact_5795_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => ( ! [X4: A] :
            ( pp(aa(A,bool,P,X4))
           => pp(aa(A,bool,Q,X4)) )
       => pp(aa(A,bool,Q,fChoice(A,P))) ) ) ).

% someI2_ex
tff(fact_5796_someI2__bex,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,member(A,X2),A5))
          & pp(aa(A,bool,P,X2)) )
     => ( ! [X4: A] :
            ( ( pp(aa(set(A),bool,member(A,X4),A5))
              & pp(aa(A,bool,P,X4)) )
           => pp(aa(A,bool,Q,X4)) )
       => pp(aa(A,bool,Q,fChoice(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_aw(set(A),fun(fun(A,bool),fun(A,bool)),A5),P)))) ) ) ).

% someI2_bex
tff(fact_5797_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( pp(aa(A,bool,P,fChoice(A,P)))
    <=> ? [X_1: A] : pp(aa(A,bool,P,X_1)) ) ).

% some_eq_ex
tff(fact_5798_some1__equality,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( ? [X2: A] :
          ( pp(aa(A,bool,P,X2))
          & ! [Y4: A] :
              ( pp(aa(A,bool,P,Y4))
             => ( Y4 = X2 ) ) )
     => ( pp(aa(A,bool,P,A3))
       => ( fChoice(A,P) = A3 ) ) ) ).

% some1_equality
tff(fact_5799_some__in__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5))),A5))
    <=> ( A5 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5800_pick__middlep__def,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: fun(B,fun(A,bool)),Q: fun(A,fun(C,bool)),A3: B,C2: C] : bNF_pick_middlep(B,A,C,P,Q,A3,C2) = fChoice(A,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),aa(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool))),aTP_Lamp_akh(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool)))),P),Q),A3),C2)) ).

% pick_middlep_def
tff(fact_5801_equiv__Eps__preserves,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),X6))),A5)) ) ) ).

% equiv_Eps_preserves
tff(fact_5802_equiv__Eps__in,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),X6))),X6)) ) ) ).

% equiv_Eps_in
tff(fact_5803_antisym__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( antisym(A,R)
     => 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% antisym_Restr
tff(fact_5804_arg__min__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F: fun(B,A),P: fun(B,bool)] : lattices_ord_arg_min(B,A,F,P) = fChoice(B,lattic501386751177426532rg_min(B,A,F,P)) ) ).

% arg_min_def
tff(fact_5805_toCard__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: set(product_prod(B,B))] : bNF_Greatest_toCard(A,B,A5,R) = fChoice(fun(A,B),bNF_Gr1419584066657907630d_pred(A,B,A5,R)) ).

% toCard_def
tff(fact_5806_proj__Eps,axiom,
    ! [A: $tType,A5: set(A),R: set(product_prod(A,A)),X6: set(A)] :
      ( equiv_equiv(A,A5,R)
     => ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A5,R)))
       => ( aa(A,set(A),equiv_proj(A,A,R),fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),X6))) = X6 ) ) ) ).

% proj_Eps
tff(fact_5807_fromCard__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),R: set(product_prod(B,B)),K: B] : bNF_Gr5436034075474128252omCard(A,B,A5,R,K) = fChoice(A,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_aki(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),A5),R),K)) ).

% fromCard_def
tff(fact_5808_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),F: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite2(A),S2))
         => ( lattic7623131987881927897min_on(A,B,F,S2) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_akj(set(A),fun(fun(A,B),fun(A,bool)),S2),F)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_5809_lenlex__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = inv_image(product_prod(nat,list(A)),list(A),lex_prod(nat,list(A),less_than,lex(A,R)),aTP_Lamp_akk(list(A),product_prod(nat,list(A)))) ).

% lenlex_def
tff(fact_5810_mlex__prod__def,axiom,
    ! [A: $tType,F: fun(A,nat),R4: set(product_prod(A,A))] : mlex_prod(A,F,R4) = inv_image(product_prod(nat,A),A,lex_prod(nat,A,less_than,R4),aTP_Lamp_akl(fun(A,nat),fun(A,product_prod(nat,A)),F)) ).

% mlex_prod_def
tff(fact_5811_in__inv__image,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: A,R: set(product_prod(B,B)),F: 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),Y3)),inv_image(B,A,R,F)))
    <=> 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,F,X)),aa(A,B,F,Y3))),R)) ) ).

% in_inv_image
tff(fact_5812_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_ko(A,fun(B,fun(A,fun(B,bool))),X),Y3))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) ).

% Eps_case_prod_eq
tff(fact_5813_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_akm(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P))) ).

% split_paired_Eps
tff(fact_5814_card__of__def,axiom,
    ! [A: $tType,A5: set(A)] : bNF_Ca6860139660246222851ard_of(A,A5) = fChoice(set(product_prod(A,A)),bNF_Ca8970107618336181345der_on(A,A5)) ).

% card_of_def
tff(fact_5815_cardSuc__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : bNF_Ca8387033319878233205ardSuc(A,R) = fChoice(set(product_prod(set(A),set(A))),bNF_Ca6246979054910435723ardSuc(A,R)) ).

% cardSuc_def
tff(fact_5816_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P)) = fChoice(product_prod(A,B),aTP_Lamp_kp(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P)) ).

% Eps_case_prod
tff(fact_5817_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,B)),F: fun(A,B)] : inv_image(B,A,R,F) = 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_akn(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R),F))) ).

% inv_image_def
tff(fact_5818_arg__min__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F: fun(B,A),S2: set(B)] : lattic7623131987881927897min_on(B,A,F,S2) = lattices_ord_arg_min(B,A,F,aTP_Lamp_agi(set(B),fun(B,bool),S2)) ) ).

% arg_min_on_def
tff(fact_5819_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_ako(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_5820_measures__def,axiom,
    ! [A: $tType,Fs: list(fun(A,nat))] : measures(A,Fs) = inv_image(list(nat),A,lex(nat,less_than),aTP_Lamp_akq(list(fun(A,nat)),fun(A,list(nat)),Fs)) ).

% measures_def
tff(fact_5821_rel__filter_Ocases,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),F5: filter(A),G5: filter(B)] :
      ( rel_filter(A,B,R4,F5,G5)
     => ~ ! [Z8: filter(product_prod(A,B))] :
            ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R4),Z8)
           => ( ( map_filter_on(product_prod(A,B),A,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),R4)),product_fst(A,B),Z8) = F5 )
             => ( map_filter_on(product_prod(A,B),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),R4)),product_snd(A,B),Z8) != G5 ) ) ) ) ).

% rel_filter.cases
tff(fact_5822_rel__filter_Ointros,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),Z3: filter(product_prod(A,B)),F5: filter(A),G5: filter(B)] :
      ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R4),Z3)
     => ( ( map_filter_on(product_prod(A,B),A,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),R4)),product_fst(A,B),Z3) = F5 )
       => ( ( map_filter_on(product_prod(A,B),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),R4)),product_snd(A,B),Z3) = G5 )
         => rel_filter(A,B,R4,F5,G5) ) ) ) ).

% rel_filter.intros
tff(fact_5823_rel__filter_Osimps,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),F5: filter(A),G5: filter(B)] :
      ( rel_filter(A,B,R4,F5,G5)
    <=> ? [Z7: filter(product_prod(A,B))] :
          ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R4),Z7)
          & ( map_filter_on(product_prod(A,B),A,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),R4)),product_fst(A,B),Z7) = F5 )
          & ( map_filter_on(product_prod(A,B),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),R4)),product_snd(A,B),Z7) = G5 ) ) ) ).

% rel_filter.simps
tff(fact_5824_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C6: set(A)] :
      ( pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
     => ( ~ pred_maxchain(A,A5,P,C6)
       => ( pp(aa(set(A),bool,pred_chain(A,A5,P),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_akr(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6))))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),C6),fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_akr(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6)))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_5825_type__copy__vimage2p__Grp__Abs,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: fun(A,B),Abs: fun(B,A),G: fun(D,C),P: fun(C,bool),H: fun(C,A)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(C,fun(A,bool)),fun(D,fun(B,bool)),bNF_vimage2p(D,C,B,A,bool,G,Abs),bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),P),H)) = bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),aa(fun(C,bool),fun(D,bool),aTP_Lamp_aks(fun(D,C),fun(fun(C,bool),fun(D,bool)),G),P)),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),Rep),H)),G)) ) ) ).

% type_copy_vimage2p_Grp_Abs
tff(fact_5826_type__copy__vimage2p__Grp__Rep,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,D),P: fun(D,bool),H: fun(D,B)] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( aa(fun(D,fun(B,bool)),fun(C,fun(A,bool)),bNF_vimage2p(C,D,A,B,bool,F,Rep),bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),P),H)) = bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),aa(fun(D,bool),fun(C,bool),aTP_Lamp_akt(fun(C,D),fun(fun(D,bool),fun(C,bool)),F),P)),aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),Abs),H)),F)) ) ) ).

% type_copy_vimage2p_Grp_Rep
tff(fact_5827_Grp__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B),X2: A,Xa3: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Grp(A,B,A5,F),X2),Xa3))
    <=> ( ( Xa3 = aa(A,B,F,X2) )
        & pp(aa(set(A),bool,member(A,X2),A5)) ) ) ).

% Grp_def
tff(fact_5828_subset_Onot__maxchain__Some,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))
     => ( ~ pred_maxchain(set(A),A5,ord_less(set(A)),C6)
       => ( pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aku(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6))))
          & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),C6),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aku(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6)))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_5829_list_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] : list_all2(A,B,bNF_Grp(A,B,A5,F)) = bNF_Grp(list(A),list(B),aa(fun(list(A),bool),set(list(A)),collect(list(A)),aTP_Lamp_akv(set(A),fun(list(A),bool),A5)),map(A,B,F)) ).

% list.rel_Grp
tff(fact_5830_fun_Orel__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,A5: set(A),F: fun(A,B)] : bNF_rel_fun(D,D,A,B,fequal(D),bNF_Grp(A,B,A5,F)) = bNF_Grp(fun(D,A),fun(D,B),aa(fun(fun(D,A),bool),set(fun(D,A)),collect(fun(D,A)),aTP_Lamp_akw(set(A),fun(fun(D,A),bool),A5)),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F)) ).

% fun.rel_Grp
tff(fact_5831_eq__le__Grp__id__iff,axiom,
    ! [A: $tType,R4: 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),R4),id(A))))
    <=> ! [X_1: A] : pp(aa(A,bool,R4,X_1)) ) ).

% eq_le_Grp_id_iff
tff(fact_5832_Collect__case__prod__Grp__eqD,axiom,
    ! [B: $tType,A: $tType,Z: product_prod(A,B),A5: set(A),F: fun(A,B)] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Z),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),bNF_Grp(A,B,A5,F)))))
     => ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),B),aa(fun(A,B),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),B)),comp(A,B,product_prod(A,B)),F),product_fst(A,B)),Z) = aa(product_prod(A,B),B,product_snd(A,B),Z) ) ) ).

% Collect_case_prod_Grp_eqD
tff(fact_5833_pred__on_Osuc__def,axiom,
    ! [A: $tType,A5: set(A),P: fun(A,fun(A,bool)),C6: set(A)] :
      ( ( ( ~ pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
          | pred_maxchain(A,A5,P,C6) )
       => ( pred_suc(A,A5,P,C6) = C6 ) )
      & ( ~ ( ~ pp(aa(set(A),bool,pred_chain(A,A5,P),C6))
            | pred_maxchain(A,A5,P,C6) )
       => ( pred_suc(A,A5,P,C6) = fChoice(set(A),aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_akr(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),A5),P),C6)) ) ) ) ).

% pred_on.suc_def
tff(fact_5834_flip__pred,axiom,
    ! [A: $tType,B: $tType,A5: set(product_prod(A,B)),R4: 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,R4)))))
     => 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_fw(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),R4)))) ) ).

% flip_pred
tff(fact_5835_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)) = linorder_sort_key(A,A,aTP_Lamp_vw(A,A),remdups(A,Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_5836_conversep__noteq,axiom,
    ! [A: $tType,X2: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),conversep(A,A,aTP_Lamp_adp(A,fun(A,bool))),X2),Xa3))
    <=> ( X2 != Xa3 ) ) ).

% conversep_noteq
tff(fact_5837_sort__upt,axiom,
    ! [M: nat,N: nat] : linorder_sort_key(nat,nat,aTP_Lamp_cp(nat,nat),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_5838_sort__upto,axiom,
    ! [I: int,J: int] : linorder_sort_key(int,int,aTP_Lamp_gf(int,int),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_5839_acyclicP__converse,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] :
      ( transitive_acyclic(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),conversep(A,A,R))))
    <=> transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).

% acyclicP_converse
tff(fact_5840_subset_Osuc__def,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))
          | pred_maxchain(set(A),A5,ord_less(set(A)),C6) )
       => ( pred_suc(set(A),A5,ord_less(set(A)),C6) = C6 ) )
      & ( ~ ( ~ pp(aa(set(set(A)),bool,pred_chain(set(A),A5,ord_less(set(A))),C6))
            | pred_maxchain(set(A),A5,ord_less(set(A)),C6) )
       => ( pred_suc(set(A),A5,ord_less(set(A)),C6) = fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aku(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),A5),C6)) ) ) ) ).

% subset.suc_def
tff(fact_5841_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : linorder_sort_key(A,B,aTP_Lamp_akx(B,fun(A,B),C2),Xs) = Xs ) ).

% sort_key_const
tff(fact_5842_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),K: B,Xs: list(A)] : filter2(A,aa(B,fun(A,bool),aTP_Lamp_aky(fun(A,B),fun(B,fun(A,bool)),F),K),linorder_sort_key(A,B,F,Xs)) = filter2(A,aa(B,fun(A,bool),aTP_Lamp_aky(fun(A,B),fun(B,fun(A,bool)),F),K),Xs) ) ).

% sort_key_stable
tff(fact_5843_conversep__converse__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X2),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),X2),Xa3)),converse(A,B,R))) ) ).

% conversep_converse_eq
tff(fact_5844_converse__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : converse(A,B,X2) = 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),X2)))) ).

% converse_def
tff(fact_5845_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_vw(A,A),Xs)) ) ).

% sorted_sort
tff(fact_5846_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( linorder_sort_key(A,A,aTP_Lamp_vw(A,A),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_5847_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linorder_sort_key(A,A,aTP_Lamp_vw(A,A),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_5848_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_akz(A,fun(list(A),A))),filter2(A,P,linorder_sort_key(A,A,aTP_Lamp_vw(A,A),Xs))) ) ).

% Bleast_code
tff(fact_5849_fun_Orel__compp__Grp,axiom,
    ! [D: $tType,B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : bNF_rel_fun(D,D,A,B,fequal(D),R4) = 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_sr(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),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_sr(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)))) ).

% fun.rel_compp_Grp
tff(fact_5850_list_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : list_all2(A,B,R4) = 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_acj(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R4)),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_acj(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R4)),map(product_prod(A,B),B,product_snd(A,B)))) ).

% list.rel_compp_Grp
tff(fact_5851_relpowp__add,axiom,
    ! [A: $tType,M: nat,N: nat,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),P) = 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),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M),P)),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P)) ).

% relpowp_add
tff(fact_5852_OO__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: fun(A,fun(C,bool)),S2: fun(C,fun(B,bool)),X2: 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),R4),S2),X2),Xa3))
    <=> ? [Y: C] :
          ( pp(aa(C,bool,aa(A,fun(C,bool),R4,X2),Y))
          & pp(aa(B,bool,aa(C,fun(B,bool),S2,Y),Xa3)) ) ) ).

% OO_def
tff(fact_5853_relcompp__relcomp__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C)),X2: 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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),aTP_Lamp_ala(set(product_prod(B,C)),fun(B,fun(C,bool)),S)),X2),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),X2),Xa3)),relcomp(A,B,C,R,S))) ) ).

% relcompp_relcomp_eq
tff(fact_5854_relcompp__SUP__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S: fun(A,fun(C,bool)),R: 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),S),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)),R),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_alb(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),S),R)),I5)) ).

% relcompp_SUP_distrib
tff(fact_5855_relcompp__SUP__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R: fun(D,fun(A,fun(C,bool))),I5: set(D),S: 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)),R),I5))),S) = 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_alc(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),R),S)),I5)) ).

% relcompp_SUP_distrib2
tff(fact_5856_relcomp__def,axiom,
    ! [A: $tType,C: $tType,B: $tType,X2: set(product_prod(A,B)),Xa3: set(product_prod(B,C))] : relcomp(A,B,C,X2,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_as(set(product_prod(A,B)),fun(A,fun(B,bool))),X2)),aTP_Lamp_ala(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa3)))) ).

% relcomp_def
tff(fact_5857_OO__Grp__alt,axiom,
    ! [A: $tType,C: $tType,B: $tType,A5: set(C),F: fun(C,A),G: fun(C,B),X2: 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,F))),bNF_Grp(C,B,A5,G)),X2),Xa3))
    <=> ? [Z5: C] :
          ( pp(aa(set(C),bool,member(C,Z5),A5))
          & ( aa(C,A,F,Z5) = X2 )
          & ( aa(C,B,G,Z5) = Xa3 ) ) ) ).

% OO_Grp_alt
tff(fact_5858_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_5859_single__valuedp__single__valued__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
      ( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R))
    <=> single_valued(A,B,R) ) ).

% single_valuedp_single_valued_eq
tff(fact_5860_univ__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X6: set(B)] : bNF_Greatest_univ(B,A,F,X6) = aa(B,A,F,fChoice(B,aTP_Lamp_agi(set(B),fun(B,bool),X6))) ).

% univ_def
tff(fact_5861_single__valuedp__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,bool))] :
      ( single_valuedp(A,B,R)
    <=> ! [X3: A] : uniq(B,aa(A,fun(B,bool),R,X3)) ) ).

% single_valuedp_iff_Uniq
tff(fact_5862_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_ald(set(set(A)),fun(set(set(A)),bool),A5)) ).

% chains_def
tff(fact_5863_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,X: A,Y3: B] : aa(product_prod(A,B),set(A),basic_fsts(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_5864_prod__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3)) = aa(set(B),set(B),insert2(B,Y3),bot_bot(set(B))) ).

% prod_set_simps(2)
tff(fact_5865_prod_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(C,bool)),R23: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R12,R23) = aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool)),aa(fun(product_prod(A,B),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),fun(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool))),relcompp(product_prod(A,B),product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D)),conversep(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ale(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R12),R23)),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D))))),bNF_Grp(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ale(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R12),R23)),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).

% prod.rel_compp_Grp
tff(fact_5866_prod_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R12: fun(A,fun(C,bool)),R23: fun(B,fun(D,bool)),A3: product_prod(A,B),B2: product_prod(C,D)] :
      ( pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,R12,R23),A3),B2))
    <=> ? [Z5: product_prod(product_prod(A,C),product_prod(B,D))] :
          ( pp(aa(set(product_prod(product_prod(A,C),product_prod(B,D))),bool,member(product_prod(product_prod(A,C),product_prod(B,D)),Z5),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ale(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R12),R23))))
          & ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)),Z5) = A3 )
          & ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)),Z5) = B2 ) ) ) ).

% prod.in_rel
tff(fact_5867_prod_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A13: set(A),F1: fun(A,C),A24: set(B),F22: fun(B,D)] : basic_rel_prod(A,C,B,D,bNF_Grp(A,C,A13,F1),bNF_Grp(B,D,A24,F22)) = bNF_Grp(product_prod(A,B),product_prod(C,D),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(B),fun(product_prod(A,B),bool),aTP_Lamp_alf(set(A),fun(set(B),fun(product_prod(A,B),bool)),A13),A24)),product_map_prod(A,C,B,D,F1,F22)) ).

% prod.rel_Grp
tff(fact_5868_rel__prod__inject,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A3: A,B2: C,C2: B,D3: D] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C2),D3)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),C2))
        & pp(aa(D,bool,aa(C,fun(D,bool),R23,B2),D3)) ) ) ).

% rel_prod_inject
tff(fact_5869_fst__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: fun(A,fun(C,bool)),B4: fun(B,fun(D,bool))] : pp(aa(fun(product_prod(C,D),C),bool,aa(fun(product_prod(A,B),A),fun(fun(product_prod(C,D),C),bool),bNF_rel_fun(product_prod(A,B),product_prod(C,D),A,C,basic_rel_prod(A,C,B,D,A5,B4),A5),product_fst(A,B)),product_fst(C,D))) ).

% fst_transfer
tff(fact_5870_snd__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A5: fun(A,fun(C,bool)),B4: fun(B,fun(D,bool))] : pp(aa(fun(product_prod(C,D),D),bool,aa(fun(product_prod(A,B),B),fun(fun(product_prod(C,D),D),bool),bNF_rel_fun(product_prod(A,B),product_prod(C,D),B,D,basic_rel_prod(A,C,B,D,A5,B4),B4),product_snd(A,B)),product_snd(C,D))) ).

% snd_transfer
tff(fact_5871_rel__prod_Ointros,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,R12: fun(A,fun(B,bool)),A3: A,B2: B,R23: fun(C,fun(D,bool)),C2: C,D3: D] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A3),B2))
     => ( pp(aa(D,bool,aa(C,fun(D,bool),R23,C2),D3))
       => pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D3))) ) ) ).

% rel_prod.intros
tff(fact_5872_rel__prod_Osimps,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A1),A22))
    <=> ? [A7: A,B7: B,C4: C,D4: D] :
          ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),C4) )
          & ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B7),D4) )
          & pp(aa(B,bool,aa(A,fun(B,bool),R12,A7),B7))
          & pp(aa(D,bool,aa(C,fun(D,bool),R23,C4),D4)) ) ) ).

% rel_prod.simps
tff(fact_5873_rel__prod_Ocases,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: fun(A,fun(B,bool)),R23: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
      ( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R12,R23),A1),A22))
     => ~ ! [A4: A,B3: B,C3: C] :
            ( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A4),C3) )
           => ! [D2: D] :
                ( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B3),D2) )
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R12,A4),B3))
                 => ~ pp(aa(D,bool,aa(C,fun(D,bool),R23,C3),D2)) ) ) ) ) ).

% rel_prod.cases
tff(fact_5874_prod_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1a: fun(A,fun(E,bool)),S2a: fun(B,fun(F2,bool)),X: product_prod(A,B),G1: fun(C,E),G22: fun(D,F2),Y3: product_prod(C,D)] :
      ( pp(aa(product_prod(E,F2),bool,aa(product_prod(A,B),fun(product_prod(E,F2),bool),basic_rel_prod(A,E,B,F2,S1a,S2a),X),aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,G1,G22),Y3)))
    <=> pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,aa(fun(C,E),fun(A,fun(C,bool)),aTP_Lamp_alg(fun(A,fun(E,bool)),fun(fun(C,E),fun(A,fun(C,bool))),S1a),G1),aa(fun(D,F2),fun(B,fun(D,bool)),aTP_Lamp_alh(fun(B,fun(F2,bool)),fun(fun(D,F2),fun(B,fun(D,bool))),S2a),G22)),X),Y3)) ) ).

% prod.rel_map(2)
tff(fact_5875_prod_Orel__map_I1_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1b: fun(E,fun(C,bool)),S2b: fun(F2,fun(D,bool)),I1: fun(A,E),I22: fun(B,F2),X: product_prod(A,B),Y3: product_prod(C,D)] :
      ( pp(aa(product_prod(C,D),bool,aa(product_prod(E,F2),fun(product_prod(C,D),bool),basic_rel_prod(E,C,F2,D,S1b,S2b),aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,I1,I22),X)),Y3))
    <=> pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,aa(fun(A,E),fun(A,fun(C,bool)),aTP_Lamp_ali(fun(E,fun(C,bool)),fun(fun(A,E),fun(A,fun(C,bool))),S1b),I1),aa(fun(B,F2),fun(B,fun(D,bool)),aTP_Lamp_alj(fun(F2,fun(D,bool)),fun(fun(B,F2),fun(B,fun(D,bool))),S2b),I22)),X),Y3)) ) ).

% prod.rel_map(1)
tff(fact_5876_rel__prod__conv,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(C,bool)),R23: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R12,R23) = aa(fun(A,fun(B,fun(product_prod(C,D),bool))),fun(product_prod(A,B),fun(product_prod(C,D),bool)),product_case_prod(A,B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_all(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),R12),R23)) ).

% rel_prod_conv
tff(fact_5877_int_Oid__abs__transfer,axiom,
    pp(aa(fun(product_prod(nat,nat),int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),int),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),int,basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat)),pcr_int),aTP_Lamp_alm(product_prod(nat,nat),product_prod(nat,nat))),abs_Integ)) ).

% int.id_abs_transfer
tff(fact_5878_convol__transfer,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,R4: fun(A,fun(C,bool)),S2: fun(B,fun(D,bool)),T4: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(C,F2),fun(C,product_prod(D,F2)))),bool,aa(fun(fun(A,B),fun(fun(A,E),fun(A,product_prod(B,E)))),fun(fun(fun(C,D),fun(fun(C,F2),fun(C,product_prod(D,F2)))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(A,E),fun(A,product_prod(B,E))),fun(fun(C,F2),fun(C,product_prod(D,F2))),bNF_rel_fun(A,C,B,D,R4,S2),bNF_rel_fun(fun(A,E),fun(C,F2),fun(A,product_prod(B,E)),fun(C,product_prod(D,F2)),bNF_rel_fun(A,C,E,F2,R4,T4),bNF_rel_fun(A,C,product_prod(B,E),product_prod(D,F2),R4,basic_rel_prod(B,D,E,F2,S2,T4)))),bNF_convol(A,B,E)),bNF_convol(C,D,F2))) ).

% convol_transfer
tff(fact_5879_Pair__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(B,fun(D,product_prod(B,D))),bool,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),bool),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),A5,bNF_rel_fun(C,D,product_prod(A,C),product_prod(B,D),B4,basic_rel_prod(A,B,C,D,A5,B4))),product_Pair(A,C)),product_Pair(B,D))) ).

% Pair_transfer
tff(fact_5880_case__prod__transfer,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A5: fun(A,fun(D,bool)),B4: fun(B,fun(E,bool)),C6: fun(C,fun(F2,bool))] : pp(aa(fun(fun(D,fun(E,F2)),fun(product_prod(D,E),F2)),bool,aa(fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)),fun(fun(fun(D,fun(E,F2)),fun(product_prod(D,E),F2)),bool),bNF_rel_fun(fun(A,fun(B,C)),fun(D,fun(E,F2)),fun(product_prod(A,B),C),fun(product_prod(D,E),F2),bNF_rel_fun(A,D,fun(B,C),fun(E,F2),A5,bNF_rel_fun(B,E,C,F2,B4,C6)),bNF_rel_fun(product_prod(A,B),product_prod(D,E),C,F2,basic_rel_prod(A,D,B,E,A5,B4),C6)),product_case_prod(A,B,C)),product_case_prod(D,E,F2))) ).

% case_prod_transfer
tff(fact_5881_Partial__order__Restr,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A5: set(A)] :
      ( order_7125193373082350890der_on(A,field2(A,R),R)
     => order_7125193373082350890der_on(A,field2(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(set(A),fun(A,set(A)),A5)))) ) ).

% Partial_order_Restr
tff(fact_5882_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R4: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F)
     => ( 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,G),top_top(set(C)))),S2))
       => finite4664212375090638736ute_on(C,B,R4,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_5883_is__singleton__altdef,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_5884_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S2,F)
     => finite4664212375090638736ute_on(A,B,S2,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_th(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F),G)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_5885_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F)
    <=> ! [X3: A,Y: A] :
          ( pp(aa(set(A),bool,member(A,X3),S2))
         => ( pp(aa(set(A),bool,member(A,Y),S2))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_5886_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B)),X: A,Y3: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F)
     => ( pp(aa(set(A),bool,member(A,X),S2))
       => ( pp(aa(set(A),bool,member(A,Y3),S2))
         => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y3)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y3)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_5887_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S2: set(A),F: fun(A,fun(B,B)),X: A,Y3: A,G: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S2,F)
     => ( pp(aa(set(A),bool,member(A,X),S2))
       => ( pp(aa(set(A),bool,member(A,Y3),S2))
         => ( aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,Y3)),aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,X)),G)) = aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,X)),aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,Y3)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_5888_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S2))
         => ( pp(aa(set(A),bool,member(A,Y4),S2))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4)) ) ) )
     => finite4664212375090638736ute_on(A,B,S2,F) ) ).

% comp_fun_commute_on.intro
tff(fact_5889_Zorns__po__lemma,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( order_7125193373082350890der_on(A,field2(A,R),R)
     => ( ! [C7: set(A)] :
            ( pp(aa(set(set(A)),bool,member(set(A),C7),chains(A,R)))
           => ? [X2: A] :
                ( pp(aa(set(A),bool,member(A,X2),field2(A,R)))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),C7))
                   => pp(aa(set(product_prod(A,A)),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),X2)),R)) ) ) )
       => ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),field2(A,R)))
            & ! [Xa3: A] :
                ( pp(aa(set(A),bool,member(A,Xa3),field2(A,R)))
               => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa3)),R))
                 => ( Xa3 = X4 ) ) ) ) ) ) ).

% Zorns_po_lemma
tff(fact_5890_pcr__int__def,axiom,
    pcr_int = aa(fun(product_prod(nat,nat),fun(int,bool)),fun(product_prod(nat,nat),fun(int,bool)),aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(int,bool)),fun(product_prod(nat,nat),fun(int,bool))),relcompp(product_prod(nat,nat),product_prod(nat,nat),int),basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat))),cr_int) ).

% pcr_int_def
tff(fact_5891_lcm__code__integer,axiom,
    ! [A3: code_integer,B2: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),A3),B2) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),A3)),aa(code_integer,code_integer,abs_abs(code_integer),B2))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_gcd(code_integer),A3),B2)) ).

% lcm_code_integer
tff(fact_5892_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,bool)] : ord_Least(A,P) = the(A,aTP_Lamp_aln(fun(A,bool),fun(A,bool),P)) ) ).

% Least_def
tff(fact_5893_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_5894_lcm__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_left_idem
tff(fact_5895_lcm__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_right_idem
tff(fact_5896_lcm_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% lcm.bottom_right_bottom
tff(fact_5897_lcm_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% lcm.bottom_left_bottom
tff(fact_5898_lcm__neg1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg1
tff(fact_5899_lcm__neg2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm_neg2
tff(fact_5900_lcm__least__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_lcm(A),A3),B2)),C2))
        <=> ( 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)) ) ) ) ).

% lcm_least_iff
tff(fact_5901_dvd__lcm2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).

% dvd_lcm2
tff(fact_5902_dvd__lcm1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_lcm(A),A3),B2))) ) ).

% dvd_lcm1
tff(fact_5903_lcm__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: num] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(num,A,numeral_numeral(A),N)) ) ).

% lcm_neg_numeral_2
tff(fact_5904_lcm__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),A3) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(num,A,numeral_numeral(A),N)),A3) ) ).

% lcm_neg_numeral_1
tff(fact_5905_lcm__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = one_one(A) )
        <=> ( 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))) ) ) ) ).

% lcm_eq_1_iff
tff(fact_5906_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_5907_gcd__dvd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).

% gcd_dvd_lcm
tff(fact_5908_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_5909_lcm__least,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))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C2)) ) ) ) ).

% lcm_least
tff(fact_5910_lcm__dvdD2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_lcm(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C2)) ) ) ).

% lcm_dvdD2
tff(fact_5911_lcm__dvdD1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_lcm(A),A3),B2)),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C2)) ) ) ).

% lcm_dvdD1
tff(fact_5912_dvd__lcmI2,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),A3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2))) ) ) ).

% dvd_lcmI2
tff(fact_5913_dvd__lcmI1,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),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2))) ) ) ).

% dvd_lcmI1
tff(fact_5914_lcm__mono,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A,D3: 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),D3))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),C2),D3))) ) ) ) ).

% lcm_mono
tff(fact_5915_lcm_Oassoc,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ).

% lcm.assoc
tff(fact_5916_lcm_Ocommute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A3) ) ).

% lcm.commute
tff(fact_5917_lcm_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2)) ) ).

% lcm.left_commute
tff(fact_5918_lcm__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% lcm_eq_0_iff
tff(fact_5919_zero__eq__lcm__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_eq_lcm_iff
tff(fact_5920_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_5921_Least__Suc2,axiom,
    ! [P: fun(nat,bool),N: nat,Q: fun(nat,bool),M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( 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_5922_lcm__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_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit1
tff(fact_5923_lcm__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_lcm(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_mult_unit2
tff(fact_5924_lcm__div__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_lcm(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A3)),C2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit1
tff(fact_5925_lcm__div__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_lcm(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),C2) ) ) ) ).

% lcm_div_unit2
tff(fact_5926_Least__Suc,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_agy(fun(nat,bool),fun(nat,bool),P))) ) ) ) ).

% Least_Suc
tff(fact_5927_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_12: A] : pp(aa(A,bool,P,X_12))
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,bool),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_5928_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,bool)] : bleast(A,S2,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_alo(set(A),fun(fun(A,bool),fun(A,bool)),S2),P)) ) ).

% Bleast_def
tff(fact_5929_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,bool)] : abort_Bleast(A,S2,P) = ord_Least(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_alo(set(A),fun(fun(A,bool),fun(A,bool)),S2),P)) ) ).

% abort_Bleast_def
tff(fact_5930_int_Opcr__cr__eq,axiom,
    pcr_int = cr_int ).

% int.pcr_cr_eq
tff(fact_5931_cr__int__def,axiom,
    ! [X2: product_prod(nat,nat)] : aa(product_prod(nat,nat),fun(int,bool),cr_int,X2) = aa(int,fun(int,bool),fequal(int),aa(product_prod(nat,nat),int,abs_Integ,X2)) ).

% cr_int_def
tff(fact_5932_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F: fun(A,B),S2: set(A)] :
          ( order_mono(A,B,F)
         => ( ? [X2: A] :
                ( pp(aa(set(A),bool,member(A,X2),S2))
                & ! [Xa4: A] :
                    ( pp(aa(set(A),bool,member(A,Xa4),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa4)) ) )
           => ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_alp(fun(A,B),fun(set(A),fun(B,bool)),F),S2)) = aa(A,B,F,ord_Least(A,aTP_Lamp_alq(set(A),fun(A,bool),S2))) ) ) ) ) ).

% Least_mono
tff(fact_5933_Lcm__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = finite_fold(A,A,gcd_lcm(A),one_one(A),A5) ) )
          & ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
           => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) ) ) ) ) ).

% Lcm_fin.eq_fold
tff(fact_5934_lcm__integer__def,axiom,
    gcd_lcm(code_integer) = aa(fun(int,fun(int,int)),fun(code_integer,fun(code_integer,code_integer)),map_fun(code_integer,int,fun(int,int),fun(code_integer,code_integer),code_int_of_integer,map_fun(code_integer,int,int,code_integer,code_int_of_integer,code_integer_of_int)),gcd_lcm(int)) ).

% lcm_integer_def
tff(fact_5935_Quotient__int,axiom,
    quotient(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ,cr_int) ).

% Quotient_int
tff(fact_5936_lcm__0__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% lcm_0_iff_nat
tff(fact_5937_lcm__proj2__if__dvd__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),Y3))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y3) = Y3 ) ) ).

% lcm_proj2_if_dvd_nat
tff(fact_5938_lcm__proj1__if__dvd__nat,axiom,
    ! [X: nat,Y3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),Y3))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),Y3),X) = Y3 ) ) ).

% lcm_proj1_if_dvd_nat
tff(fact_5939_lcm__proj2__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = N )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),N)) ) ).

% lcm_proj2_iff_nat
tff(fact_5940_lcm__proj1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),N),M)) ) ).

% lcm_proj1_iff_nat
tff(fact_5941_lcm__1__iff__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% lcm_1_iff_nat
tff(fact_5942_lcm__0__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = zero_zero(int) )
    <=> ( ( M = zero_zero(int) )
        | ( N = zero_zero(int) ) ) ) ).

% lcm_0_iff_int
tff(fact_5943_lcm__int__int__eq,axiom,
    ! [M: nat,N: nat] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N)) ).

% lcm_int_int_eq
tff(fact_5944_abs__lcm__int,axiom,
    ! [I: int,J: int] : aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),I),J)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),I),J) ).

% abs_lcm_int
tff(fact_5945_lcm__abs1__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,abs_abs(int),X)),Y3) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) ).

% lcm_abs1_int
tff(fact_5946_lcm__abs2__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,abs_abs(int),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) ).

% lcm_abs2_int
tff(fact_5947_Lcm__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) ) ) ) ).

% Lcm_fin.infinite
tff(fact_5948_Lcm__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_fin.empty
tff(fact_5949_is__unit__Lcm__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),one_one(A)))
        <=> ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = one_one(A) ) ) ) ).

% is_unit_Lcm_fin_iff
tff(fact_5950_lcm__1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) )
        & ( ( N = one_one(int) )
          | ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_5951_Lcm__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) ) ).

% Lcm_fin.insert
tff(fact_5952_lcm__proj2__if__dvd__int,axiom,
    ! [X: int,Y3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),Y3))
     => ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) = aa(int,int,abs_abs(int),Y3) ) ) ).

% lcm_proj2_if_dvd_int
tff(fact_5953_lcm__proj1__if__dvd__int,axiom,
    ! [X: int,Y3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),X),Y3))
     => ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Y3),X) = aa(int,int,abs_abs(int),Y3) ) ) ).

% lcm_proj1_if_dvd_int
tff(fact_5954_lcm__proj2__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = aa(int,int,abs_abs(int),N) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),N)) ) ).

% lcm_proj2_iff_int
tff(fact_5955_lcm__proj1__iff__int,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N) = aa(int,int,abs_abs(int),M) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),N),M)) ) ).

% lcm_proj1_iff_int
tff(fact_5956_lcm__nat__abs__right__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),N),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ).

% lcm_nat_abs_right_eq
tff(fact_5957_lcm__nat__abs__left__eq,axiom,
    ! [K: int,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),N) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),K),aa(nat,int,semiring_1_of_nat(int),N))) ).

% lcm_nat_abs_left_eq
tff(fact_5958_lcm__ge__0__int,axiom,
    ! [X: int,Y3: 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_lcm(int),X),Y3))) ).

% lcm_ge_0_int
tff(fact_5959_Inf__nat__def,axiom,
    ! [X6: set(nat)] : aa(set(nat),nat,complete_Inf_Inf(nat),X6) = ord_Least(nat,aTP_Lamp_alr(set(nat),fun(nat,bool),X6)) ).

% Inf_nat_def
tff(fact_5960_prod__gcd__lcm__nat,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N)) ).

% prod_gcd_lcm_nat
tff(fact_5961_lcm__neg1__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y3) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) ).

% lcm_neg1_int
tff(fact_5962_lcm__neg2__int,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y3)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) ).

% lcm_neg2_int
tff(fact_5963_dvd__lcm__I2__int,axiom,
    ! [I: int,N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),I),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),I),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N))) ) ).

% dvd_lcm_I2_int
tff(fact_5964_dvd__lcm__I1__int,axiom,
    ! [I: int,M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),I),M))
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),I),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N))) ) ).

% dvd_lcm_I1_int
tff(fact_5965_lcm__unique__nat,axiom,
    ! [A3: nat,D3: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),D3))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),D3))
        & ! [E4: nat] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),A3),E4))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),B2),E4)) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),D3),E4)) ) )
    <=> ( D3 = aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),A3),B2) ) ) ).

% lcm_unique_nat
tff(fact_5966_dvd__lcm__I2__nat,axiom,
    ! [K: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N))) ) ).

% dvd_lcm_I2_nat
tff(fact_5967_dvd__lcm__I1__nat,axiom,
    ! [K: nat,M: nat,N: 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),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N))) ) ).

% dvd_lcm_I1_nat
tff(fact_5968_Lcm__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) ) ) ) ).

% Lcm_fin.in_idem
tff(fact_5969_lcm__int__def,axiom,
    ! [X: int,Y3: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),X))),aa(int,nat,nat2,aa(int,int,abs_abs(int),Y3)))) ).

% lcm_int_def
tff(fact_5970_lcm__pos__nat,axiom,
    ! [M: nat,N: 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)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M),N))) ) ) ).

% lcm_pos_nat
tff(fact_5971_lcm__pos__int,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( ( N != zero_zero(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_lcm(int),M),N))) ) ) ).

% lcm_pos_int
tff(fact_5972_Quotient__cr__rel,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
     => ! [X2: A,Xa3: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X2),Xa3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X2),X2))
            & ( aa(A,B,Abs,X2) = Xa3 ) ) ) ) ).

% Quotient_cr_rel
tff(fact_5973_Quotient__def,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
    <=> ( ! [A7: B] : aa(A,B,Abs,aa(B,A,Rep,A7)) = A7
        & ! [A7: B] : pp(aa(A,bool,aa(A,fun(A,bool),R4,aa(B,A,Rep,A7)),aa(B,A,Rep,A7)))
        & ! [R5: A,S7: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),R4,R5),S7))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,R5),R5))
              & pp(aa(A,bool,aa(A,fun(A,bool),R4,S7),S7))
              & ( aa(A,B,Abs,R5) = aa(A,B,Abs,S7) ) ) )
        & ! [X3: A,Xa2: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X3),Xa2))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X3),X3))
              & ( aa(A,B,Abs,X3) = Xa2 ) ) ) ) ) ).

% Quotient_def
tff(fact_5974_QuotientI,axiom,
    ! [A: $tType,B: $tType,Abs: fun(B,A),Rep: fun(A,B),R4: fun(B,fun(B,bool)),T4: fun(B,fun(A,bool))] :
      ( ! [A4: A] : aa(B,A,Abs,aa(A,B,Rep,A4)) = A4
     => ( ! [A4: A] : pp(aa(B,bool,aa(B,fun(B,bool),R4,aa(A,B,Rep,A4)),aa(A,B,Rep,A4)))
       => ( ! [R2: B,S4: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),R4,R2),S4))
            <=> ( pp(aa(B,bool,aa(B,fun(B,bool),R4,R2),R2))
                & pp(aa(B,bool,aa(B,fun(B,bool),R4,S4),S4))
                & ( aa(B,A,Abs,R2) = aa(B,A,Abs,S4) ) ) )
         => ( ! [X4: B,Xa4: A] :
                ( pp(aa(A,bool,aa(B,fun(A,bool),T4,X4),Xa4))
              <=> ( pp(aa(B,bool,aa(B,fun(B,bool),R4,X4),X4))
                  & ( aa(B,A,Abs,X4) = Xa4 ) ) )
           => quotient(B,A,R4,Abs,Rep,T4) ) ) ) ) ).

% QuotientI
tff(fact_5975_dvd__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5))) ) ) ).

% dvd_Lcm_fin
tff(fact_5976_Quotient__crel__typecopy,axiom,
    ! [A: $tType,B: $tType,Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,fequal(A),Abs,Rep,T4)
     => ! [X2: A,Xa3: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X2),Xa3))
        <=> ( X2 = aa(B,A,Rep,Xa3) ) ) ) ).

% Quotient_crel_typecopy
tff(fact_5977_Lcm__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = zero_zero(A) )
          <=> pp(aa(set(A),bool,member(A,zero_zero(A)),A5)) ) ) ) ).

% Lcm_fin_0_iff
tff(fact_5978_Lcm__fin__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),A3: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),A3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),A3)) ) ) ) ).

% Lcm_fin_least
tff(fact_5979_Lcm__fin__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),B2))
          <=> ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),B2)) ) ) ) ) ).

% Lcm_fin_dvd_iff
tff(fact_5980_UNIV__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),T4: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,top_top(set(B)))
     => ( ! [X4: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T4,X4),Xa4))
          <=> ( X4 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,fequal(B),Abs,Rep,T4) ) ) ).

% UNIV_typedef_to_Quotient
tff(fact_5981_lcm__list__least,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Bs: list(A),A3: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,member(A,B3),aa(list(A),set(A),set2(A),Bs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),A3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Bs))),A3)) ) ) ).

% lcm_list_least
tff(fact_5982_lcm__list__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs))),B2))
        <=> ! [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),dvd_dvd(A),X3),B2)) ) ) ) ).

% lcm_list_dvd_iff
tff(fact_5983_lcm__integer_Orsp,axiom,
    pp(aa(fun(int,fun(int,int)),bool,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),gcd_lcm(int)),gcd_lcm(int))) ).

% lcm_integer.rsp
tff(fact_5984_Lcm__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_lcm(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),B4)),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) ) ) ) ).

% Lcm_fin.subset
tff(fact_5985_lcm__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),X),Xa)) = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(code_integer,int,code_int_of_integer,X)),aa(code_integer,int,code_int_of_integer,Xa)) ).

% lcm_integer.rep_eq
tff(fact_5986_lcm__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),gcd_lcm(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,X)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),Xa),X)) ).

% lcm_integer.abs_eq
tff(fact_5987_Lcm__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B4: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B4)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),aa(set(A),A,semiring_gcd_Lcm_fin(A),B4)) ) ).

% Lcm_fin.union
tff(fact_5988_lcm__cases__int,axiom,
    ! [X: int,Y3: 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)),Y3))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3))) ) )
     => ( ( 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),Y3),zero_zero(int)))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),aa(int,int,uminus_uminus(int),Y3)))) ) )
       => ( ( 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)),Y3))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),Y3))) ) )
         => ( ( 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),Y3),zero_zero(int)))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y3)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_lcm(int),X),Y3))) ) ) ) ) ).

% lcm_cases_int
tff(fact_5989_lcm__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),A3),D3))
        & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),D3))
        & ! [E4: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),A3),E4))
              & pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B2),E4)) )
           => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),D3),E4)) ) )
    <=> ( D3 = aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) ) ) ).

% lcm_unique_int
tff(fact_5990_lcm__nat__def,axiom,
    ! [X: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),X),Y3) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y3)) ).

% lcm_nat_def
tff(fact_5991_Quotient__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),T1: fun(A,fun(B,bool)),R23: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),T22: fun(B,fun(C,bool))] :
      ( quotient(A,B,R12,Abs1,Rep1,T1)
     => ( quotient(B,C,R23,Abs22,Rep22,T22)
       => quotient(A,C,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),T1),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),R23),conversep(A,B,T1))),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22),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),T1),T22)) ) ) ).

% Quotient_compose
tff(fact_5992_prod__gcd__lcm__int,axiom,
    ! [M: int,N: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),M)),aa(int,int,abs_abs(int),N)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)),aa(int,int,aa(int,fun(int,int),gcd_lcm(int),M),N)) ).

% prod_gcd_lcm_int
tff(fact_5993_Lcm__fin__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = one_one(A) )
        <=> ( ! [X3: A] :
                ( pp(aa(set(A),bool,member(A,X3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),one_one(A))) )
            & pp(aa(set(A),bool,finite_finite2(A),A5)) ) ) ) ).

% Lcm_fin_1_iff
tff(fact_5994_lcm__altdef__int,axiom,
    ! [A3: int,B2: int] : aa(int,int,aa(int,fun(int,int),gcd_lcm(int),A3),B2) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),B2))),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B2)) ).

% lcm_altdef_int
tff(fact_5995_Lcm__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ) ) ).

% Lcm_fin.remove
tff(fact_5996_Lcm__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A5: set(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))))) ) ).

% Lcm_fin.insert_remove
tff(fact_5997_Lcm__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).

% Lcm_fin.set_eq_fold
tff(fact_5998_lcm__integer_Otransfer,axiom,
    pp(aa(fun(code_integer,fun(code_integer,code_integer)),bool,aa(fun(int,fun(int,int)),fun(fun(code_integer,fun(code_integer,code_integer)),bool),bNF_rel_fun(int,code_integer,fun(int,int),fun(code_integer,code_integer),code_pcr_integer,bNF_rel_fun(int,code_integer,int,code_integer,code_pcr_integer,code_pcr_integer)),gcd_lcm(int)),gcd_lcm(code_integer))) ).

% lcm_integer.transfer
tff(fact_5999_Lcm__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Lcm_fin(A) = bounde2362111253966948842tice_F(A,gcd_lcm(A),one_one(A),zero_zero(A)) ) ) ).

% Lcm_fin_def
tff(fact_6000_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list(int)] : gcd_Lcm(int,aa(list(int),set(int),set2(int),Xs)) = aa(int,int,fold(int,int,gcd_lcm(int),Xs),one_one(int)) ).

% Lcm_int_set_eq_fold
tff(fact_6001_Lcm__eq__Max__nat,axiom,
    ! [M2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M2))
     => ( ( M2 != bot_bot(set(nat)) )
       => ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M2))
         => ( ! [M4: nat,N2: nat] :
                ( pp(aa(set(nat),bool,member(nat,M4),M2))
               => ( pp(aa(set(nat),bool,member(nat,N2),M2))
                 => pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M4),N2)),M2)) ) )
           => ( gcd_Lcm(nat,M2) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),M2) ) ) ) ) ) ).

% Lcm_eq_Max_nat
tff(fact_6002_map__of__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_zr(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),map_of(A,C,Xs)) ).

% map_of_map
tff(fact_6003_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_6004_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_6005_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: option(A)] :
      ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),A3) = none(B) )
    <=> ( A3 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_6006_map__option__eq__Some,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),Xo: option(B),Y3: A] :
      ( ( aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Xo) = aa(A,option(A),some(A),Y3) )
    <=> ? [Z5: B] :
          ( ( Xo = aa(B,option(B),some(B),Z5) )
          & ( aa(B,A,F,Z5) = Y3 ) ) ) ).

% map_option_eq_Some
tff(fact_6007_Lcm__eq__0__I__nat,axiom,
    ! [A5: set(nat)] :
      ( pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5))
     => ( gcd_Lcm(nat,A5) = zero_zero(nat) ) ) ).

% Lcm_eq_0_I_nat
tff(fact_6008_abs__Lcm__eq,axiom,
    ! [K5: set(int)] : aa(int,int,abs_abs(int),gcd_Lcm(int,K5)) = gcd_Lcm(int,K5) ).

% abs_Lcm_eq
tff(fact_6009_Lcm__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,top_top(set(A))) = zero_zero(A) ) ) ).

% Lcm_UNIV
tff(fact_6010_Lcm__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Lcm(A,bot_bot(set(A))) = one_one(A) ) ) ).

% Lcm_empty
tff(fact_6011_Lcm__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( gcd_Lcm(A,A5) = one_one(A) )
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),one_one(A))) ) ) ) ).

% Lcm_1_iff
tff(fact_6012_Lcm__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] : gcd_Lcm(A,aa(set(A),set(A),insert2(A,A3),A5)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),gcd_Lcm(A,A5)) ) ).

% Lcm_insert
tff(fact_6013_case__map__option,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F: fun(C,B),X: option(C)] : aa(option(B),A,aa(fun(B,A),fun(option(B),A),aa(A,fun(fun(B,A),fun(option(B),A)),case_option(A,B),G),H),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F),X)) = aa(option(C),A,aa(fun(C,A),fun(option(C),A),aa(A,fun(fun(C,A),fun(option(C),A)),case_option(A,C),G),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),F)),X) ).

% case_map_option
tff(fact_6014_Lcm__fin__eq__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) = gcd_Lcm(A,A5) ) ) ) ).

% Lcm_fin_eq_Lcm
tff(fact_6015_Lcm__0__iff__nat,axiom,
    ! [A5: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),A5))
     => ( ( gcd_Lcm(nat,A5) = zero_zero(nat) )
      <=> pp(aa(set(nat),bool,member(nat,zero_zero(nat)),A5)) ) ) ).

% Lcm_0_iff_nat
tff(fact_6016_dom__map__option__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(C,B),M: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),G)),M)) = dom(A,C,M) ).

% dom_map_option_comp
tff(fact_6017_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,B),X2: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),aTP_Lamp_aik(A,option(C))),X2) = none(B) ).

% map_option_o_empty
tff(fact_6018_Lcm__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B2: A] : gcd_Lcm(A,aa(set(A),set(A),insert2(A,A3),aa(set(A),set(A),insert2(A,B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% Lcm_2
tff(fact_6019_Lcm__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Lcm(int,aa(set(int),set(int),image2(int,int,abs_abs(int)),K5)) = gcd_Lcm(int,K5) ).

% Lcm_abs_eq
tff(fact_6020_map__option__o__map__upd,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),M: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),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)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F)),M),A3,aa(B,option(B),some(B),aa(C,B,F,B2))) ).

% map_option_o_map_upd
tff(fact_6021_Lcm__int__eq,axiom,
    ! [N5: set(nat)] : gcd_Lcm(int,aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),N5)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N5)) ).

% Lcm_int_eq
tff(fact_6022_Lcm__nat__abs__eq,axiom,
    ! [K5: set(int)] : gcd_Lcm(nat,aa(set(int),set(nat),image2(int,nat,aTP_Lamp_nn(int,nat)),K5)) = aa(int,nat,nat2,gcd_Lcm(int,K5)) ).

% Lcm_nat_abs_eq
tff(fact_6023_Lcm__nat__insert,axiom,
    ! [N: nat,M2: set(nat)] : gcd_Lcm(nat,aa(set(nat),set(nat),insert2(nat,N),M2)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),N),gcd_Lcm(nat,M2)) ).

% Lcm_nat_insert
tff(fact_6024_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B)] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),none(A)) = none(B) ).

% option.simps(8)
tff(fact_6025_Lcm__least__int,axiom,
    ! [A5: set(int),A3: int] :
      ( ! [B3: int] :
          ( pp(aa(set(int),bool,member(int,B3),A5))
         => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),B3),A3)) )
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),gcd_Lcm(int,A5)),A3)) ) ).

% Lcm_least_int
tff(fact_6026_dvd__Lcm__int,axiom,
    ! [M: int,M2: set(int)] :
      ( pp(aa(set(int),bool,member(int,M),M2))
     => pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),M),gcd_Lcm(int,M2))) ) ).

% dvd_Lcm_int
tff(fact_6027_Lcm__dvd__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Lcm(A,A5)),X))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),X)) ) ) ) ).

% Lcm_dvd_iff
tff(fact_6028_Lcm__least,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),A3: A] :
          ( ! [B3: A] :
              ( pp(aa(set(A),bool,member(A,B3),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),A3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Lcm(A,A5)),A3)) ) ) ).

% Lcm_least
tff(fact_6029_Lcm__dvdD,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Lcm(A,A5)),X))
         => ( pp(aa(set(A),bool,member(A,Y3),A5))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y3),X)) ) ) ) ).

% Lcm_dvdD
tff(fact_6030_dvd__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,A3),A5))
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),gcd_Lcm(A,A5))) ) ) ).

% dvd_Lcm
tff(fact_6031_option_Omap__ident,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_ak(A,A)),T2) = T2 ).

% option.map_ident
tff(fact_6032_dvd__Lcm__nat,axiom,
    ! [M: nat,M2: set(nat)] :
      ( pp(aa(set(nat),bool,member(nat,M),M2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),M),gcd_Lcm(nat,M2))) ) ).

% dvd_Lcm_nat
tff(fact_6033_Lcm__dvd__nat,axiom,
    ! [M2: set(nat),N: nat] :
      ( ! [X4: nat] :
          ( pp(aa(set(nat),bool,member(nat,X4),M2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X4),N)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),gcd_Lcm(nat,M2)),N)) ) ).

% Lcm_dvd_nat
tff(fact_6034_Lcm__eq__0__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,member(A,zero_zero(A)),A5))
         => ( gcd_Lcm(A,A5) = zero_zero(A) ) ) ) ).

% Lcm_eq_0_I
tff(fact_6035_map__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y3: option(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Y3 )
     => ( ! [A4: A] :
            ( ( Y3 = aa(A,option(A),some(A),A4) )
           => ( aa(A,B,F,A4) = aa(A,B,G,A4) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Y3) ) ) ) ).

% map_option_cong
tff(fact_6036_option_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X22: A] : aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),aa(A,option(A),some(A),X22)) = aa(B,option(B),some(B),aa(A,B,F,X22)) ).

% option.simps(9)
tff(fact_6037_dom__map__option,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,fun(C,B)),M: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aTP_Lamp_als(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),F),M)) = dom(A,C,M) ).

% dom_map_option
tff(fact_6038_map__option_Oidentity,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),aTP_Lamp_ak(A,A)) = id(option(A)) ).

% map_option.identity
tff(fact_6039_option_Omap__id,axiom,
    ! [A: $tType,T2: option(A)] : aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)),T2) = T2 ).

% option.map_id
tff(fact_6040_option_Omap__id0,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),id(A)) = id(option(A)) ).

% option.map_id0
tff(fact_6041_map__option_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),aa(fun(option(B),option(C)),fun(fun(option(A),option(B)),fun(option(A),option(C))),comp(option(B),option(C),option(A)),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).

% map_option.comp
tff(fact_6042_option_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),V)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).

% option.map_comp
tff(fact_6043_map__option_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Option: option(A)] : aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),F),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Option)) = aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Option) ).

% map_option.compositionality
tff(fact_6044_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A3: option(A),F: fun(A,B)] :
      ( ( A3 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),A3)) = aa(A,B,F,aa(option(A),A,the2(A),A3)) ) ) ).

% option.map_sel
tff(fact_6045_Lcm__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_Lcm(int,K5))) ).

% Lcm_int_greater_eq_0
tff(fact_6046_Lcm__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( gcd_Lcm(A,A5) = zero_zero(A) )
          <=> pp(aa(set(A),bool,member(A,zero_zero(A)),A5)) ) ) ) ).

% Lcm_0_iff
tff(fact_6047_Lcm__0__iff_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( gcd_Lcm(A,A5) = zero_zero(A) )
        <=> ~ ? [L4: A] :
                ( ( L4 != zero_zero(A) )
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),L4)) ) ) ) ) ).

% Lcm_0_iff'
tff(fact_6048_Lcm__no__multiple,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ! [M4: A] :
              ( ( M4 != zero_zero(A) )
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,member(A,X2),A5))
                  & ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X2),M4)) ) )
         => ( gcd_Lcm(A,A5) = zero_zero(A) ) ) ) ).

% Lcm_no_multiple
tff(fact_6049_Lcm__nat__infinite,axiom,
    ! [M2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite2(nat),M2))
     => ( gcd_Lcm(nat,M2) = zero_zero(nat) ) ) ).

% Lcm_nat_infinite
tff(fact_6050_Lcm__subset,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(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),dvd_dvd(A),gcd_Lcm(A,A5)),gcd_Lcm(A,B4))) ) ) ).

% Lcm_subset
tff(fact_6051_Lcm__nat__empty,axiom,
    gcd_Lcm(nat,bot_bot(set(nat))) = one_one(nat) ).

% Lcm_nat_empty
tff(fact_6052_Lcm__Un,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),B4: set(A)] : gcd_Lcm(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_lcm(A),gcd_Lcm(A,A5)),gcd_Lcm(A,B4)) ) ).

% Lcm_Un
tff(fact_6053_ran__map__option,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),M: fun(B,option(C))] : ran(B,A,aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_alt(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),F),M)) = aa(set(C),set(A),image2(C,A,F),ran(B,C,M)) ).

% ran_map_option
tff(fact_6054_Lcm__in__lcm__closed__set__nat,axiom,
    ! [M2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite2(nat),M2))
     => ( ( M2 != bot_bot(set(nat)) )
       => ( ! [M4: nat,N2: nat] :
              ( pp(aa(set(nat),bool,member(nat,M4),M2))
             => ( pp(aa(set(nat),bool,member(nat,N2),M2))
               => pp(aa(set(nat),bool,member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_lcm(nat),M4),N2)),M2)) ) )
         => pp(aa(set(nat),bool,member(nat,gcd_Lcm(nat,M2)),M2)) ) ) ) ).

% Lcm_in_lcm_closed_set_nat
tff(fact_6055_Lcm__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(B),F: fun(B,A),G: fun(B,A)] :
          ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(B,A,F,X4)),aa(B,A,G,X4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),gcd_Lcm(A,aa(set(B),set(A),image2(B,A,F),A5))),gcd_Lcm(A,aa(set(B),set(A),image2(B,A,G),A5)))) ) ) ).

% Lcm_mono
tff(fact_6056_option_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),aa(fun(option(B),nat),fun(fun(option(A),option(B)),fun(option(A),nat)),comp(option(B),nat,option(A)),size_option(B,F)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G)) = size_option(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).

% option.size_gen_o_map
tff(fact_6057_Gcd__nat__def,axiom,
    ! [M2: set(nat)] : gcd_Gcd(nat,M2) = gcd_Lcm(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_alu(set(nat),fun(nat,bool),M2))) ).

% Gcd_nat_def
tff(fact_6058_Lcm__int__def,axiom,
    ! [K5: set(int)] : gcd_Lcm(int,K5) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))),K5))) ).

% Lcm_int_def
tff(fact_6059_map__option__case,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Y3: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),Y3) = aa(option(B),option(A),aa(fun(B,option(A)),fun(option(B),option(A)),aa(option(A),fun(fun(B,option(A)),fun(option(B),option(A))),case_option(option(A),B),none(A)),aTP_Lamp_alv(fun(B,A),fun(B,option(A)),F)),Y3) ).

% map_option_case
tff(fact_6060_option_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),top_top(set(option(A)))) ) ).

% option.inj_map
tff(fact_6061_Lcm__no__units,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : gcd_Lcm(A,A5) = gcd_Lcm(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alw(A,bool)))) ) ).

% Lcm_no_units
tff(fact_6062_Gcd__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : gcd_Gcd(A,A5) = gcd_Lcm(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_alx(set(A),fun(A,bool),A5))) ) ).

% Gcd_Lcm
tff(fact_6063_Lcm__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : gcd_Lcm(A,A5) = gcd_Gcd(A,aa(fun(A,bool),set(A),collect(A),aTP_Lamp_aly(set(A),fun(A,bool),A5))) ) ).

% Lcm_Gcd
tff(fact_6064_Lcm__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Lcm(A,aa(list(A),set(A),set2(A),Xs)) = aa(A,A,fold(A,A,gcd_lcm(A),Xs),one_one(A)) ) ).

% Lcm_set_eq_fold
tff(fact_6065_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Lcm(nat,aa(list(nat),set(nat),set2(nat),Xs)) = aa(nat,nat,fold(nat,nat,gcd_lcm(nat),Xs),one_one(nat)) ).

% Lcm_nat_set_eq_fold
tff(fact_6066_Lcm__nat__def,axiom,
    ! [M2: set(nat)] :
      ( ( pp(aa(set(nat),bool,finite_finite2(nat),M2))
       => ( gcd_Lcm(nat,M2) = lattic5214292709420241887eutr_F(nat,gcd_lcm(nat),one_one(nat),M2) ) )
      & ( ~ pp(aa(set(nat),bool,finite_finite2(nat),M2))
       => ( gcd_Lcm(nat,M2) = zero_zero(nat) ) ) ) ).

% Lcm_nat_def
tff(fact_6067_aboveS__def,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_aboveS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_alz(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).

% aboveS_def
tff(fact_6068_option_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,C),F: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),aa(fun(option(B),C),fun(fun(option(A),option(B)),fun(option(A),C)),comp(option(B),C,option(A)),aa(fun(B,C),fun(option(B),C),aa(C,fun(fun(B,C),fun(option(B),C)),rec_option(C,B),G),Ga)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F)) = aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),G),aa(fun(A,B),fun(A,C),aTP_Lamp_an(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F)) ).

% option.rec_o_map
tff(fact_6069_prod__list__def,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A) = groups_monoid_F(A,times_times(A),one_one(A)) ) ) ).

% prod_list_def
tff(fact_6070_option_Osimps_I6_J,axiom,
    ! [A: $tType,C: $tType,F1: C,F22: fun(A,C)] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),none(A)) = F1 ).

% option.simps(6)
tff(fact_6071_option_Osimps_I7_J,axiom,
    ! [C: $tType,A: $tType,F1: C,F22: fun(A,C),X22: A] : aa(option(A),C,aa(fun(A,C),fun(option(A),C),aa(C,fun(fun(A,C),fun(option(A),C)),rec_option(C,A),F1),F22),aa(A,option(A),some(A),X22)) = aa(A,C,F22,X22) ).

% option.simps(7)
tff(fact_6072_sum__list__def,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A) = groups_monoid_F(A,plus_plus(A),zero_zero(A)) ) ) ).

% sum_list_def
tff(fact_6073_scomp__unfold,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,X2: fun(A,product_prod(B,C)),Xa3: fun(B,fun(C,D)),Xb: A] : aa(A,D,product_scomp(A,B,C,D,X2,Xa3),Xb) = 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),X2,Xb))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X2,Xb))) ).

% scomp_unfold
tff(fact_6074_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q2))
     => ( algebr8660921524188924756oprime(int,P2,Q2)
       => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) ) ) ) ).

% normalize_stable
tff(fact_6075_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_6076_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_6077_coprime__minus__right__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,uminus_uminus(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_right_iff
tff(fact_6078_coprime__minus__left__iff,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,uminus_uminus(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_minus_left_iff
tff(fact_6079_scomp__apply,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(B,product_prod(C,D)),G: fun(C,fun(D,A)),X: B] : aa(B,A,product_scomp(B,C,D,A,F,G),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),G),aa(B,product_prod(C,D),F,X)) ).

% scomp_apply
tff(fact_6080_coprime__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_self
tff(fact_6081_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A3: A] :
          ( ( B2 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A3,B2),B2)
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_left_iff
tff(fact_6082_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A3,modulo_modulo(A,B2,A3))
          <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ) ).

% coprime_mod_right_iff
tff(fact_6083_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,N: nat,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),B2)
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_6084_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,N: nat] :
          ( algebr8660921524188924756oprime(A,A3,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N))
        <=> ( algebr8660921524188924756oprime(A,A3,B2)
            | ( N = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_6085_coprime__imp__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_imp_gcd_eq_1
tff(fact_6086_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_left_iff
tff(fact_6087_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% coprime_0_right_iff
tff(fact_6088_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_6089_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_6090_is__unit__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2)),one_one(A)))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_gcd
tff(fact_6091_prod__list__coprime__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A3: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => algebr8660921524188924756oprime(A,A3,X4) )
         => algebr8660921524188924756oprime(A,A3,aa(list(A),A,groups5270119922927024881d_list(A),Xs)) ) ) ).

% prod_list_coprime_right
tff(fact_6092_prod__list__coprime__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A),A3: A] :
          ( ! [X4: A] :
              ( pp(aa(set(A),bool,member(A,X4),aa(list(A),set(A),set2(A),Xs)))
             => algebr8660921524188924756oprime(A,X4,A3) )
         => algebr8660921524188924756oprime(A,aa(list(A),A,groups5270119922927024881d_list(A),Xs),A3) ) ) ).

% prod_list_coprime_left
tff(fact_6093_normalize__coprime,axiom,
    ! [R: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => algebr8660921524188924756oprime(int,P2,Q2) ) ).

% normalize_coprime
tff(fact_6094_scomp__def,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(B,fun(C,D)),X2: A] : aa(A,D,product_scomp(A,B,C,D,F,G),X2) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G),aa(A,product_prod(B,C),F,X2)) ).

% scomp_def
tff(fact_6095_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_6096_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_6097_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_6098_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_6099_gcd__eq__1__imp__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% gcd_eq_1_imp_coprime
tff(fact_6100_coprime__iff__gcd__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = one_one(A) ) ) ) ).

% coprime_iff_gcd_eq_1
tff(fact_6101_is__unit__right__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_right_imp_coprime
tff(fact_6102_is__unit__left__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% is_unit_left_imp_coprime
tff(fact_6103_coprime__common__divisor,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A))) ) ) ) ) ).

% coprime_common_divisor
tff(fact_6104_coprime__absorb__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Y3: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y3),X))
         => ( algebr8660921524188924756oprime(A,X,Y3)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Y3),one_one(A))) ) ) ) ).

% coprime_absorb_right
tff(fact_6105_coprime__imp__coprime,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,D3: A,A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,C2,D3)
         => ( ! [E3: A] :
                ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),one_one(A)))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),B2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),C2)) ) ) )
           => ( ! [E3: A] :
                  ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),one_one(A)))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),B2))
                     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E3),D3)) ) ) )
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_imp_coprime
tff(fact_6106_coprime__absorb__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,Y3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),Y3))
         => ( algebr8660921524188924756oprime(A,X,Y3)
          <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X),one_one(A))) ) ) ) ).

% coprime_absorb_left
tff(fact_6107_not__coprimeI,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),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),one_one(A)))
             => ~ algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% not_coprimeI
tff(fact_6108_not__coprimeE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ~ algebr8660921524188924756oprime(A,A3,B2)
         => ~ ! [C3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) ) ) ) ).

% not_coprimeE
tff(fact_6109_coprime__def,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
        <=> ! [C4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C4),one_one(A))) ) ) ) ) ).

% coprime_def
tff(fact_6110_coprimeI,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B2: A] :
          ( ! [C3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),one_one(A))) ) )
         => algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprimeI
tff(fact_6111_coprime__add__one__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% coprime_add_one_right
tff(fact_6112_coprime__add__one__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),A3) ) ).

% coprime_add_one_left
tff(fact_6113_coprime__diff__one__left,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A)),A3) ) ).

% coprime_diff_one_left
tff(fact_6114_coprime__doff__one__right,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),one_one(A))) ) ).

% coprime_doff_one_right
tff(fact_6115_coprime__1__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,A3,one_one(A)) ) ).

% coprime_1_right
tff(fact_6116_coprime__1__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] : algebr8660921524188924756oprime(A,one_one(A),A3) ) ).

% coprime_1_left
tff(fact_6117_coprime__divisors,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B2: A,D3: 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),D3))
           => ( algebr8660921524188924756oprime(A,C2,D3)
             => algebr8660921524188924756oprime(A,A3,B2) ) ) ) ) ).

% coprime_divisors
tff(fact_6118_coprime__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A3: A] :
          ( algebr8660921524188924756oprime(A,B2,A3)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_commute
tff(fact_6119_scomp__scomp,axiom,
    ! [A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,product_prod(E,F2)),G: fun(E,fun(F2,product_prod(C,D))),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,product_scomp(A,E,F2,product_prod(C,D),F,G),H) = product_scomp(A,E,F2,B,F,aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_ama(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),G),H)) ).

% scomp_scomp
tff(fact_6120_prod__coprime__left,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(B),F: fun(B,A),A3: A] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => algebr8660921524188924756oprime(A,aa(B,A,F,I2),A3) )
         => algebr8660921524188924756oprime(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5),A3) ) ) ).

% prod_coprime_left
tff(fact_6121_prod__coprime__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(B),A3: A,F: fun(B,A)] :
          ( ! [I2: B] :
              ( pp(aa(set(B),bool,member(B,I2),A5))
             => algebr8660921524188924756oprime(A,A3,aa(B,A,F,I2)) )
         => algebr8660921524188924756oprime(A,A3,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A5)) ) ) ).

% prod_coprime_right
tff(fact_6122_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_6123_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_6124_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_6125_mult__mod__cancel__left,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [N: A,A3: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),A3),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),N),B2),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_left
tff(fact_6126_mult__mod__cancel__right,axiom,
    ! [A: $tType] :
      ( ( euclid8851590272496341667cancel(A)
        & semiring_gcd(A) )
     => ! [A3: A,N: A,M: A,B2: A] :
          ( ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),N),M) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),N),M) )
         => ( algebr8660921524188924756oprime(A,M,N)
           => ( modulo_modulo(A,A3,M) = modulo_modulo(A,B2,M) ) ) ) ) ).

% mult_mod_cancel_right
tff(fact_6127_coprime__crossproduct__int,axiom,
    ! [A3: int,D3: int,B2: int,C2: int] :
      ( algebr8660921524188924756oprime(int,A3,D3)
     => ( algebr8660921524188924756oprime(int,B2,C2)
       => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),A3)),aa(int,int,abs_abs(int),C2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),B2)),aa(int,int,abs_abs(int),D3)) )
        <=> ( ( aa(int,int,abs_abs(int),A3) = aa(int,int,abs_abs(int),B2) )
            & ( aa(int,int,abs_abs(int),C2) = aa(int,int,abs_abs(int),D3) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_6128_Pair__scomp,axiom,
    ! [A: $tType,B: $tType,C: $tType,X: C,F: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X),F) = aa(C,fun(A,B),F,X) ).

% Pair_scomp
tff(fact_6129_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_6130_quotient__of__coprime,axiom,
    ! [R: rat,P2: int,Q2: int] :
      ( ( quotient_of(R) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => algebr8660921524188924756oprime(int,P2,Q2) ) ).

% quotient_of_coprime
tff(fact_6131_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_6132_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) != zero_zero(A) )
         => ? [A14: A,B6: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A14),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),B6),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
              & algebr8660921524188924756oprime(A,A14,B6) ) ) ) ).

% gcd_coprime_exists
tff(fact_6133_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A,A6: A,B5: 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),A6),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),B5),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) )
             => algebr8660921524188924756oprime(A,A6,B5) ) ) ) ) ).

% gcd_coprime
tff(fact_6134_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( A3 != zero_zero(A) )
            | ( B2 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ) ).

% div_gcd_coprime
tff(fact_6135_quotient__of__def,axiom,
    ! [X: rat] : quotient_of(X) = the(product_prod(int,int),aTP_Lamp_amb(rat,fun(product_prod(int,int),bool),X)) ).

% quotient_of_def
tff(fact_6136_iterate_Osimps,axiom,
    ! [A: $tType,B: $tType,K: code_natural,F: 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,F),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,F),X) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F,X),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F)) ) ) ) ).

% iterate.simps
tff(fact_6137_coprime__common__divisor__nat,axiom,
    ! [A3: nat,B2: nat,X: nat] :
      ( algebr8660921524188924756oprime(nat,A3,B2)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),A3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),X),B2))
         => ( X = one_one(nat) ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_6138_coprime__Suc__left__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,N),N) ).

% coprime_Suc_left_nat
tff(fact_6139_coprime__Suc__right__nat,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,N)) ).

% coprime_Suc_right_nat
tff(fact_6140_coprime__crossproduct__nat,axiom,
    ! [A3: nat,D3: nat,B2: nat,C2: nat] :
      ( algebr8660921524188924756oprime(nat,A3,D3)
     => ( algebr8660921524188924756oprime(nat,B2,C2)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),C2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),D3) )
        <=> ( ( A3 = B2 )
            & ( C2 = D3 ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_6141_coprime__Suc__0__left,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),N) ).

% coprime_Suc_0_left
tff(fact_6142_coprime__Suc__0__right,axiom,
    ! [N: nat] : algebr8660921524188924756oprime(nat,N,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_6143_quotient__of__eq,axiom,
    ! [A3: int,B2: int,P2: int,Q2: 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),P2),Q2) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q2) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% quotient_of_eq
tff(fact_6144_normalize__eq,axiom,
    ! [A3: int,B2: int,P2: int,Q2: int] :
      ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),B2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => ( aa(int,rat,aa(int,fun(int,rat),fract,P2),Q2) = aa(int,rat,aa(int,fun(int,rat),fract,A3),B2) ) ) ).

% normalize_eq
tff(fact_6145_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_6146_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,N,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_6147_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => algebr8660921524188924756oprime(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_6148_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] : aa(int,rat,aa(int,fun(int,rat),fract,Xa),X) = aa(product_prod(int,int),rat,abs_Rat,aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),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),Xa),X))) ).

% Fract.abs_eq
tff(fact_6149_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_tf(int,fun(int,product_prod(int,int)))),fract)) ).

% Fract.transfer
tff(fact_6150_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_tf(int,fun(int,product_prod(int,int)))) ).

% Fract_def
tff(fact_6151_mult__inj__if__coprime__nat,axiom,
    ! [A: $tType,B: $tType,F: fun(A,nat),A5: set(A),G: fun(B,nat),B4: set(B)] :
      ( inj_on(A,nat,F,A5)
     => ( inj_on(B,nat,G,B4)
       => ( ! [A4: A,B3: B] :
              ( pp(aa(set(A),bool,member(A,A4),A5))
             => ( pp(aa(set(B),bool,member(B,B3),B4))
               => algebr8660921524188924756oprime(nat,aa(A,nat,F,A4),aa(B,nat,G,B3)) ) )
         => 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_amc(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F),G)),product_Sigma(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4))) ) ) ) ).

% mult_inj_if_coprime_nat
tff(fact_6152_iterate_Oelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb2: B,Y3: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb2) = Y3 )
     => ( ( ( X = zero_zero(code_natural) )
         => ( Y3 = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb2) ) )
        & ( ( X != zero_zero(code_natural) )
         => ( Y3 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa)) ) ) ) ) ).

% iterate.elims
tff(fact_6153_iterate_Opelims,axiom,
    ! [A: $tType,B: $tType,X: code_natural,Xa: fun(B,fun(A,product_prod(B,A))),Xb2: B,Y3: fun(A,product_prod(B,A))] :
      ( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X,Xa),Xb2) = Y3 )
     => ( 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),Xa),Xb2))))
       => ~ ( ( ( ( X = zero_zero(code_natural) )
               => ( Y3 = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb2) ) )
              & ( ( X != zero_zero(code_natural) )
               => ( Y3 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa,Xb2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X),one_one(code_natural)),Xa)) ) ) )
           => ~ 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),Xa),Xb2)))) ) ) ) ).

% iterate.pelims
tff(fact_6154_Lcm__coprime_H,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),A5) != zero_zero(nat) )
         => ( ! [A4: A,B3: A] :
                ( pp(aa(set(A),bool,member(A,A4),A5))
               => ( pp(aa(set(A),bool,member(A,B3),A5))
                 => ( ( A4 != B3 )
                   => algebr8660921524188924756oprime(A,A4,B3) ) ) )
           => ( gcd_Lcm(A,A5) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_amd(A,A)),A5)) ) ) ) ) ).

% Lcm_coprime'
tff(fact_6155_normalize__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,normal6383669964737779283malize(A),A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% normalize_idem
tff(fact_6156_of__int__eq__id,axiom,
    ring_1_of_int(int) = id(int) ).

% of_int_eq_id
tff(fact_6157_lcm_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% lcm.normalize_bottom
tff(fact_6158_normalize__0,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),zero_zero(A)) = zero_zero(A) ) ) ).

% normalize_0
tff(fact_6159_normalize__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% normalize_eq_0_iff
tff(fact_6160_normalize__mult__normalize__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,normal6383669964737779283malize(A),B2))) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_right
tff(fact_6161_normalize__mult__normalize__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% normalize_mult_normalize_left
tff(fact_6162_gcd_Onormalize__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% gcd.normalize_bottom
tff(fact_6163_normalize__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( aa(A,A,normal6383669964737779283malize(A),one_one(A)) = one_one(A) ) ) ).

% normalize_1
tff(fact_6164_dvd__normalize__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% dvd_normalize_iff
tff(fact_6165_normalize__dvd__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% normalize_dvd_iff
tff(fact_6166_gcd_Oidem__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.idem_normalize
tff(fact_6167_gcd_Onormalize__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd.normalize_idem
tff(fact_6168_gcd_Onormalize__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd.normalize_left_idem
tff(fact_6169_gcd_Onormalize__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ).

% gcd.normalize_right_idem
tff(fact_6170_lcm_Oidem__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.idem_normalize
tff(fact_6171_lcm_Onormalize__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm.normalize_idem
tff(fact_6172_lcm_Onormalize__left__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,normal6383669964737779283malize(A),A3)),B2) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm.normalize_left_idem
tff(fact_6173_lcm_Onormalize__right__idem,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),aa(A,A,normal6383669964737779283malize(A),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ).

% lcm.normalize_right_idem
tff(fact_6174_coprime__normalize__right__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,aa(A,A,normal6383669964737779283malize(A),B2))
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_right_iff
tff(fact_6175_coprime__normalize__left__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,aa(A,A,normal6383669964737779283malize(A),A3),B2)
        <=> algebr8660921524188924756oprime(A,A3,B2) ) ) ).

% coprime_normalize_left_iff
tff(fact_6176_normalize__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : aa(A,A,normal6383669964737779283malize(A),gcd_Lcm(A,A5)) = gcd_Lcm(A,A5) ) ).

% normalize_Lcm
tff(fact_6177_normalize__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : aa(A,A,normal6383669964737779283malize(A),gcd_Gcd(A,A5)) = gcd_Gcd(A,A5) ) ).

% normalize_Gcd
tff(fact_6178_Lcm__fin_Onormalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] : aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) = aa(set(A),A,semiring_gcd_Lcm_fin(A),A5) ) ).

% Lcm_fin.normalize
tff(fact_6179_Gcd__fin_Onormalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] : aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A5) ) ).

% Gcd_fin.normalize
tff(fact_6180_gcd_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),zero_zero(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_right_normalize
tff(fact_6181_gcd_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),zero_zero(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% gcd.top_left_normalize
tff(fact_6182_lcm_Otop__right__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),one_one(A)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_right_normalize
tff(fact_6183_lcm_Otop__left__normalize,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),one_one(A)),A3) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% lcm.top_left_normalize
tff(fact_6184_Gcd__image__normalize,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,normal6383669964737779283malize(A)),A5)) = gcd_Gcd(A,A5) ) ).

% Gcd_image_normalize
tff(fact_6185_normalize__mult__unit__right,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).

% normalize_mult_unit_right
tff(fact_6186_normalize__mult__unit__left,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% normalize_mult_unit_left
tff(fact_6187_gcd__mult__lcm,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% gcd_mult_lcm
tff(fact_6188_lcm__mult__gcd,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% lcm_mult_gcd
tff(fact_6189_Lcm__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A] : gcd_Lcm(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% Lcm_singleton
tff(fact_6190_Gcd__singleton,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A] : gcd_Gcd(A,aa(set(A),set(A),insert2(A,A3),bot_bot(set(A)))) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% Gcd_singleton
tff(fact_6191_coprime__crossproduct,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,D3: A,B2: A,C2: A] :
          ( algebr8660921524188924756oprime(A,A3,D3)
         => ( algebr8660921524188924756oprime(A,B2,C2)
           => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),B2)),aa(A,A,normal6383669964737779283malize(A),D3)) )
            <=> ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
                & ( aa(A,A,normal6383669964737779283malize(A),C2) = aa(A,A,normal6383669964737779283malize(A),D3) ) ) ) ) ) ) ).

% coprime_crossproduct
tff(fact_6192_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,F3: fun(B,fun(A,product_prod(B,A))),X4: 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),F3),X4)) ).

% iterate.cases
tff(fact_6193_lcm__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% lcm_mult_distrib'
tff(fact_6194_lcm__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),B2),A3)),C2)) ) ).

% lcm_mult_right
tff(fact_6195_lcm__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ).

% lcm_mult_left
tff(fact_6196_lcm__proj2__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% lcm_proj2_if_dvd
tff(fact_6197_lcm__proj1__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).

% lcm_proj1_if_dvd
tff(fact_6198_lcm__proj2__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),M),N) = aa(A,A,normal6383669964737779283malize(A),N) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),N)) ) ) ).

% lcm_proj2_iff
tff(fact_6199_lcm__proj1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),M),N) = aa(A,A,normal6383669964737779283malize(A),M) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),N),M)) ) ) ).

% lcm_proj1_iff
tff(fact_6200_lcm__unique,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,D3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),D3))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D3))
            & ( aa(A,A,normal6383669964737779283malize(A),D3) = D3 )
            & ! [E4: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),E4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),E4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),E4)) ) )
        <=> ( D3 = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ) ) ).

% lcm_unique
tff(fact_6201_lcmI,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))
           => ( ! [D2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),D2))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),D2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),D2)) ) )
             => ( ( aa(A,A,normal6383669964737779283malize(A),C2) = C2 )
               => ( C2 = aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) ) ) ) ) ) ) ).

% lcmI
tff(fact_6202_gcd__proj2__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ).

% gcd_proj2_if_dvd
tff(fact_6203_gcd__proj1__if__dvd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(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),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),A3) ) ) ) ).

% gcd_proj1_if_dvd
tff(fact_6204_gcd__proj2__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) = aa(A,A,normal6383669964737779283malize(A),N) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),N),M)) ) ) ).

% gcd_proj2_iff
tff(fact_6205_gcd__proj1__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) = aa(A,A,normal6383669964737779283malize(A),M) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),M),N)) ) ) ).

% gcd_proj1_iff
tff(fact_6206_gcd__unique,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [D3: A,A3: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),A3))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D3),B2))
            & ( aa(A,A,normal6383669964737779283malize(A),D3) = D3 )
            & ! [E4: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E4),B2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),E4),D3)) ) )
        <=> ( D3 = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ).

% gcd_unique
tff(fact_6207_gcdI,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C2),B2))
           => ( ! [D2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),B2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),D2),C2)) ) )
             => ( ( aa(A,A,normal6383669964737779283malize(A),C2) = C2 )
               => ( C2 = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) ) ) ) ) ) ) ).

% gcdI
tff(fact_6208_gcd__mult__distrib_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% gcd_mult_distrib'
tff(fact_6209_gcd__mult__right,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),B2),A3)),C2)) ) ).

% gcd_mult_right
tff(fact_6210_gcd__mult__left,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).

% gcd_mult_left
tff(fact_6211_associatedI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) ) ) ) ) ).

% associatedI
tff(fact_6212_associatedD1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2)) ) ) ).

% associatedD1
tff(fact_6213_associatedD2,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ).

% associatedD2
tff(fact_6214_associated__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
           => ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
             => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
               => ( A3 = B2 ) ) ) ) ) ) ).

% associated_eqI
tff(fact_6215_associated__iff__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3)) ) ) ) ).

% associated_iff_dvd
tff(fact_6216_normalize__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ).

% normalize_mult
tff(fact_6217_dvd__normalize__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),aa(A,A,normal6383669964737779283malize(A),B2)) ) ) ) ).

% dvd_normalize_div
tff(fact_6218_normalize__idem__imp__is__unit__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
          <=> ( A3 = one_one(A) ) ) ) ) ).

% normalize_idem_imp_is_unit_iff
tff(fact_6219_is__unit__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) ) ) ) ).

% is_unit_normalize
tff(fact_6220_normalize__1__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A))) ) ) ).

% normalize_1_iff
tff(fact_6221_associated__unit,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),one_one(A))) ) ) ) ).

% associated_unit
tff(fact_6222_gcd__exp__weak,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,N: nat,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),N)) = aa(A,A,normal6383669964737779283malize(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),N)) ) ).

% gcd_exp_weak
tff(fact_6223_Gcd__eqI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),B3)) )
           => ( ! [C3: A] :
                  ( ! [B11: A] :
                      ( pp(aa(set(A),bool,member(A,B11),A5))
                     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B11)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A3)) )
             => ( gcd_Gcd(A,A5) = A3 ) ) ) ) ) ).

% Gcd_eqI
tff(fact_6224_GcdI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),B2: A] :
          ( ! [A4: A] :
              ( pp(aa(set(A),bool,member(A,A4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A4)) )
         => ( ! [C3: A] :
                ( ! [A17: A] :
                    ( pp(aa(set(A),bool,member(A,A17),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),A17)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),C3),B2)) )
           => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
             => ( B2 = gcd_Gcd(A,A5) ) ) ) ) ) ).

% GcdI
tff(fact_6225_LcmI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),B2: A] :
          ( ! [A4: A] :
              ( pp(aa(set(A),bool,member(A,A4),A5))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A4),B2)) )
         => ( ! [C3: A] :
                ( ! [A17: A] :
                    ( pp(aa(set(A),bool,member(A,A17),A5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A17),C3)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),C3)) )
           => ( ( aa(A,A,normal6383669964737779283malize(A),B2) = B2 )
             => ( B2 = gcd_Lcm(A,A5) ) ) ) ) ) ).

% LcmI
tff(fact_6226_Lcm__eqI,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A5: set(A)] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( ! [B3: A] :
                ( pp(aa(set(A),bool,member(A,B3),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B3),A3)) )
           => ( ! [C3: A] :
                  ( ! [B11: A] :
                      ( pp(aa(set(A),bool,member(A,B11),A5))
                     => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B11),C3)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),C3)) )
             => ( gcd_Lcm(A,A5) = A3 ) ) ) ) ) ).

% Lcm_eqI
tff(fact_6227_lcm__gcd__prod,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ).

% lcm_gcd_prod
tff(fact_6228_Gcd__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [C2: A,A5: set(A)] : gcd_Gcd(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Gcd(A,A5))) ) ).

% Gcd_mult
tff(fact_6229_lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( algebr8660921524188924756oprime(A,A3,B2)
         => ( aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) ) ) ) ).

% lcm_coprime
tff(fact_6230_Lcm__mult,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A),C2: A] :
          ( ( A5 != bot_bot(set(A)) )
         => ( gcd_Lcm(A,aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C2)),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),gcd_Lcm(A,A5))) ) ) ) ).

% Lcm_mult
tff(fact_6231_lcm__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2))) ) ).

% lcm_gcd
tff(fact_6232_Lcm__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B2: A] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(A),A,semiring_gcd_Lcm_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5))) ) ) ) ).

% Lcm_fin_mult
tff(fact_6233_Gcd__fin__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B2: A] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),B2)),A5)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5))) ) ) ) ).

% Gcd_fin_mult
tff(fact_6234_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% Gcd_fin.bounded_quasi_semilattice_set_axioms
tff(fact_6235_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde6485984586167503788ce_set(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% Lcm_fin.bounded_quasi_semilattice_set_axioms
tff(fact_6236_gcd__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2))) ) ) ) ) ).

% gcd_lcm
tff(fact_6237_Lcm__coprime,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( pp(aa(set(A),bool,finite_finite2(A),A5))
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A,B3: A] :
                  ( pp(aa(set(A),bool,member(A,A4),A5))
                 => ( pp(aa(set(A),bool,member(A,B3),A5))
                   => ( ( A4 != B3 )
                     => algebr8660921524188924756oprime(A,A4,B3) ) ) )
             => ( gcd_Lcm(A,A5) = aa(A,A,normal6383669964737779283malize(A),aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_amd(A,A)),A5)) ) ) ) ) ) ).

% Lcm_coprime
tff(fact_6238_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_6239_lcm_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_lcm(A),one_one(A),zero_zero(A),normal6383669964737779283malize(A)) ) ).

% lcm.bounded_quasi_semilattice_axioms
tff(fact_6240_bounded__quasi__semilattice__set__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
    <=> bounde8507323023520639062attice(A,F,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set_def
tff(fact_6241_bounded__quasi__semilattice__set_Oaxioms,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize)
     => bounde8507323023520639062attice(A,F,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set.axioms
tff(fact_6242_bounded__quasi__semilattice__set_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => bounde6485984586167503788ce_set(A,F,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice_set.intro
tff(fact_6243_bounded__quasi__semilattice_Oleft__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,B2: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% bounded_quasi_semilattice.left_idem
tff(fact_6244_bounded__quasi__semilattice_Oright__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,B2: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% bounded_quasi_semilattice.right_idem
tff(fact_6245_bounded__quasi__semilattice_Onormalize__top,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,Top) = Top ) ) ).

% bounded_quasi_semilattice.normalize_top
tff(fact_6246_bounded__quasi__semilattice_Oidem__normalize,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),A3) = aa(A,A,Normalize,A3) ) ) ).

% bounded_quasi_semilattice.idem_normalize
tff(fact_6247_bounded__quasi__semilattice_Onormalize__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,B2: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,aa(A,A,aa(A,fun(A,A),F,A3),B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% bounded_quasi_semilattice.normalize_idem
tff(fact_6248_bounded__quasi__semilattice_Onormalize__bottom,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,Normalize,Bot) = Bot ) ) ).

% bounded_quasi_semilattice.normalize_bottom
tff(fact_6249_bounded__quasi__semilattice_Obottom__left__bottom,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,Bot),A3) = Bot ) ) ).

% bounded_quasi_semilattice.bottom_left_bottom
tff(fact_6250_bounded__quasi__semilattice_Otop__left__normalize,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,Top),A3) = aa(A,A,Normalize,A3) ) ) ).

% bounded_quasi_semilattice.top_left_normalize
tff(fact_6251_bounded__quasi__semilattice_Obottom__right__bottom,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Bot) = Bot ) ) ).

% bounded_quasi_semilattice.bottom_right_bottom
tff(fact_6252_bounded__quasi__semilattice_Onormalize__left__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,B2: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,Normalize,A3)),B2) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% bounded_quasi_semilattice.normalize_left_idem
tff(fact_6253_bounded__quasi__semilattice_Otop__right__normalize,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Top) = aa(A,A,Normalize,A3) ) ) ).

% bounded_quasi_semilattice.top_right_normalize
tff(fact_6254_bounded__quasi__semilattice_Onormalize__right__idem,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,B2: A] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,Normalize,B2)) = aa(A,A,aa(A,fun(A,A),F,A3),B2) ) ) ).

% bounded_quasi_semilattice.normalize_right_idem
tff(fact_6255_select__weight__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(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_ame(code_natural,fun(A,bool))),Xs)) = select_weight(A,Xs) ).

% select_weight_drop_zero
tff(fact_6256_gcd_Obounded__quasi__semilattice__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => bounde8507323023520639062attice(A,gcd_gcd(A),zero_zero(A),one_one(A),normal6383669964737779283malize(A)) ) ).

% gcd.bounded_quasi_semilattice_axioms
tff(fact_6257_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_6258_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_zero(code_natural) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(code_natural,nat,code_nat_of_natural,N)),aa(nat,nat,suc,zero_zero(nat)))),aa(code_natural,nat,code_nat_of_natural,N))) ) ).

% natural_decr
tff(fact_6259_plus__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa: code_natural] : aa(code_natural,nat,code_nat_of_natural,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),X),Xa)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_natural,nat,code_nat_of_natural,X)),aa(code_natural,nat,code_nat_of_natural,Xa)) ).

% plus_natural.rep_eq
tff(fact_6260_zero__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,zero_zero(code_natural)) = zero_zero(nat) ).

% zero_natural.rep_eq
tff(fact_6261_one__natural_Orep__eq,axiom,
    aa(code_natural,nat,code_nat_of_natural,one_one(code_natural)) = one_one(nat) ).

% one_natural.rep_eq
tff(fact_6262_pick__drop__zero,axiom,
    ! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(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_ame(code_natural,fun(A,bool))),Xs)) = pick(A,Xs) ).

% pick_drop_zero
tff(fact_6263_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)),aa(nat,code_natural,code_natural_of_nat,L)) = aa(nat,A,nth(A,Xs),L) ) ) ).

% pick_same
tff(fact_6264_plus__natural__def,axiom,
    plus_plus(code_natural) = aa(fun(nat,fun(nat,nat)),fun(code_natural,fun(code_natural,code_natural)),map_fun(code_natural,nat,fun(nat,nat),fun(code_natural,code_natural),code_nat_of_natural,map_fun(code_natural,nat,nat,code_natural,code_nat_of_natural,code_natural_of_nat)),plus_plus(nat)) ).

% plus_natural_def
tff(fact_6265_one__natural__def,axiom,
    one_one(code_natural) = aa(nat,code_natural,code_natural_of_nat,one_one(nat)) ).

% one_natural_def
tff(fact_6266_zero__natural__def,axiom,
    zero_zero(code_natural) = aa(nat,code_natural,code_natural_of_nat,zero_zero(nat)) ).

% zero_natural_def
tff(fact_6267_plus__natural_Oabs__eq,axiom,
    ! [Xa: nat,X: nat] : aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),aa(nat,code_natural,code_natural_of_nat,Xa)),aa(nat,code_natural,code_natural_of_nat,X)) = aa(nat,code_natural,code_natural_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),X)) ).

% plus_natural.abs_eq
tff(fact_6268_select__def,axiom,
    ! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(aa(nat,code_natural,code_natural_of_nat,aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_amf(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_6269_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(aa(list(code_natural),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))),aTP_Lamp_amg(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_6270_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I)) ) ) ).

% nth_drop
tff(fact_6271_antisymp__antisym__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( antisymp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> antisym(A,R) ) ).

% antisymp_antisym_eq
tff(fact_6272_drop0,axiom,
    ! [A: $tType,X2: list(A)] : drop(A,zero_zero(nat),X2) = X2 ).

% drop0
tff(fact_6273_drop__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,drop(A,M,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs) ).

% drop_drop
tff(fact_6274_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] : drop(nat,M,upt(I,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M),J) ).

% drop_upt
tff(fact_6275_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6276_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6277_take__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : take(A,N,drop(A,M,Xs)) = drop(A,M,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs)) ).

% take_drop
tff(fact_6278_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_6279_take__add,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),take(A,J,drop(A,I,Xs))) ).

% take_add
tff(fact_6280_drop__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,list(A),aa(fun(nat,list(A)),fun(nat,list(A)),aa(list(A),fun(fun(nat,list(A)),fun(nat,list(A))),case_nat(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aTP_Lamp_amh(list(A),fun(nat,list(A)),Xs)),N) ).

% drop_Cons
tff(fact_6281_nths__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I5: set(nat)] : nths(A,drop(A,N,Xs),I5) = nths(A,Xs,aa(set(nat),set(nat),image2(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I5)) ).

% nths_drop
tff(fact_6282_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)),Xs) ) ) ) ).

% drop_Cons'
tff(fact_6283_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_vw(A,A),X,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A)),X),Xs) ) ) ) ).

% insort_insert_insort
tff(fact_6284_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))),pow2(A,A5)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_6285_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_vw(A,A),X,Xs) = Xs ) ) ) ).

% insort_insert_triv
tff(fact_6286_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_ami(set(A),fun(set(A),bool),A5)) ).

% Fpow_def
tff(fact_6287_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_vw(A,A),X,Xs)) = aa(set(A),set(A),insert2(A,X),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_6288_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_vw(A,A),X,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_6289_bind__singleton__conv__image,axiom,
    ! [A: $tType,B: $tType,A5: set(B),F: fun(B,A)] : bind4(B,A,A5,aTP_Lamp_pa(fun(B,A),fun(B,set(A)),F)) = aa(set(B),set(A),image2(B,A,F),A5) ).

% bind_singleton_conv_image
tff(fact_6290_and_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => semilattice_neutr(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.semilattice_neutr_axioms
tff(fact_6291_nonempty__bind__const,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B4: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( bind4(A,B,A5,aTP_Lamp_xa(set(B),fun(A,set(B)),B4)) = B4 ) ) ).

% nonempty_bind_const
tff(fact_6292_bind__const,axiom,
    ! [B: $tType,A: $tType,A5: set(B),B4: set(A)] :
      ( ( ( A5 = bot_bot(set(B)) )
       => ( bind4(B,A,A5,aTP_Lamp_nh(set(A),fun(B,set(A)),B4)) = bot_bot(set(A)) ) )
      & ( ( A5 != bot_bot(set(B)) )
       => ( bind4(B,A,A5,aTP_Lamp_nh(set(A),fun(B,set(A)),B4)) = B4 ) ) ) ).

% bind_const
tff(fact_6293_Set_Obind__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,A5: set(A),B4: fun(A,set(C)),C6: fun(C,set(B))] : bind4(C,B,bind4(A,C,A5,B4),C6) = bind4(A,B,A5,aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_amj(fun(A,set(C)),fun(fun(C,set(B)),fun(A,set(B))),B4),C6)) ).

% Set.bind_bind
tff(fact_6294_bind__image,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),A5: set(C),G: fun(B,set(A))] : bind4(B,A,aa(set(C),set(B),image2(C,B,F),A5),G) = bind4(C,A,A5,aa(fun(C,B),fun(C,set(A)),aa(fun(B,set(A)),fun(fun(C,B),fun(C,set(A))),comp(B,set(A),C),G),F)) ).

% bind_image
tff(fact_6295_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.semilattice_neutr_axioms
tff(fact_6296_or_Osemilattice__neutr__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => semilattice_neutr(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.semilattice_neutr_axioms
tff(fact_6297_max__nat_Osemilattice__neutr__axioms,axiom,
    semilattice_neutr(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.semilattice_neutr_axioms
tff(fact_6298_image__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,A),A5: set(C),G: fun(C,set(B))] : aa(set(B),set(A),image2(B,A,F),bind4(C,B,A5,G)) = bind4(C,A,A5,aa(fun(C,set(B)),fun(C,set(A)),aa(fun(set(B),set(A)),fun(fun(C,set(B)),fun(C,set(A))),comp(set(B),set(A),C),image2(B,A,F)),G)) ).

% image_bind
tff(fact_6299_Set_Obind__def,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,set(B))] : bind4(A,B,A5,F) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_amk(set(A),fun(fun(A,set(B)),fun(B,bool)),A5),F)) ).

% Set.bind_def
tff(fact_6300_fun_Opred__set,axiom,
    ! [D: $tType,A: $tType,P: fun(A,bool),X2: 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_aml(D,bool)),P),X2))
    <=> ! [Xa2: A] :
          ( pp(aa(set(A),bool,member(A,Xa2),aa(set(D),set(A),image2(D,A,X2),top_top(set(D)))))
         => pp(aa(A,bool,P,Xa2)) ) ) ).

% fun.pred_set
tff(fact_6301_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_vw(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6302_fun_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,D: $tType,X: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: 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_aml(D,bool)),aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
       => ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).

% fun.map_cong_pred
tff(fact_6303_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_aml(D,bool)),P)),aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_aml(D,bool)),Pa))) ) ).

% fun.pred_mono
tff(fact_6304_pred__fun__def,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,bool),B4: fun(B,bool),X2: 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),X2))
    <=> ! [Xa2: A] :
          ( pp(aa(A,bool,A5,Xa2))
         => pp(aa(B,bool,B4,aa(A,B,X2,Xa2))) ) ) ).

% pred_fun_def
tff(fact_6305_fun_Opred__True,axiom,
    ! [A: $tType,D: $tType,X2: 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_aml(D,bool)),aTP_Lamp_aq(A,bool)),X2)) ).

% fun.pred_True
tff(fact_6306_pred__fun__True__id,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: fun(B,bool),F: fun(C,B)] :
      ( nO_MATCH(fun(A,A),fun(B,bool),id(A),P2)
     => ( pp(aa(fun(C,B),bool,aa(fun(B,bool),fun(fun(C,B),bool),basic_pred_fun(C,B,aTP_Lamp_amm(C,bool)),P2),F))
      <=> pp(aa(fun(C,bool),bool,aa(fun(bool,bool),fun(fun(C,bool),bool),basic_pred_fun(C,bool,aTP_Lamp_amm(C,bool)),id(bool)),aa(fun(C,B),fun(C,bool),aa(fun(B,bool),fun(fun(C,B),fun(C,bool)),comp(B,bool,C),P2),F))) ) ) ).

% pred_fun_True_id
tff(fact_6307_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_aml(D,bool)),P),X))
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(set(D),set(A),image2(D,A,X),top_top(set(D)))))
           => ( pp(aa(A,bool,P,Z4))
             => pp(aa(A,bool,Pa,Z4)) ) )
       => pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_aml(D,bool)),Pa),X)) ) ) ).

% fun.pred_mono_strong
tff(fact_6308_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 )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(set(D),set(A),image2(D,A,Ya),top_top(set(D)))))
           => ( pp(aa(A,bool,P,Z4))
            <=> pp(aa(A,bool,Pa,Z4)) ) )
       => ( pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_aml(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_aml(D,bool)),Pa),Ya)) ) ) ) ).

% fun.pred_cong
tff(fact_6309_fun_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R4: 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,R4,fequal(bool)),bNF_rel_fun(fun(D,A),fun(D,B),bool,bool,bNF_rel_fun(D,D,A,B,fequal(D),R4),fequal(bool))),basic_pred_fun(D,A,aTP_Lamp_aml(D,bool))),basic_pred_fun(D,B,aTP_Lamp_aml(D,bool)))) ).

% fun.pred_transfer
tff(fact_6310_fun_Opred__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,Q: fun(B,bool),F: 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_aml(D,bool)),Q),aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X)))
    <=> pp(aa(fun(D,A),bool,aa(fun(A,bool),fun(fun(D,A),bool),basic_pred_fun(D,A,aTP_Lamp_aml(D,bool)),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X)) ) ).

% fun.pred_map
tff(fact_6311_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F: fun(B,A),X: B,Y3: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F)
     => ( pp(aa(set(B),bool,member(B,X),S2))
       => ( pp(aa(set(B),bool,member(B,Y3),S2))
         => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),insort_key(A,B,Less_eq,F,Y3)),insort_key(A,B,Less_eq,F,X)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),insort_key(A,B,Less_eq,F,X)),insort_key(A,B,Less_eq,F,Y3)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_6312_o__prs_I2_J,axiom,
    ! [F2: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H6: $tType,G2: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F2),Rep32: fun(F2,E)] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(C,D,R23,Abs22,Rep22)
       => ( quotient3(E,F2,R32,Abs32,Rep32)
         => ( aa(fun(fun(G2,H6),fun(fun(A,G2),fun(A,H6))),fun(fun(G2,H6),fun(fun(B,G2),fun(B,H6))),map_fun(fun(G2,H6),fun(G2,H6),fun(fun(A,G2),fun(A,H6)),fun(fun(B,G2),fun(B,H6)),id(fun(G2,H6)),map_fun(fun(B,G2),fun(A,G2),fun(A,H6),fun(B,H6),map_fun(A,B,G2,G2,Abs1,id(G2)),map_fun(B,A,H6,H6,Rep1,id(H6)))),comp(G2,H6,A)) = comp(G2,H6,B) ) ) ) ) ).

% o_prs(2)
tff(fact_6313_lambda__prs,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),F: fun(B,D)] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(C,D,R23,Abs22,Rep22)
       => ( aa(fun(A,C),fun(B,D),map_fun(B,A,C,D,Rep1,Abs22),aa(fun(B,D),fun(A,C),aa(fun(D,C),fun(fun(B,D),fun(A,C)),aTP_Lamp_amn(fun(A,B),fun(fun(D,C),fun(fun(B,D),fun(A,C))),Abs1),Rep22),F)) = F ) ) ) ).

% lambda_prs
tff(fact_6314_lambda__prs1,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),F: fun(B,D)] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(C,D,R23,Abs22,Rep22)
       => ( aa(fun(A,C),fun(B,D),map_fun(B,A,C,D,Rep1,Abs22),aa(fun(B,D),fun(A,C),map_fun(A,B,D,C,Abs1,Rep22),F)) = F ) ) ) ).

% lambda_prs1
tff(fact_6315_o__prs_I1_J,axiom,
    ! [C: $tType,E: $tType,A: $tType,B: $tType,F2: $tType,D: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F2),Rep32: fun(F2,E)] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(C,D,R23,Abs22,Rep22)
       => ( quotient3(E,F2,R32,Abs32,Rep32)
         => ( aa(fun(fun(C,E),fun(fun(A,C),fun(A,E))),fun(fun(D,F2),fun(fun(B,D),fun(B,F2))),map_fun(fun(D,F2),fun(C,E),fun(fun(A,C),fun(A,E)),fun(fun(B,D),fun(B,F2)),map_fun(C,D,F2,E,Abs22,Rep32),map_fun(fun(B,D),fun(A,C),fun(A,E),fun(B,F2),map_fun(A,B,D,C,Abs1,Rep22),map_fun(B,A,E,F2,Rep1,Abs32))),comp(C,E,A)) = comp(D,F2,B) ) ) ) ) ).

% o_prs(1)
tff(fact_6316_abs__o__rep,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A)] :
      ( quotient3(A,B,R4,Abs,Rep)
     => ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Abs),Rep) = id(B) ) ) ).

% abs_o_rep
tff(fact_6317_Quotient3__to__Quotient,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient3(A,B,R4,Abs,Rep)
     => ( ! [X4: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X4),Xa4))
          <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X4),X4))
              & ( aa(A,B,Abs,X4) = Xa4 ) ) )
       => quotient(A,B,R4,Abs,Rep,T4) ) ) ).

% Quotient3_to_Quotient
tff(fact_6318_OOO__quotient3,axiom,
    ! [A: $tType,B: $tType,C: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R23: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),R24: fun(A,fun(A,bool))] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(B,C,R23,Abs22,Rep22)
       => ( ! [X4: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),R24,X4),Y4))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),R12,X4),X4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),R12,Y4),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),R23,aa(A,B,Abs1,X4)),aa(A,B,Abs1,Y4))) ) ) )
         => ( ! [X4: B,Y4: B] :
                ( pp(aa(B,bool,aa(B,fun(B,bool),R23,X4),Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),R24,aa(B,A,Rep1,X4)),aa(B,A,Rep1,Y4))) )
           => quotient3(A,C,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),R12),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),R24),R12)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ) ) ).

% OOO_quotient3
tff(fact_6319_OOO__eq__quotient3,axiom,
    ! [A: $tType,B: $tType,C: $tType,R12: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),Abs22: fun(B,C),Rep22: fun(C,B)] :
      ( quotient3(A,B,R12,Abs1,Rep1)
     => ( quotient3(B,C,fequal(B),Abs22,Rep22)
       => quotient3(A,C,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),R12),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),fequal(A)),R12)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ).

% OOO_eq_quotient3
tff(fact_6320_Quotient3__int,axiom,
    quotient3(product_prod(nat,nat),int,intrel,abs_Integ,rep_Integ) ).

% Quotient3_int
tff(fact_6321_String_Oless__literal__def,axiom,
    ord_less(literal) = aa(fun(list(char),fun(list(char),bool)),fun(literal,fun(literal,bool)),map_fun(literal,list(char),fun(list(char),bool),fun(literal,bool),explode,map_fun(literal,list(char),bool,bool,explode,id(bool))),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool)))) ).

% String.less_literal_def
tff(fact_6322_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B2: A,R: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R)) = aa(set(A),set(A),insert2(A,B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R)) ).

% Range_insert
tff(fact_6323_Range__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,bool))] : aa(set(product_prod(B,A)),set(A),range2(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(A,bool),set(A),collect(A),aTP_Lamp_amp(fun(B,fun(A,bool)),fun(A,bool),P)) ).

% Range_Collect_case_prod
tff(fact_6324_Range_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(B),bool,member(B,A3),aa(set(product_prod(A,B)),set(B),range2(A,B),R)))
     => ~ ! [A4: 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),A4),A3)),R)) ) ).

% Range.cases
tff(fact_6325_Range_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(B),bool,member(B,A3),aa(set(product_prod(A,B)),set(B),range2(A,B),R)))
    <=> ? [A7: A,B7: B] :
          ( ( A3 = B7 )
          & 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),A7),B7)),R)) ) ) ).

% Range.simps
tff(fact_6326_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => pp(aa(set(B),bool,member(B,B2),aa(set(product_prod(A,B)),set(B),range2(A,B),R))) ) ).

% Range.intros
tff(fact_6327_RangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R)))
     => ~ ! [A4: 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),A4),B2)),R)) ) ).

% RangeE
tff(fact_6328_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(B,A)),set(A),range2(B,A),R)))
    <=> ? [Y: 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),Y),A3)),R)) ) ).

% Range_iff
tff(fact_6329_Range__Union,axiom,
    ! [A: $tType,B: $tType,S2: set(set(product_prod(B,A)))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(B,A))),set(set(A)),image2(set(product_prod(B,A)),set(A),range2(B,A)),S2)) ).

% Range_Union
tff(fact_6330_less__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( pp(aa(literal,bool,aa(literal,fun(literal,bool),ord_less(literal),X),Xa))
    <=> pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool))),aa(literal,list(char),explode,X)),aa(literal,list(char),explode,Xa))) ) ).

% less_literal.rep_eq
tff(fact_6331_String_Oless__eq__literal__def,axiom,
    ord_less_eq(literal) = aa(fun(list(char),fun(list(char),bool)),fun(literal,fun(literal,bool)),map_fun(literal,list(char),fun(list(char),bool),fun(literal,bool),explode,map_fun(literal,list(char),bool,bool,explode,id(bool))),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool)))) ).

% String.less_eq_literal_def
tff(fact_6332_wf__UN,axiom,
    ! [B: $tType,A: $tType,I5: set(A),R: fun(A,set(product_prod(B,B)))] :
      ( ! [I2: A] :
          ( pp(aa(set(A),bool,member(A,I2),I5))
         => wf(B,aa(A,set(product_prod(B,B)),R,I2)) )
     => ( ! [I2: A,J2: A] :
            ( pp(aa(set(A),bool,member(A,I2),I5))
           => ( pp(aa(set(A),bool,member(A,J2),I5))
             => ( ( aa(A,set(product_prod(B,B)),R,I2) != aa(A,set(product_prod(B,B)),R,J2) )
               => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(product_prod(B,B)),set(B),domain(B,B),aa(A,set(product_prod(B,B)),R,I2))),aa(set(product_prod(B,B)),set(B),range2(B,B),aa(A,set(product_prod(B,B)),R,J2))) = bot_bot(set(B)) ) ) ) )
       => wf(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R),I5))) ) ) ).

% wf_UN
tff(fact_6333_Domain__Collect__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool))] : aa(set(product_prod(A,B)),set(A),domain(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),P))) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_um(fun(A,fun(B,bool)),fun(A,bool),P)) ).

% Domain_Collect_case_prod
tff(fact_6334_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R)) = aa(set(A),set(A),insert2(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)) ).

% Domain_insert
tff(fact_6335_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
    <=> ? [Y: 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),Y)),R)) ) ).

% Domain_iff
tff(fact_6336_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
     => ~ ! [B3: 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),B3)),R)) ) ).

% DomainE
tff(fact_6337_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
     => pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R))) ) ).

% Domain.DomainI
tff(fact_6338_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
    <=> ? [A7: A,B7: B] :
          ( ( A3 = A7 )
          & 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),A7),B7)),R)) ) ) ).

% Domain.simps
tff(fact_6339_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
      ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
     => ~ ! [B3: 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),B3)),R)) ) ).

% Domain.cases
tff(fact_6340_Not__Domain__rtrancl,axiom,
    ! [A: $tType,X: A,R4: set(product_prod(A,A)),Y3: A] :
      ( ~ pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),domain(A,A),R4)))
     => ( pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,R4)))
      <=> ( X = Y3 ) ) ) ).

% Not_Domain_rtrancl
tff(fact_6341_Domain__unfold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_amq(set(product_prod(A,B)),fun(A,bool),R)) ).

% Domain_unfold
tff(fact_6342_Domain__Union,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B)))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,B))),set(set(A)),image2(set(product_prod(A,B)),set(A),domain(A,B)),S2)) ).

% Domain_Union
tff(fact_6343_less__eq__literal_Orep__eq,axiom,
    ! [X: literal,Xa: literal] :
      ( pp(aa(literal,bool,aa(literal,fun(literal,bool),ord_less_eq(literal),X),Xa))
    <=> pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool))),aa(literal,list(char),explode,X)),aa(literal,list(char),explode,Xa))) ) ).

% less_eq_literal.rep_eq
tff(fact_6344_less__eq__literal_Otransfer,axiom,
    pp(aa(fun(literal,fun(literal,bool)),bool,aa(fun(list(char),fun(list(char),bool)),fun(fun(literal,fun(literal,bool)),bool),bNF_rel_fun(list(char),literal,fun(list(char),bool),fun(literal,bool),pcr_literal,bNF_rel_fun(list(char),literal,bool,bool,pcr_literal,fequal(bool))),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool)))),ord_less_eq(literal))) ).

% less_eq_literal.transfer
tff(fact_6345_less__literal_Otransfer,axiom,
    pp(aa(fun(literal,fun(literal,bool)),bool,aa(fun(list(char),fun(list(char),bool)),fun(fun(literal,fun(literal,bool)),bool),bNF_rel_fun(list(char),literal,fun(list(char),bool),fun(literal,bool),pcr_literal,bNF_rel_fun(list(char),literal,bool,bool,pcr_literal,fequal(bool))),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool)))),ord_less(literal))) ).

% less_literal.transfer
tff(fact_6346_literal_Orep__transfer,axiom,
    pp(aa(fun(literal,list(char)),bool,aa(fun(list(char),list(char)),fun(fun(literal,list(char)),bool),bNF_rel_fun(list(char),literal,list(char),list(char),pcr_literal,list_all2(char,char,fequal(char))),aTP_Lamp_amr(list(char),list(char))),explode)) ).

% literal.rep_transfer
tff(fact_6347_Range__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(B),range2(A,B),X2) = aa(fun(B,bool),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),X2))) ).

% Range_def
tff(fact_6348_Rangep__Range__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X2: B] :
      ( pp(aa(B,bool,rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X2))
    <=> pp(aa(set(B),bool,member(B,X2),aa(set(product_prod(A,B)),set(B),range2(A,B),R))) ) ).

% Rangep_Range_eq
tff(fact_6349_String_Ocr__literal__def,axiom,
    ! [X2: list(char),Xa3: literal] :
      ( pp(aa(literal,bool,aa(list(char),fun(literal,bool),cr_literal,X2),Xa3))
    <=> ( X2 = aa(literal,list(char),explode,Xa3) ) ) ).

% String.cr_literal_def
tff(fact_6350_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_ams(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M) ).

% map_add_map_of_foldr
tff(fact_6351_map__add__None,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = none(A) )
    <=> ( ( aa(B,option(A),N,K) = none(A) )
        & ( aa(B,option(A),M,K) = none(A) ) ) ) ).

% map_add_None
tff(fact_6352_map__add__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ! [X3: A] : aa(A,option(B),map_add(A,B,F,G),X3) = none(B)
    <=> ( ! [X3: A] : aa(A,option(B),F,X3) = none(B)
        & ! [X3: A] : aa(A,option(B),G,X3) = none(B) ) ) ).

% map_add_eq_empty_iff
tff(fact_6353_map__add__find__right,axiom,
    ! [B: $tType,A: $tType,N: fun(B,option(A)),K: B,Xx: A,M: fun(B,option(A))] :
      ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),Xx) )
     => ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),Xx) ) ) ).

% map_add_find_right
tff(fact_6354_empty__eq__map__add__iff,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( aTP_Lamp_acc(A,option(B)) = map_add(A,B,F,G) )
    <=> ( ! [X3: A] : aa(A,option(B),F,X3) = none(B)
        & ! [X3: A] : aa(A,option(B),G,X3) = none(B) ) ) ).

% empty_eq_map_add_iff
tff(fact_6355_map__add__empty,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,M,aTP_Lamp_acc(A,option(B))) = M ).

% map_add_empty
tff(fact_6356_empty__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_add(A,B,aTP_Lamp_acc(A,option(B)),M) = M ).

% empty_map_add
tff(fact_6357_map__add__upd,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B)),G: fun(A,option(B)),X: A,Y3: B] : map_add(A,B,F,fun_upd(A,option(B),G,X,aa(B,option(B),some(B),Y3))) = fun_upd(A,option(B),map_add(A,B,F,G),X,aa(B,option(B),some(B),Y3)) ).

% map_add_upd
tff(fact_6358_map__add__Some__iff,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X) )
    <=> ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_Some_iff
tff(fact_6359_map__add__SomeD,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),N: fun(B,option(A)),K: B,X: A] :
      ( ( aa(B,option(A),map_add(B,A,M,N),K) = aa(A,option(A),some(A),X) )
     => ( ( aa(B,option(A),N,K) = aa(A,option(A),some(A),X) )
        | ( ( aa(B,option(A),N,K) = none(A) )
          & ( aa(B,option(A),M,K) = aa(A,option(A),some(A),X) ) ) ) ) ).

% map_add_SomeD
tff(fact_6360_map__add__def,axiom,
    ! [B: $tType,A: $tType,M12: fun(A,option(B)),M23: fun(A,option(B)),X2: A] : aa(A,option(B),map_add(A,B,M12,M23),X2) = aa(option(B),option(B),aa(fun(B,option(B)),fun(option(B),option(B)),aa(option(B),fun(fun(B,option(B)),fun(option(B),option(B))),case_option(option(B),B),aa(A,option(B),M12,X2)),some(B)),aa(A,option(B),M23,X2)) ).

% map_add_def
tff(fact_6361_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_6362_inv__o__cancel,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).

% inv_o_cancel
tff(fact_6363_cr__natural__def,axiom,
    ! [X2: nat,Xa3: code_natural] :
      ( code_cr_natural(X2,Xa3)
    <=> ( X2 = aa(code_natural,nat,code_nat_of_natural,Xa3) ) ) ).

% cr_natural_def
tff(fact_6364_inv__identity,axiom,
    ! [A: $tType,X2: A] : aa(A,A,hilbert_inv_into(A,A,top_top(set(A)),aTP_Lamp_ak(A,A)),X2) = X2 ).

% inv_identity
tff(fact_6365_o__inv__o__cancel,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),hilbert_inv_into(A,B,top_top(set(A)),F))),F) = G ) ) ).

% o_inv_o_cancel
tff(fact_6366_inv__def,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),X2: A] : aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X2) = fChoice(B,aa(A,fun(B,bool),aTP_Lamp_amt(fun(B,A),fun(A,fun(B,bool)),F),X2)) ).

% inv_def
tff(fact_6367_inv__into__def2,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X: B] : aa(B,A,hilbert_inv_into(A,B,A5,F),X) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F),X)) ).

% inv_into_def2
tff(fact_6368_inv__into__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(A,B),X2: B] : aa(B,A,hilbert_inv_into(A,B,A5,F),X2) = fChoice(A,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),A5),F),X2)) ).

% inv_into_def
tff(fact_6369_inv__into__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A5: set(C),X: B] :
      ( inj_on(A,B,F,aa(set(C),set(A),image2(C,A,G),A5))
     => ( inj_on(C,A,G,A5)
       => ( pp(aa(set(B),bool,member(B,X),aa(set(A),set(B),image2(A,B,F),aa(set(C),set(A),image2(C,A,G),A5))))
         => ( aa(B,C,hilbert_inv_into(C,B,A5,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,A5,G)),hilbert_inv_into(A,B,aa(set(C),set(A),image2(C,A,G),A5),F)),X) ) ) ) ) ).

% inv_into_comp
tff(fact_6370_inv__unique__comp,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
      ( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
     => ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),G),F) = id(B) )
       => ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).

% inv_unique_comp
tff(fact_6371_o__inv__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,A)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( bij_betw(C,A,G,top_top(set(C)),top_top(set(A)))
       => ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)) = aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F)) ) ) ) ).

% o_inv_distrib
tff(fact_6372_iso__backward,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: A,R3: set(product_prod(A,A)),R: set(product_prod(B,B)),F: fun(B,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))
     => ( bNF_Wellorder_iso(B,A,R,R3,F)
       => 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,hilbert_inv_into(B,A,field2(B,R),F),X)),aa(A,B,hilbert_inv_into(B,A,field2(B,R),F),Y3))),R)) ) ) ).

% iso_backward
tff(fact_6373_bij__image__Collect__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),P: fun(A,bool)] :
      ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),set(B),image2(A,B,F),aa(fun(A,bool),set(A),collect(A),P)) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aTP_Lamp_amu(fun(A,B),fun(fun(A,bool),fun(B,bool)),F),P)) ) ) ).

% bij_image_Collect_eq
tff(fact_6374_surj__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A)] :
      ( ( aa(set(B),set(A),image2(B,A,F),top_top(set(B))) = top_top(set(A)) )
    <=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),hilbert_inv_into(B,A,top_top(set(B)),F)) = id(A) ) ) ).

% surj_iff
tff(fact_6375_inj__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
    <=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).

% inj_iff
tff(fact_6376_inv__fn__o__fn__is__id,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => ! [X2: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F))),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),X2) = X2 ) ).

% inv_fn_o_fn_is_id
tff(fact_6377_fn__o__inv__fn__is__id,axiom,
    ! [A: $tType,F: fun(A,A),N: nat] :
      ( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
     => ! [X2: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),hilbert_inv_into(A,A,top_top(set(A)),F))),X2) = X2 ) ).

% fn_o_inv_fn_is_id
tff(fact_6378_bijection_Oinv__comp__right,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),hilbert_inv_into(A,A,top_top(set(A)),F)) = id(A) ) ) ).

% bijection.inv_comp_right
tff(fact_6379_bijection_Oinv__comp__left,axiom,
    ! [A: $tType,F: fun(A,A)] :
      ( hilbert_bijection(A,F)
     => ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),hilbert_inv_into(A,A,top_top(set(A)),F)),F) = id(A) ) ) ).

% bijection.inv_comp_left
tff(fact_6380_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_6381_splice__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat] : splice(A,replicate(A,M,X),replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),X) ).

% splice_replicate
tff(fact_6382_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_splice
tff(fact_6383_and_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.monoid_axioms
tff(fact_6384_cr__integer__def,axiom,
    ! [X2: int,Xa3: code_integer] :
      ( code_cr_integer(X2,Xa3)
    <=> ( X2 = aa(code_integer,int,code_int_of_integer,Xa3) ) ) ).

% cr_integer_def
tff(fact_6385_monoid_Oright__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,A3: A] :
      ( monoid(A,F,Z)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Z) = A3 ) ) ).

% monoid.right_neutral
tff(fact_6386_monoid_Oleft__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,A3: A] :
      ( monoid(A,F,Z)
     => ( aa(A,A,aa(A,fun(A,A),F,Z),A3) = A3 ) ) ).

% monoid.left_neutral
tff(fact_6387_add_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.monoid_axioms
tff(fact_6388_mult_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => monoid(A,times_times(A),one_one(A)) ) ).

% mult.monoid_axioms
tff(fact_6389_gcd__nat_Omonoid__axioms,axiom,
    monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.monoid_axioms
tff(fact_6390_or_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.monoid_axioms
tff(fact_6391_xor_Omonoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.monoid_axioms
tff(fact_6392_max__nat_Omonoid__axioms,axiom,
    monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.monoid_axioms
tff(fact_6393_ndepth__Push__Node__aux,axiom,
    ! [A: $tType,I: nat,F: fun(nat,sum_sum(A,nat)),K: nat] :
      ( ( aa(nat,sum_sum(A,nat),aa(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat)),aa(sum_sum(A,nat),fun(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat))),case_nat(sum_sum(A,nat)),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,I))),F),K) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_amv(fun(nat,sum_sum(A,nat)),fun(nat,bool),F)))),K)) ) ).

% ndepth_Push_Node_aux
tff(fact_6394_Finite__Set_Ofold__def,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F: fun(A,fun(B,B)),Z: B] :
      ( ( pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( finite_fold(A,B,F,Z,A5) = the(B,finite_fold_graph(A,B,F,Z,A5)) ) )
      & ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( finite_fold(A,B,F,Z,A5) = Z ) ) ) ).

% Finite_Set.fold_def
tff(fact_6395_sum_Osize_I4_J,axiom,
    ! [B: $tType,A: $tType,X22: B] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% sum.size(4)
tff(fact_6396_fold__graph__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A5: set(A),F: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A5)
     => ( finite_fold_graph(B,C,F,Z,aa(set(A),set(B),image2(A,B,G),A5)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),aa(fun(B,fun(C,C)),fun(fun(A,B),fun(A,fun(C,C))),comp(B,fun(C,C),A),F),G),Z,A5) ) ) ).

% fold_graph_image
tff(fact_6397_sum_Osize__gen_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: fun(A,nat),X: fun(B,nat),X22: B] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa,X),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(2)
tff(fact_6398_Node__def,axiom,
    ! [A: $tType,B: $tType] : old_Node(B,A) = aa(fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),set(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),collect(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),aTP_Lamp_amw(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool)) ).

% Node_def
tff(fact_6399_Node__K0__I,axiom,
    ! [B: $tType,A: $tType,A3: sum_sum(B,nat)] : pp(aa(set(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),bool,member(product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aa(fun(nat,sum_sum(A,nat)),fun(sum_sum(B,nat),product_prod(fun(nat,sum_sum(A,nat)),sum_sum(B,nat))),product_Pair(fun(nat,sum_sum(A,nat)),sum_sum(B,nat)),aTP_Lamp_amx(nat,sum_sum(A,nat))),A3)),old_Node(A,B))) ).

% Node_K0_I
tff(fact_6400_Push__neq__K0,axiom,
    ! [A: $tType,K: nat,F: fun(nat,sum_sum(A,nat))] :
      ~ ! [X4: nat] : aa(nat,sum_sum(A,nat),aa(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat)),aa(sum_sum(A,nat),fun(fun(nat,sum_sum(A,nat)),fun(nat,sum_sum(A,nat))),old_Push(A),aa(nat,sum_sum(A,nat),sum_Inr(nat,A),aa(nat,nat,suc,K))),F),X4) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).

% Push_neq_K0
tff(fact_6401_sum__set__simps_I4_J,axiom,
    ! [E: $tType,A: $tType,X: A] : aa(sum_sum(E,A),set(A),basic_setr(E,A),aa(A,sum_sum(E,A),sum_Inr(A,E),X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).

% sum_set_simps(4)
tff(fact_6402_Push__def,axiom,
    ! [B: $tType] : old_Push(B) = case_nat(sum_sum(B,nat)) ).

% Push_def
tff(fact_6403_prod_OPlus,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [A5: set(B),B4: set(C),G: fun(sum_sum(B,C),A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7121269368397514597t_prod(sum_sum(B,C),A),G),sum_Plus(B,C,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),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A5)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B4)) ) ) ) ) ).

% prod.Plus
tff(fact_6404_sum_OPlus,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(B),B4: set(C),G: fun(sum_sum(B,C),A)] :
          ( pp(aa(set(B),bool,finite_finite2(B),A5))
         => ( pp(aa(set(C),bool,finite_finite2(C),B4))
           => ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7311177749621191930dd_sum(sum_sum(B,C),A),G),sum_Plus(B,C,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),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A5)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B4)) ) ) ) ) ).

% sum.Plus
tff(fact_6405_Inr__Inl__False,axiom,
    ! [B: $tType,A: $tType,X: B,Y3: A] : aa(B,sum_sum(A,B),sum_Inr(B,A),X) != aa(A,sum_sum(A,B),sum_Inl(A,B),Y3) ).

% Inr_Inl_False
tff(fact_6406_Inl__Inr__False,axiom,
    ! [A: $tType,B: $tType,X: A,Y3: B] : aa(A,sum_sum(A,B),sum_Inl(A,B),X) != aa(B,sum_sum(A,B),sum_Inr(B,A),Y3) ).

% Inl_Inr_False
tff(fact_6407_sum__set__simps_I3_J,axiom,
    ! [A: $tType,D: $tType,X: A] : aa(sum_sum(A,D),set(D),basic_setr(A,D),aa(A,sum_sum(A,D),sum_Inl(A,D),X)) = bot_bot(set(D)) ).

% sum_set_simps(3)
tff(fact_6408_obj__sumE__f,axiom,
    ! [A: $tType,C: $tType,B: $tType,S: B,F: fun(sum_sum(A,C),B),P: bool] :
      ( ! [X4: A] :
          ( ( S = aa(sum_sum(A,C),B,F,aa(A,sum_sum(A,C),sum_Inl(A,C),X4)) )
         => pp(P) )
     => ( ! [X4: C] :
            ( ( S = aa(sum_sum(A,C),B,F,aa(C,sum_sum(A,C),sum_Inr(C,A),X4)) )
           => pp(P) )
       => ! [X2: sum_sum(A,C)] :
            ( ( S = aa(sum_sum(A,C),B,F,X2) )
           => pp(P) ) ) ) ).

% obj_sumE_f
tff(fact_6409_sum_Osize_I3_J,axiom,
    ! [A: $tType,B: $tType,X1: A] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,suc,zero_zero(nat)) ).

% sum.size(3)
tff(fact_6410_sum_Osize__gen_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa: fun(A,nat),X: fun(B,nat),X1: A] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa,X),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X1)),aa(nat,nat,suc,zero_zero(nat))) ).

% sum.size_gen(1)
tff(fact_6411_sum__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,X: A] : aa(sum_sum(A,B),set(A),basic_setl(A,B),aa(A,sum_sum(A,B),sum_Inl(A,B),X)) = aa(set(A),set(A),insert2(A,X),bot_bot(set(A))) ).

% sum_set_simps(1)
tff(fact_6412_ndepth__def,axiom,
    ! [B: $tType,A: $tType,N: old_node(A,B)] : old_ndepth(A,B,N) = aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat,aa(fun(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat),product_case_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),nat),aTP_Lamp_amz(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,N)) ).

% ndepth_def
tff(fact_6413_sum__set__simps_I2_J,axiom,
    ! [A: $tType,C: $tType,X: A] : aa(sum_sum(C,A),set(C),basic_setl(C,A),aa(A,sum_sum(C,A),sum_Inr(A,C),X)) = bot_bot(set(C)) ).

% sum_set_simps(2)
tff(fact_6414_ndepth__K0,axiom,
    ! [A: $tType,B: $tType,X: sum_sum(A,nat)] : old_ndepth(A,B,old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_ana(nat,sum_sum(B,nat))),X))) = zero_zero(nat) ).

% ndepth_K0
tff(fact_6415_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_6416_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_6417_Atom__def,axiom,
    ! [B: $tType,A: $tType,X2: sum_sum(A,nat)] : aa(sum_sum(A,nat),set(old_node(A,B)),old_Atom(A,B),X2) = aa(set(old_node(A,B)),set(old_node(A,B)),insert2(old_node(A,B),old_Abs_Node(B,A,aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aTP_Lamp_ana(nat,sum_sum(B,nat))),X2))),bot_bot(set(old_node(A,B)))) ).

% Atom_def
tff(fact_6418_Push__Node__def,axiom,
    ! [B: $tType,A: $tType,X2: sum_sum(B,nat),Xa3: old_node(A,B)] : aa(old_node(A,B),old_node(A,B),old_Push_Node(B,A,X2),Xa3) = old_Abs_Node(B,A,aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),product_apfst(fun(nat,sum_sum(B,nat)),fun(nat,sum_sum(B,nat)),sum_sum(A,nat),aa(sum_sum(B,nat),fun(fun(nat,sum_sum(B,nat)),fun(nat,sum_sum(B,nat))),old_Push(B),X2)),old_Rep_Node(A,B,Xa3))) ).

% Push_Node_def
tff(fact_6419_Lim__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,set(old_node(A,B)))] : old_Lim(B,A,F) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(fun(set(old_node(A,B)),bool),set(set(old_node(A,B))),collect(set(old_node(A,B))),aTP_Lamp_anb(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),bool),F))) ).

% Lim_def
tff(fact_6420_Scons__def,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B)),N5: set(old_node(A,B))] : old_Scons(A,B,M2,N5) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),sup_sup(set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),one_one(nat)))),M2)),aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,one_one(nat))))),N5)) ).

% Scons_def
tff(fact_6421_Scons__UN1__y,axiom,
    ! [A: $tType,B: $tType,C: $tType,M2: set(old_node(A,B)),F: fun(C,set(old_node(A,B)))] : old_Scons(A,B,M2,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_anc(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),M2),F)),top_top(set(C)))) ).

% Scons_UN1_y
tff(fact_6422_Scons__UN1__x,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B))),M2: set(old_node(A,B))] : old_Scons(A,B,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C)))),M2) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_and(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),F),M2)),top_top(set(C)))) ).

% Scons_UN1_x
tff(fact_6423_Split__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,C2: fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)),M2: set(old_node(A,B))] : aa(set(old_node(A,B)),C,old_Split(A,B,C,C2),M2) = the(C,aa(set(old_node(A,B)),fun(C,bool),aTP_Lamp_ane(fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)),fun(set(old_node(A,B)),fun(C,bool)),C2),M2)) ).

% Split_def
tff(fact_6424_Leaf__def,axiom,
    ! [B: $tType,A: $tType] : old_Leaf(A,B) = aa(fun(A,sum_sum(A,nat)),fun(A,set(old_node(A,B))),aa(fun(sum_sum(A,nat),set(old_node(A,B))),fun(fun(A,sum_sum(A,nat)),fun(A,set(old_node(A,B)))),comp(sum_sum(A,nat),set(old_node(A,B)),A),old_Atom(A,B)),sum_Inl(A,nat)) ).

% Leaf_def
tff(fact_6425_ntrunc__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,K: nat,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,K),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_anf(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),K),F)),top_top(set(C)))) ).

% ntrunc_UN1
tff(fact_6426_In1__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : old_In1(A,B,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aTP_Lamp_ang(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).

% In1_UN1
tff(fact_6427_ntrunc__0,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,zero_zero(nat)),M2) = bot_bot(set(old_node(A,B))) ).

% ntrunc_0
tff(fact_6428_ntrunc__one__In1,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat))),old_In1(A,B,M2)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In1
tff(fact_6429_ntrunc__o__equality,axiom,
    ! [C: $tType,B: $tType,A: $tType,H1: fun(A,set(old_node(B,C))),H22: fun(A,set(old_node(B,C)))] :
      ( ! [K2: nat] : aa(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C))),aa(fun(set(old_node(B,C)),set(old_node(B,C))),fun(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C)))),comp(set(old_node(B,C)),set(old_node(B,C)),A),old_ntrunc(B,C,K2)),H1) = aa(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C))),aa(fun(set(old_node(B,C)),set(old_node(B,C))),fun(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C)))),comp(set(old_node(B,C)),set(old_node(B,C)),A),old_ntrunc(B,C,K2)),H22)
     => ( H1 = H22 ) ) ).

% ntrunc_o_equality
tff(fact_6430_ntrunc__def,axiom,
    ! [B: $tType,A: $tType,K: nat,N5: set(old_node(A,B))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,K),N5) = aa(fun(old_node(A,B),bool),set(old_node(A,B)),collect(old_node(A,B)),aa(set(old_node(A,B)),fun(old_node(A,B),bool),aTP_Lamp_anh(nat,fun(set(old_node(A,B)),fun(old_node(A,B),bool)),K),N5)) ).

% ntrunc_def
tff(fact_6431_ntrunc__one__In0,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,aa(nat,nat,suc,zero_zero(nat))),old_In0(A,B,M2)) = bot_bot(set(old_node(A,B))) ).

% ntrunc_one_In0
tff(fact_6432_In1__def,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B))] : old_In1(A,B,M2) = old_Scons(A,B,aa(nat,set(old_node(A,B)),old_Numb(A,B),one_one(nat)),M2) ).

% In1_def
tff(fact_6433_In0__def,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B))] : old_In0(A,B,M2) = old_Scons(A,B,aa(nat,set(old_node(A,B)),old_Numb(A,B),zero_zero(nat)),M2) ).

% In0_def
tff(fact_6434_Numb__def,axiom,
    ! [B: $tType,A: $tType] : old_Numb(A,B) = aa(fun(nat,sum_sum(A,nat)),fun(nat,set(old_node(A,B))),aa(fun(sum_sum(A,nat),set(old_node(A,B))),fun(fun(nat,sum_sum(A,nat)),fun(nat,set(old_node(A,B)))),comp(sum_sum(A,nat),set(old_node(A,B)),nat),old_Atom(A,B)),sum_Inr(nat,A)) ).

% Numb_def
tff(fact_6435_In0__UN1,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : old_In0(A,B,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),image2(C,set(old_node(A,B)),aTP_Lamp_ani(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).

% In0_UN1
tff(fact_6436_Case__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,C2: fun(set(old_node(A,B)),C),D3: fun(set(old_node(A,B)),C),M2: set(old_node(A,B))] : old_Case(A,B,C,C2,D3,M2) = the(C,aa(set(old_node(A,B)),fun(C,bool),aa(fun(set(old_node(A,B)),C),fun(set(old_node(A,B)),fun(C,bool)),aTP_Lamp_anj(fun(set(old_node(A,B)),C),fun(fun(set(old_node(A,B)),C),fun(set(old_node(A,B)),fun(C,bool))),C2),D3),M2)) ).

% Case_def
tff(fact_6437_dsum__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dsum(A,B,R,S) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),sup_sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_ank(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),R))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_anl(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),S))) ).

% dsum_def
tff(fact_6438_dsum__In1I,axiom,
    ! [B: $tType,A: $tType,N5: set(old_node(A,B)),N11: set(old_node(A,B)),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N5),N11)),S))
     => pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,N5)),old_In1(A,B,N11))),old_dsum(A,B,R,S))) ) ).

% dsum_In1I
tff(fact_6439_dsum__In0I,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B)),M9: set(old_node(A,B)),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M2),M9)),R))
     => pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,M2)),old_In0(A,B,M9))),old_dsum(A,B,R,S))) ) ).

% dsum_In0I
tff(fact_6440_dsumE,axiom,
    ! [B: $tType,A: $tType,W: product_prod(set(old_node(A,B)),set(old_node(A,B))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),W),old_dsum(A,B,R,S)))
     => ( ! [X4: set(old_node(A,B)),X8: set(old_node(A,B))] :
            ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X4),X8)),R))
           => ( W != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,X4)),old_In0(A,B,X8)) ) )
       => ~ ! [Y4: set(old_node(A,B)),Y9: set(old_node(A,B))] :
              ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y4),Y9)),S))
             => ( W != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Y4)),old_In1(A,B,Y9)) ) ) ) ) ).

% dsumE
tff(fact_6441_dprod__def,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dprod(A,B,R,S) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_ann(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),S))),R)) ).

% dprod_def
tff(fact_6442_dprodI,axiom,
    ! [B: $tType,A: $tType,M2: set(old_node(A,B)),M9: set(old_node(A,B)),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),N5: set(old_node(A,B)),N11: set(old_node(A,B)),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),M2),M9)),R))
     => ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),N5),N11)),S))
       => pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,M2,N5)),old_Scons(A,B,M9,N11))),old_dprod(A,B,R,S))) ) ) ).

% dprodI
tff(fact_6443_dprodE,axiom,
    ! [B: $tType,A: $tType,C2: product_prod(set(old_node(A,B)),set(old_node(A,B))),R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),C2),old_dprod(A,B,R,S)))
     => ~ ! [X4: set(old_node(A,B)),Y4: set(old_node(A,B)),X8: set(old_node(A,B))] :
            ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),X4),X8)),R))
           => ! [Y9: set(old_node(A,B))] :
                ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,member(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),Y4),Y9)),S))
               => ( C2 != aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,X4,Y4)),old_Scons(A,B,X8,Y9)) ) ) ) ) ).

% dprodE
tff(fact_6444_Powp__Pow__eq,axiom,
    ! [A: $tType,A5: set(A),X2: set(A)] :
      ( powp(A,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),A5),X2)
    <=> pp(aa(set(set(A)),bool,member(set(A),X2),pow2(A,A5))) ) ).

% Powp_Pow_eq
tff(fact_6445_combine__options__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: option(A),Y3: option(A)] : combine_options(A,F,X,Y3) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),Y3),aa(option(A),fun(A,option(A)),aTP_Lamp_anp(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),F),Y3)),X) ).

% combine_options_def
tff(fact_6446_combine__options__simps_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: option(A)] : combine_options(A,F,X,none(A)) = X ).

% combine_options_simps(2)
tff(fact_6447_combine__options__simps_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Y3: option(A)] : combine_options(A,F,none(A),Y3) = Y3 ).

% combine_options_simps(1)
tff(fact_6448_combine__options__simps_I3_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] : combine_options(A,F,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),F,A3),B2)) ).

% combine_options_simps(3)
tff(fact_6449_combine__options__left__commute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Y3: option(A),X: option(A),Z: option(A)] :
      ( ! [X4: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F,X4),Y4) = aa(A,A,aa(A,fun(A,A),F,Y4),X4)
     => ( ! [X4: A,Y4: A,Z4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,X4),Y4)),Z4) = aa(A,A,aa(A,fun(A,A),F,X4),aa(A,A,aa(A,fun(A,A),F,Y4),Z4))
       => ( combine_options(A,F,Y3,combine_options(A,F,X,Z)) = combine_options(A,F,X,combine_options(A,F,Y3,Z)) ) ) ) ).

% combine_options_left_commute
tff(fact_6450_combine__options__commute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: option(A),Y3: option(A)] :
      ( ! [X4: A,Y4: A] : aa(A,A,aa(A,fun(A,A),F,X4),Y4) = aa(A,A,aa(A,fun(A,A),F,Y4),X4)
     => ( combine_options(A,F,X,Y3) = combine_options(A,F,Y3,X) ) ) ).

% combine_options_commute
tff(fact_6451_combine__options__assoc,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),X: option(A),Y3: option(A),Z: option(A)] :
      ( ! [X4: A,Y4: A,Z4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,X4),Y4)),Z4) = aa(A,A,aa(A,fun(A,A),F,X4),aa(A,A,aa(A,fun(A,A),F,Y4),Z4))
     => ( combine_options(A,F,combine_options(A,F,X,Y3),Z) = combine_options(A,F,X,combine_options(A,F,Y3,Z)) ) ) ).

% combine_options_assoc
tff(fact_6452_Powp__def,axiom,
    ! [A: $tType,A5: fun(A,bool),X2: set(A)] :
      ( powp(A,A5,X2)
    <=> ! [Xa2: A] :
          ( pp(aa(set(A),bool,member(A,Xa2),X2))
         => pp(aa(A,bool,A5,Xa2)) ) ) ).

% Powp_def
tff(fact_6453_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y3: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y3 )
     => ( 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),Xa)))
       => ( ( ( X = nil(A) )
           => ( ( Y3 = aa(set(list(A)),set(list(A)),insert2(list(A),Xa),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)),Xa))) ) )
         => ( ( ( Xa = nil(A) )
             => ( ( Y3 = aa(set(list(A)),set(list(A)),insert2(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)))) ) )
           => ~ ! [X4: A,Xs3: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                 => ! [Y4: A,Ys3: list(A)] :
                      ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                     => ( ( Y3 = 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),X4)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3),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),X4),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_6454_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y3: A,Ys: 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),Y3),Ys))))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys)))),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),X),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_6455_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)) )
       => ( ! [Xs3: 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)),Xs3),nil(A))))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs3),nil(A))) )
         => ( ! [X4: A,Xs3: list(A),Y4: 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),X4),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3))))
               => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),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),X4),Xs3)),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),X4),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3))) ) ) )
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).

% shuffles.pinduct
tff(fact_6456_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: 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)),Ys)))
     => ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),insert2(list(A),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_6457_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)),insert2(list(A),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_6458_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y3: list(A)] :
      ( ( splice(A,X,Xa) = Y3 )
     => ( 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),Xa)))
       => ( ( ( X = nil(A) )
           => ( ( Y3 = Xa )
             => ~ 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)),Xa))) ) )
         => ~ ! [X4: A,Xs3: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
               => ( ( Y3 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),splice(A,Xa,Xs3)) )
                 => ~ 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),X4),Xs3)),Xa))) ) ) ) ) ) ).

% splice.pelims
tff(fact_6459_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: 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)),Ys)))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_6460_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)) )
       => ( ! [X4: A,Xs3: 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),X4),Xs3)),Ys3)))
             => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys3),Xs3))
               => 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),X4),Xs3)),Ys3)) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).

% splice.pinduct
tff(fact_6461_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: 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)),Ys)))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),splice(A,Ys,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_6462_equivp__equiv,axiom,
    ! [A: $tType,A5: set(product_prod(A,A))] :
      ( equiv_equiv(A,top_top(set(A)),A5)
    <=> equiv_equivp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),A5)) ) ).

% equivp_equiv
tff(fact_6463_Shift__def,axiom,
    ! [A: $tType,Kl: set(list(A)),K: A] : bNF_Greatest_Shift(A,Kl,K) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(A,fun(list(A),bool),aTP_Lamp_anq(set(list(A)),fun(A,fun(list(A),bool)),Kl),K)) ).

% Shift_def
tff(fact_6464_int__equivp,axiom,
    equiv_equivp(product_prod(nat,nat),intrel) ).

% int_equivp
tff(fact_6465_Quotient__crel__quotient,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
     => ( equiv_equivp(A,R4)
       => ! [X2: A] : aa(A,fun(B,bool),T4,X2) = aa(B,fun(B,bool),fequal(B),aa(A,B,Abs,X2)) ) ) ).

% Quotient_crel_quotient
tff(fact_6466_Quotient3__to__Quotient__equivp,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient3(A,B,R4,Abs,Rep)
     => ( ! [X4: A] : aa(A,fun(B,bool),T4,X4) = aa(B,fun(B,bool),fequal(B),aa(A,B,Abs,X4))
       => ( equiv_equivp(A,R4)
         => quotient(A,B,R4,Abs,Rep,T4) ) ) ) ).

% Quotient3_to_Quotient_equivp
tff(fact_6467_cut__def,axiom,
    ! [A: $tType,B: $tType,X: A,R4: set(product_prod(A,A)),F: fun(A,B),X2: 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),X2),X)),R4))
       => ( aa(A,B,cut(A,B,F,R4,X),X2) = aa(A,B,F,X2) ) )
      & ( ~ pp(aa(set(product_prod(A,A)),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),X)),R4))
       => ( aa(A,B,cut(A,B,F,R4,X),X2) = undefined(B) ) ) ) ).

% cut_def
tff(fact_6468_map__conv__bind__option,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: option(B)] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),X),aa(fun(B,A),fun(B,option(A)),aa(fun(A,option(A)),fun(fun(B,A),fun(B,option(A))),comp(A,option(A),B),some(A)),F)) ).

% map_conv_bind_option
tff(fact_6469_bind__assoc,axiom,
    ! [C: $tType,A: $tType,B: $tType,X: option(C),F: fun(C,option(B)),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind2(C,B),X),F)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind2(C,A),X),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_anr(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),F),G)) ).

% bind_assoc
tff(fact_6470_bind__runit,axiom,
    ! [A: $tType,X: option(A)] : aa(fun(A,option(A)),option(A),aa(option(A),fun(fun(A,option(A)),option(A)),bind2(A,A),X),some(A)) = X ).

% bind_runit
tff(fact_6471_bind__rzero,axiom,
    ! [B: $tType,A: $tType,X: option(B)] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),X),aTP_Lamp_acu(B,option(A))) = none(A) ).

% bind_rzero
tff(fact_6472_bind_Obind__lunit,axiom,
    ! [B: $tType,A: $tType,X: A,F: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),aa(A,option(A),some(A),X)),F) = aa(A,option(B),F,X) ).

% bind.bind_lunit
tff(fact_6473_Option_Obind__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y3: option(A),F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( X = Y3 )
     => ( ! [A4: A] :
            ( ( Y3 = aa(A,option(A),some(A),A4) )
           => ( aa(A,option(B),F,A4) = aa(A,option(B),G,A4) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),X),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),Y3),G) ) ) ) ).

% Option.bind_cong
tff(fact_6474_bind__eq__Some__conv,axiom,
    ! [B: $tType,A: $tType,F: option(B),G: fun(B,option(A)),X: A] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),F),G) = aa(A,option(A),some(A),X) )
    <=> ? [Y: B] :
          ( ( F = aa(B,option(B),some(B),Y) )
          & ( aa(B,option(A),G,Y) = aa(A,option(A),some(A),X) ) ) ) ).

% bind_eq_Some_conv
tff(fact_6475_bind__option__cong__code,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y3: option(A),F: fun(A,option(B))] :
      ( ( X = Y3 )
     => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),X),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),Y3),F) ) ) ).

% bind_option_cong_code
tff(fact_6476_bind_Obind__lzero,axiom,
    ! [A: $tType,B: $tType,F: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),none(A)),F) = none(B) ).

% bind.bind_lzero
tff(fact_6477_cuts__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),R4: set(product_prod(A,A)),X: A,G: fun(A,B)] :
      ( ( cut(A,B,F,R4,X) = cut(A,B,G,R4,X) )
    <=> ! [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),Y),X)),R4))
         => ( aa(A,B,F,Y) = aa(A,B,G,Y) ) ) ) ).

% cuts_eq
tff(fact_6478_cut__apply,axiom,
    ! [B: $tType,A: $tType,X: A,A3: A,R4: set(product_prod(A,A)),F: 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),A3)),R4))
     => ( aa(A,B,cut(A,B,F,R4,A3),X) = aa(A,B,F,X) ) ) ).

% cut_apply
tff(fact_6479_bind__split__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),bool),M: option(B),F: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),M),F)))
    <=> ~ ( ( ( M = none(B) )
            & ~ pp(aa(option(A),bool,P,none(A))) )
          | ? [X3: B] :
              ( ( M = aa(B,option(B),some(B),X3) )
              & ~ pp(aa(option(A),bool,P,aa(B,option(A),F,X3))) ) ) ) ).

% bind_split_asm
tff(fact_6480_bind__split,axiom,
    ! [A: $tType,B: $tType,P: fun(option(A),bool),M: option(B),F: fun(B,option(A))] :
      ( pp(aa(option(A),bool,P,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),M),F)))
    <=> ( ( ( M = none(B) )
         => pp(aa(option(A),bool,P,none(A))) )
        & ! [V7: B] :
            ( ( M = aa(B,option(B),some(B),V7) )
           => pp(aa(option(A),bool,P,aa(B,option(A),F,V7))) ) ) ) ).

% bind_split
tff(fact_6481_bind__eq__None__conv,axiom,
    ! [B: $tType,A: $tType,A3: option(B),F: fun(B,option(A))] :
      ( ( aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),A3),F) = none(A) )
    <=> ( ( A3 = none(B) )
        | ( aa(B,option(A),F,aa(option(B),B,the2(B),A3)) = none(A) ) ) ) ).

% bind_eq_None_conv
tff(fact_6482_bind__map__option,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X: option(C),G: fun(B,option(A))] : aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),F),X)),G) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind2(C,A),X),aa(fun(C,B),fun(C,option(A)),aa(fun(B,option(A)),fun(fun(C,B),fun(C,option(A))),comp(B,option(A),C),G),F)) ).

% bind_map_option
tff(fact_6483_map__option__bind,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,A),X: option(C),G: fun(C,option(B))] : aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),aa(fun(C,option(B)),option(B),aa(option(C),fun(fun(C,option(B)),option(B)),bind2(C,B),X),G)) = aa(fun(C,option(A)),option(A),aa(option(C),fun(fun(C,option(A)),option(A)),bind2(C,A),X),aa(fun(C,option(B)),fun(C,option(A)),aa(fun(option(B),option(A)),fun(fun(C,option(B)),fun(C,option(A))),comp(option(B),option(A),C),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F)),G)) ).

% map_option_bind
tff(fact_6484_adm__lemma,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,A)),F5: fun(fun(A,B),fun(A,B))] : adm_wf(A,B,R4,aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aTP_Lamp_ans(set(product_prod(A,A)),fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),R4),F5)) ).

% adm_lemma
tff(fact_6485_set__bind__option,axiom,
    ! [A: $tType,B: $tType,X: option(B),F: fun(B,option(A))] : aa(option(A),set(A),set_option(A),aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),X),F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,option(A)),fun(B,set(A)),aa(fun(option(A),set(A)),fun(fun(B,option(A)),fun(B,set(A))),comp(option(A),set(A),B),set_option(A)),F)),aa(option(B),set(B),set_option(B),X))) ).

% set_bind_option
tff(fact_6486_elem__set,axiom,
    ! [A: $tType,X: A,Xo: option(A)] :
      ( pp(aa(set(A),bool,member(A,X),aa(option(A),set(A),set_option(A),Xo)))
    <=> ( Xo = aa(A,option(A),some(A),X) ) ) ).

% elem_set
tff(fact_6487_set__empty__eq,axiom,
    ! [A: $tType,Xo: option(A)] :
      ( ( aa(option(A),set(A),set_option(A),Xo) = bot_bot(set(A)) )
    <=> ( Xo = none(A) ) ) ).

% set_empty_eq
tff(fact_6488_bind__option__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Y3: option(A),F: fun(A,option(B)),G: fun(A,option(B))] :
      ( ( X = Y3 )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),Y3)))
           => ( aa(A,option(B),F,Z4) = aa(A,option(B),G,Z4) ) )
       => ( aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),X),F) = aa(fun(A,option(B)),option(B),aa(option(A),fun(fun(A,option(B)),option(B)),bind2(A,B),Y3),G) ) ) ) ).

% bind_option_cong
tff(fact_6489_option_Oset__cases,axiom,
    ! [A: $tType,E2: A,A3: option(A)] :
      ( pp(aa(set(A),bool,member(A,E2),aa(option(A),set(A),set_option(A),A3)))
     => ( A3 = aa(A,option(A),some(A),E2) ) ) ).

% option.set_cases
tff(fact_6490_option_Oset__intros,axiom,
    ! [A: $tType,X22: A] : pp(aa(set(A),bool,member(A,X22),aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22)))) ).

% option.set_intros
tff(fact_6491_ospec,axiom,
    ! [A: $tType,A5: option(A),P: fun(A,bool),X: A] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(option(A),set(A),set_option(A),A5)))
         => pp(aa(A,bool,P,X4)) )
     => ( ( A5 = aa(A,option(A),some(A),X) )
       => pp(aa(A,bool,P,X)) ) ) ).

% ospec
tff(fact_6492_map__option__idI,axiom,
    ! [A: $tType,X: option(A),F: fun(A,A)] :
      ( ! [Y4: A] :
          ( pp(aa(set(A),bool,member(A,Y4),aa(option(A),set(A),set_option(A),X)))
         => ( aa(A,A,F,Y4) = Y4 ) )
     => ( aa(option(A),option(A),aa(fun(A,A),fun(option(A),option(A)),map_option(A,A),F),X) = X ) ) ).

% map_option_idI
tff(fact_6493_option_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Xa: option(A),F: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z4: A,Za: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),X)))
         => ( pp(aa(set(A),bool,member(A,Za),aa(option(A),set(A),set_option(A),Xa)))
           => ( ( aa(A,B,F,Z4) = aa(A,B,Fa,Za) )
             => ( Z4 = Za ) ) ) )
     => ( ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Fa),Xa) )
       => ( X = Xa ) ) ) ).

% option.inj_map_strong
tff(fact_6494_option_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: option(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [Z4: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),X)))
         => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
     => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),X) ) ) ).

% option.map_cong0
tff(fact_6495_option_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),Ya)))
           => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).

% option.map_cong
tff(fact_6496_adm__wf__def,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,A)),F5: fun(fun(A,B),fun(A,B))] :
      ( adm_wf(A,B,R4,F5)
    <=> ! [F6: fun(A,B),G6: fun(A,B),X3: 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),Z5),X3)),R4))
             => ( aa(A,B,F6,Z5) = aa(A,B,G6,Z5) ) )
         => ( aa(A,B,aa(fun(A,B),fun(A,B),F5,F6),X3) = aa(A,B,aa(fun(A,B),fun(A,B),F5,G6),X3) ) ) ) ).

% adm_wf_def
tff(fact_6497_option_Osimps_I14_J,axiom,
    ! [A: $tType] : aa(option(A),set(A),set_option(A),none(A)) = bot_bot(set(A)) ).

% option.simps(14)
tff(fact_6498_option_Oset__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),V: option(A)] : aa(option(B),set(B),set_option(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),V)) = aa(set(A),set(B),image2(A,B,F),aa(option(A),set(A),set_option(A),V)) ).

% option.set_map
tff(fact_6499_option_Oset__sel,axiom,
    ! [A: $tType,A3: option(A)] :
      ( ( A3 != none(A) )
     => pp(aa(set(A),bool,member(A,aa(option(A),A,the2(A),A3)),aa(option(A),set(A),set_option(A),A3))) ) ).

% option.set_sel
tff(fact_6500_option_Osimps_I15_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),set(A),set_option(A),aa(A,option(A),some(A),X22)) = aa(set(A),set(A),insert2(A,X22),bot_bot(set(A))) ).

% option.simps(15)
tff(fact_6501_option_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),bNF_Grp(A,B,A5,F)) = bNF_Grp(option(A),option(B),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_ant(set(A),fun(option(A),bool),A5)),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F)) ).

% option.rel_Grp
tff(fact_6502_option_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4) = aa(fun(option(product_prod(A,B)),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(product_prod(A,B)),bool)),fun(fun(option(product_prod(A,B)),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),relcompp(option(A),option(product_prod(A,B)),option(B)),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_anu(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B))))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_anu(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)))) ).

% option.rel_compp_Grp
tff(fact_6503_rel__option__None1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),none(A)),X))
    <=> ( X = none(B) ) ) ).

% rel_option_None1
tff(fact_6504_rel__option__None2,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),X: option(A)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X),none(B)))
    <=> ( X = none(A) ) ) ).

% rel_option_None2
tff(fact_6505_option_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: option(A),Ya: option(A),Y3: option(B),Xa: option(B),R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z4: A,Yb: B] :
              ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),Ya)))
             => ( pp(aa(set(B),bool,member(B,Yb),aa(option(B),set(B),set_option(B),Xa)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z4),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z4),Yb)) ) ) )
         => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X),Y3))
          <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),Ya),Xa)) ) ) ) ) ).

% option.rel_cong
tff(fact_6506_option_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: option(A),Y3: option(B),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X),Y3))
     => ( ! [Z4: A,Yb: B] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(set(B),bool,member(B,Yb),aa(option(B),set(B),set_option(B),Y3)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z4),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z4),Yb)) ) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2),X),Y3)) ) ) ).

% option.rel_mono_strong
tff(fact_6507_option_Orel__refl__strong,axiom,
    ! [A: $tType,X: option(A),Ra2: fun(A,fun(A,bool))] :
      ( ! [Z4: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z4),Z4)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),Ra2),X),X)) ) ).

% option.rel_refl_strong
tff(fact_6508_rel__option__reflI,axiom,
    ! [A: $tType,Y3: option(A),P: fun(A,fun(A,bool))] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,member(A,X4),aa(option(A),set(A),set_option(A),Y3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),X4)) )
     => pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),P),Y3),Y3)) ) ).

% rel_option_reflI
tff(fact_6509_option_Orec__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R4: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S2,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R4,S2),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),S2))),rec_option(C,A)),rec_option(D,B))) ).

% option.rec_transfer
tff(fact_6510_option_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F2,bool))] : pp(aa(fun(fun(E,F2),fun(option(E),option(F2))),bool,aa(fun(fun(A,B),fun(option(A),option(B))),fun(fun(fun(E,F2),fun(option(E),option(F2))),bool),bNF_rel_fun(fun(A,B),fun(E,F2),fun(option(A),option(B)),fun(option(E),option(F2)),bNF_rel_fun(A,E,B,F2,Rb2,Sd),bNF_rel_fun(option(A),option(E),option(B),option(F2),aa(fun(A,fun(E,bool)),fun(option(A),fun(option(E),bool)),rel_option(A,E),Rb2),aa(fun(B,fun(F2,bool)),fun(option(B),fun(option(F2),bool)),rel_option(B,F2),Sd))),map_option(A,B)),map_option(E,F2))) ).

% option.map_transfer
tff(fact_6511_option_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(option(A),fun(option(B),bool)),fun(option(C),fun(option(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(option(A),option(C),fun(option(B),bool),fun(option(D),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),bNF_rel_fun(option(B),option(D),bool,bool,aa(fun(B,fun(D,bool)),fun(option(B),fun(option(D),bool)),rel_option(B,D),Sc),fequal(bool)))),rel_option(A,B)),rel_option(C,D))) ).

% option.rel_transfer
tff(fact_6512_option_Octr__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(B,option(B)),bool,aa(fun(A,option(A)),fun(fun(B,option(B)),bool),bNF_rel_fun(A,B,option(A),option(B),R4,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)),some(A)),some(B))) ).

% option.ctr_transfer(2)
tff(fact_6513_option_Odisc__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool)),aTP_Lamp_anv(option(A),bool)),aTP_Lamp_anw(option(B),bool))) ).

% option.disc_transfer(1)
tff(fact_6514_option_Odisc__transfer_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(option(B),bool),bool,aa(fun(option(A),bool),fun(fun(option(B),bool),bool),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool)),aTP_Lamp_anx(option(A),bool)),aTP_Lamp_any(option(B),bool))) ).

% option.disc_transfer(2)
tff(fact_6515_option_Ocase__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: fun(C,fun(D,bool)),R4: fun(A,fun(B,bool))] : pp(aa(fun(D,fun(fun(B,D),fun(option(B),D))),bool,aa(fun(C,fun(fun(A,C),fun(option(A),C))),fun(fun(D,fun(fun(B,D),fun(option(B),D))),bool),bNF_rel_fun(C,D,fun(fun(A,C),fun(option(A),C)),fun(fun(B,D),fun(option(B),D)),S2,bNF_rel_fun(fun(A,C),fun(B,D),fun(option(A),C),fun(option(B),D),bNF_rel_fun(A,B,C,D,R4,S2),bNF_rel_fun(option(A),option(B),C,D,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),S2))),case_option(C,A)),case_option(D,B))) ).

% option.case_transfer
tff(fact_6516_option_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R4: 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))),R4),Ra2))
     => pp(aa(fun(option(A),fun(option(B),bool)),bool,aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),bool),ord_less_eq(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),Ra2))) ) ).

% option.rel_mono
tff(fact_6517_option_OQuotient,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
     => quotient(option(A),option(B),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R4),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),Abs),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),Rep),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),T4)) ) ).

% option.Quotient
tff(fact_6518_option_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: option(A),G: fun(B,C),Y3: option(B)] :
      ( pp(aa(option(C),bool,aa(option(A),fun(option(C),bool),aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),Sa),X),aa(option(B),option(C),aa(fun(B,C),fun(option(B),option(C)),map_option(B,C),G),Y3)))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_st(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y3)) ) ).

% option.rel_map(2)
tff(fact_6519_option_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: option(A),Y3: option(B)] :
      ( pp(aa(option(B),bool,aa(option(C),fun(option(B),bool),aa(fun(C,fun(B,bool)),fun(option(C),fun(option(B),bool)),rel_option(C,B),Sb),aa(option(A),option(C),aa(fun(A,C),fun(option(A),option(C)),map_option(A,C),I),X)),Y3))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_su(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y3)) ) ).

% option.rel_map(1)
tff(fact_6520_option_Orel__compp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(option(A),fun(option(C),bool)),rel_option(A,C),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),R4),S2)) = aa(fun(option(B),fun(option(C),bool)),fun(option(A),fun(option(C),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(B),fun(option(C),bool)),fun(option(A),fun(option(C),bool))),relcompp(option(A),option(B),option(C)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)),aa(fun(B,fun(C,bool)),fun(option(B),fun(option(C),bool)),rel_option(B,C),S2)) ).

% option.rel_compp
tff(fact_6521_option_Orel__flip,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(B),B2: option(A)] :
      ( pp(aa(option(A),bool,aa(option(B),fun(option(A),bool),aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R4)),A3),B2))
    <=> pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),B2),A3)) ) ).

% option.rel_flip
tff(fact_6522_option_Orel__conversep,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(option(B),fun(option(A),bool)),rel_option(B,A),conversep(A,B,R4)) = conversep(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ).

% option.rel_conversep
tff(fact_6523_option_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
    <=> ( ( ( A3 = none(A) )
        <=> ( B2 = none(B) ) )
        & ( ( A3 != none(A) )
         => ( ( B2 != none(B) )
           => pp(aa(B,bool,aa(A,fun(B,bool),R4,aa(option(A),A,the2(A),A3)),aa(option(B),B,the2(B),B2))) ) ) ) ) ).

% option.rel_sel
tff(fact_6524_option_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X22: A,Y2: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2)))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),R4,X22),Y2)) ) ).

% option.rel_inject(2)
tff(fact_6525_option_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X22: A,Y2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R4,X22),Y2))
     => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),X22)),aa(B,option(B),some(B),Y2))) ) ).

% option.rel_intros(2)
tff(fact_6526_option__rel__Some1,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),X: A,Y3: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A5),aa(A,option(A),some(A),X)),Y3))
    <=> ? [Y10: B] :
          ( ( Y3 = aa(B,option(B),some(B),Y10) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A5,X),Y10)) ) ) ).

% option_rel_Some1
tff(fact_6527_option__rel__Some2,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool)),X: option(A),Y3: B] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A5),X),aa(B,option(B),some(B),Y3)))
    <=> ? [X11: A] :
          ( ( X = aa(A,option(A),some(A),X11) )
          & pp(aa(B,bool,aa(A,fun(B,bool),A5,X11),Y3)) ) ) ).

% option_rel_Some2
tff(fact_6528_option_Orel__refl,axiom,
    ! [B: $tType,Ra2: fun(B,fun(B,bool)),X: option(B)] :
      ( ! [X4: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X4),X4))
     => pp(aa(option(B),bool,aa(option(B),fun(option(B),bool),aa(fun(B,fun(B,bool)),fun(option(B),fun(option(B),bool)),rel_option(B,B),Ra2),X),X)) ) ).

% option.rel_refl
tff(fact_6529_option_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),fequal(A)) = fequal(option(A)) ).

% option.rel_eq
tff(fact_6530_option_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),none(A)),none(B))) ).

% option.ctr_transfer(1)
tff(fact_6531_option_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),Y2: A] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),aa(A,option(A),some(A),Y2)),none(B))) ).

% option.rel_distinct(2)
tff(fact_6532_option_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),Y2: B] : ~ pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),none(A)),aa(B,option(B),some(B),Y2))) ).

% option.rel_distinct(1)
tff(fact_6533_option_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
     => ( ( ( A3 = none(A) )
         => ( B2 != none(B) ) )
       => ~ ! [X4: A] :
              ( ( A3 = aa(A,option(A),some(A),X4) )
             => ! [Y4: B] :
                  ( ( B2 = aa(B,option(B),some(B),Y4) )
                 => ~ pp(aa(B,bool,aa(A,fun(B,bool),R4,X4),Y4)) ) ) ) ) ).

% option.rel_cases
tff(fact_6534_option_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: option(A),Y3: option(B),Q: fun(option(A),fun(option(B),bool))] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X),Y3))
     => ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,none(A)),none(B)))
       => ( ! [A25: A,B23: B] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R4,A25),B23))
             => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,aa(A,option(A),some(A),A25)),aa(B,option(B),some(B),B23))) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),Q,X),Y3)) ) ) ) ).

% option.rel_induct
tff(fact_6535_rel__option__inf,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,bool)),B4: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(A),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),inf_inf(fun(option(A),fun(option(B),bool))),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A5)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),B4)) = aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),A5),B4)) ).

% rel_option_inf
tff(fact_6536_rel__option__iff,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: option(A),Y3: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X),Y3))
    <=> pp(aa(product_prod(option(A),option(B)),bool,aa(fun(option(A),fun(option(B),bool)),fun(product_prod(option(A),option(B)),bool),product_case_prod(option(A),option(B),bool),aTP_Lamp_aob(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),R4)),aa(option(B),product_prod(option(A),option(B)),aa(option(A),fun(option(B),product_prod(option(A),option(B))),product_Pair(option(A),option(B)),X),Y3))) ) ).

% rel_option_iff
tff(fact_6537_option__bind__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(option(B),fun(fun(B,option(D)),option(D))),bool,aa(fun(option(A),fun(fun(A,option(C)),option(C))),fun(fun(option(B),fun(fun(B,option(D)),option(D))),bool),bNF_rel_fun(option(A),option(B),fun(fun(A,option(C)),option(C)),fun(fun(B,option(D)),option(D)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A5),bNF_rel_fun(fun(A,option(C)),fun(B,option(D)),option(C),option(D),bNF_rel_fun(A,B,option(C),option(D),A5,aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B4)),aa(fun(C,fun(D,bool)),fun(option(C),fun(option(D),bool)),rel_option(C,D),B4))),bind2(A,C)),bind2(B,D))) ).

% option_bind_transfer
tff(fact_6538_option_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),A3),B2))
    <=> ? [Z5: option(product_prod(A,B))] :
          ( pp(aa(set(option(product_prod(A,B))),bool,member(option(product_prod(A,B)),Z5),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_anu(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R4))))
          & ( aa(option(product_prod(A,B)),option(A),aa(fun(product_prod(A,B),A),fun(option(product_prod(A,B)),option(A)),map_option(product_prod(A,B),A),product_fst(A,B)),Z5) = A3 )
          & ( aa(option(product_prod(A,B)),option(B),aa(fun(product_prod(A,B),B),fun(option(product_prod(A,B)),option(B)),map_option(product_prod(A,B),B),product_snd(A,B)),Z5) = B2 ) ) ) ).

% option.in_rel
tff(fact_6539_listrel__subset,axiom,
    ! [A: $tType,R: 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))),R),product_Sigma(A,A,A5,aTP_Lamp_wx(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,R)),product_Sigma(list(A),list(A),lists(A,A5),aTP_Lamp_aoc(set(A),fun(list(A),set(list(A))),A5)))) ) ).

% listrel_subset
tff(fact_6540_normalize__div,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,normal6383669964737779283malize(A),A3)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) ) ).

% normalize_div
tff(fact_6541_unit__factor__simps_I1_J,axiom,
    unit_f5069060285200089521factor(nat,zero_zero(nat)) = zero_zero(nat) ).

% unit_factor_simps(1)
tff(fact_6542_unit__factor__idem,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : unit_f5069060285200089521factor(A,unit_f5069060285200089521factor(A,A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% unit_factor_idem
tff(fact_6543_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : unit_f5069060285200089521factor(nat,aa(nat,nat,suc,N)) = one_one(nat) ).

% unit_factor_simps(2)
tff(fact_6544_unit__factor__0,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ( unit_f5069060285200089521factor(A,zero_zero(A)) = zero_zero(A) ) ) ).

% unit_factor_0
tff(fact_6545_unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( unit_f5069060285200089521factor(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% unit_factor_eq_0_iff
tff(fact_6546_unit__factor__1,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ( unit_f5069060285200089521factor(A,one_one(A)) = one_one(A) ) ) ).

% unit_factor_1
tff(fact_6547_normalize__mult__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,normal6383669964737779283malize(A),A3)),unit_f5069060285200089521factor(A,A3)) = A3 ) ).

% normalize_mult_unit_factor
tff(fact_6548_unit__factor__mult__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),aa(A,A,normal6383669964737779283malize(A),A3)) = A3 ) ).

% unit_factor_mult_normalize
tff(fact_6549_div__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,normal6383669964737779283malize(A),A3)) = unit_f5069060285200089521factor(A,A3) ) ).

% div_normalize
tff(fact_6550_div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,A3)) = aa(A,A,normal6383669964737779283malize(A),A3) ) ).

% div_unit_factor
tff(fact_6551_inv__unit__factor__eq__0__iff,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,A3)) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inv_unit_factor_eq_0_iff
tff(fact_6552_unit__factor__mult__unit__left,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% unit_factor_mult_unit_left
tff(fact_6553_unit__factor__mult__unit__right,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,B2)),A3) ) ) ) ).

% unit_factor_mult_unit_right
tff(fact_6554_mult__one__div__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),unit_f5069060285200089521factor(A,B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),unit_f5069060285200089521factor(A,B2)) ) ).

% mult_one_div_unit_factor
tff(fact_6555_unit__factor__lcm,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                | ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_lcm
tff(fact_6556_normalize__unit__factor,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,normal6383669964737779283malize(A),unit_f5069060285200089521factor(A,A3)) = one_one(A) ) ) ) ).

% normalize_unit_factor
tff(fact_6557_unit__factor__normalize,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( unit_f5069060285200089521factor(A,aa(A,A,normal6383669964737779283malize(A),A3)) = one_one(A) ) ) ) ).

% unit_factor_normalize
tff(fact_6558_normalize__unit__factor__eqI,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = aa(A,A,normal6383669964737779283malize(A),B2) )
         => ( ( unit_f5069060285200089521factor(A,A3) = unit_f5069060285200089521factor(A,B2) )
           => ( A3 = B2 ) ) ) ) ).

% normalize_unit_factor_eqI
tff(fact_6559_unit__factor__nat__def,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( unit_f5069060285200089521factor(nat,N) = one_one(nat) ) ) ) ).

% unit_factor_nat_def
tff(fact_6560_mult__gcd__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).

% mult_gcd_left
tff(fact_6561_mult__gcd__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).

% mult_gcd_right
tff(fact_6562_gcd__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).

% gcd_mult_distrib
tff(fact_6563_unit__factor__dvd,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A,B2: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),B2)) ) ) ).

% unit_factor_dvd
tff(fact_6564_unit__factor__mult,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [A3: A,B2: A] : unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ).

% unit_factor_mult
tff(fact_6565_unit__factor__self,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),A3)) ) ).

% unit_factor_self
tff(fact_6566_dvd__unit__factor__div,axiom,
    ! [A: $tType] :
      ( normal6328177297339901930cative(A)
     => ! [B2: A,A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),B2),A3))
         => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),unit_f5069060285200089521factor(A,A3)),unit_f5069060285200089521factor(A,B2)) ) ) ) ).

% dvd_unit_factor_div
tff(fact_6567_is__unit__unit__factor,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),A3),one_one(A)))
         => ( unit_f5069060285200089521factor(A,A3) = A3 ) ) ) ).

% is_unit_unit_factor
tff(fact_6568_mult__lcm__left,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [C2: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),unit_f5069060285200089521factor(A,C2)),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ).

% mult_lcm_left
tff(fact_6569_mult__lcm__right,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),unit_f5069060285200089521factor(A,C2)) ) ).

% mult_lcm_right
tff(fact_6570_lcm__mult__distrib,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [K: A,A3: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),A3),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),gcd_lcm(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2))),unit_f5069060285200089521factor(A,K)) ) ).

% lcm_mult_distrib
tff(fact_6571_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_akv(set(A),fun(list(A),bool),A5)) ).

% lists_eq_set
tff(fact_6572_unit__factor__is__unit,axiom,
    ! [A: $tType] :
      ( semido2269285787275462019factor(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),unit_f5069060285200089521factor(A,A3)),one_one(A))) ) ) ).

% unit_factor_is_unit
tff(fact_6573_unit__factor__gcd,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B2: A] :
          ( ( ( ( A3 = zero_zero(A) )
              & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = zero_zero(A) ) )
          & ( ~ ( ( A3 = zero_zero(A) )
                & ( B2 = zero_zero(A) ) )
           => ( unit_f5069060285200089521factor(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B2)) = one_one(A) ) ) ) ) ).

% unit_factor_gcd
tff(fact_6574_coprime__crossproduct_H,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [B2: A,D3: A,A3: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( unit_f5069060285200089521factor(A,B2) = unit_f5069060285200089521factor(A,D3) )
           => ( algebr8660921524188924756oprime(A,A3,B2)
             => ( algebr8660921524188924756oprime(A,C2,D3)
               => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),D3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
                <=> ( ( A3 = C2 )
                    & ( B2 = D3 ) ) ) ) ) ) ) ) ).

% coprime_crossproduct'
tff(fact_6575_unit__factor__Lcm,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( ( gcd_Lcm(A,A5) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A5)) = zero_zero(A) ) )
          & ( ( gcd_Lcm(A,A5) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Lcm(A,A5)) = one_one(A) ) ) ) ) ).

% unit_factor_Lcm
tff(fact_6576_unit__factor__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( ( gcd_Gcd(A,A5) = zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A5)) = zero_zero(A) ) )
          & ( ( gcd_Gcd(A,A5) != zero_zero(A) )
           => ( unit_f5069060285200089521factor(A,gcd_Gcd(A,A5)) = one_one(A) ) ) ) ) ).

% unit_factor_Gcd
tff(fact_6577_normalize__idem__imp__unit__factor__eq,axiom,
    ! [A: $tType] :
      ( normal8620421768224518004emidom(A)
     => ! [A3: A] :
          ( ( aa(A,A,normal6383669964737779283malize(A),A3) = A3 )
         => ( unit_f5069060285200089521factor(A,A3) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A3),zero_zero(A)))) ) ) ) ).

% normalize_idem_imp_unit_factor_eq
tff(fact_6578_unit__factor__Lcm__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Lcm_fin(A),A5)),zero_zero(A)))) ) ).

% unit_factor_Lcm_fin
tff(fact_6579_unit__factor__Gcd__fin,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] : unit_f5069060285200089521factor(A,aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A5)),zero_zero(A)))) ) ).

% unit_factor_Gcd_fin
tff(fact_6580_pred__option__parametric,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,A5,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),A5),fequal(bool))),pred_option(A)),pred_option(B))) ).

% pred_option_parametric
tff(fact_6581_option_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(option(B),bool)),bool,aa(fun(fun(A,bool),fun(option(A),bool)),fun(fun(fun(B,bool),fun(option(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(option(A),bool),fun(option(B),bool),bNF_rel_fun(A,B,bool,bool,R4,fequal(bool)),bNF_rel_fun(option(A),option(B),bool,bool,aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),fequal(bool))),pred_option(A)),pred_option(B))) ).

% option.pred_transfer
tff(fact_6582_option_Opred__inject_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),A3: A] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),aa(A,option(A),some(A),A3)))
    <=> pp(aa(A,bool,P,A3)) ) ).

% option.pred_inject(2)
tff(fact_6583_option_Opred__mono,axiom,
    ! [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(option(A),bool),bool,aa(fun(option(A),bool),fun(fun(option(A),bool),bool),ord_less_eq(fun(option(A),bool)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P)),aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa))) ) ).

% option.pred_mono
tff(fact_6584_option_Opred__inject_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),none(A))) ).

% option.pred_inject(1)
tff(fact_6585_option_Opred__True,axiom,
    ! [A: $tType,X2: option(A)] : pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aTP_Lamp_aq(A,bool)),X2)) ).

% option.pred_True
tff(fact_6586_option_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: option(A),Ya: option(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
       => ( aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X) = aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),G),Ya) ) ) ) ).

% option.map_cong_pred
tff(fact_6587_option_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,bool),X: option(A),Pa: fun(A,bool)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),X)))
           => ( pp(aa(A,bool,P,Z4))
             => pp(aa(A,bool,Pa,Z4)) ) )
       => pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),X)) ) ) ).

% option.pred_mono_strong
tff(fact_6588_option_Opred__cong,axiom,
    ! [A: $tType,X: option(A),Ya: option(A),P: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(option(A),set(A),set_option(A),Ya)))
           => ( pp(aa(A,bool,P,Z4))
            <=> pp(aa(A,bool,Pa,Z4)) ) )
       => ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
        <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),Pa),Ya)) ) ) ) ).

% option.pred_cong
tff(fact_6589_option_Opred__set,axiom,
    ! [A: $tType,P: fun(A,bool),X2: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X2))
    <=> ! [Xa2: A] :
          ( pp(aa(set(A),bool,member(A,Xa2),aa(option(A),set(A),set_option(A),X2)))
         => pp(aa(A,bool,P,Xa2)) ) ) ).

% option.pred_set
tff(fact_6590_option_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: option(A)] :
      ( pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),pred_option(B),Q),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X)))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X)) ) ).

% option.pred_map
tff(fact_6591_ID_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: A,B2: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),bNF_id_bnf(fun(A,fun(B,bool))),R4),A3),B2))
    <=> ? [Z5: product_prod(A,B)] :
          ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Z5),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_aod(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),R4))))
          & ( aa(product_prod(A,B),A,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),A),bNF_id_bnf(fun(product_prod(A,B),A)),product_fst(A,B)),Z5) = A3 )
          & ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),B),bNF_id_bnf(fun(product_prod(A,B),B)),product_snd(A,B)),Z5) = B2 ) ) ) ).

% ID.in_rel
tff(fact_6592_map__le__imp__upd__le,axiom,
    ! [A: $tType,B: $tType,M12: fun(A,option(B)),M23: fun(A,option(B)),X: A,Y3: B] :
      ( map_le(A,B,M12,M23)
     => map_le(A,B,fun_upd(A,option(B),M12,X,none(B)),fun_upd(A,option(B),M23,X,aa(B,option(B),some(B),Y3))) ) ).

% map_le_imp_upd_le
tff(fact_6593_ID_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: A,Ya: A,F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),bNF_id_bnf(fun(A,bool)),aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
       => ( aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),F),X) = aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),G),Ya) ) ) ) ).

% ID.map_cong_pred
tff(fact_6594_map__le__empty,axiom,
    ! [B: $tType,A: $tType,G: fun(A,option(B))] : map_le(A,B,aTP_Lamp_acc(A,option(B)),G) ).

% map_le_empty
tff(fact_6595_ID_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: A] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),bNF_id_bnf(fun(B,bool)),Q),aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),F),X)))
    <=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),bNF_id_bnf(fun(A,bool)),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X)) ) ).

% ID.pred_map
tff(fact_6596_upd__None__map__le,axiom,
    ! [B: $tType,A: $tType,F: fun(A,option(B)),X: A] : map_le(A,B,fun_upd(A,option(B),F,X,none(B)),F) ).

% upd_None_map_le
tff(fact_6597_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),basic_BNF_ctor_rec(fun(C,B),F)),G) = basic_BNF_ctor_rec(fun(A,B),aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),aa(fun(A,A),fun(A,C),aa(fun(A,C),fun(fun(A,A),fun(A,C)),comp(A,C,A),aa(fun(A,C),fun(A,C),aa(fun(C,C),fun(fun(A,C),fun(A,C)),comp(C,C,A),bNF_id_bnf(C)),G)),bNF_id_bnf(A)))) ).

% Basic_BNF_LFPs.ctor_rec_o_map
tff(fact_6598_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : F = basic_BNF_ctor_rec(fun(A,B),aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),bNF_id_bnf(A))) ).

% Basic_BNF_LFPs.ctor_rec_def_alt
tff(fact_6599_ID_Opred__True,axiom,
    ! [A: $tType,X2: A] : pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),bNF_id_bnf(fun(A,bool)),aTP_Lamp_aq(A,bool)),X2)) ).

% ID.pred_True
tff(fact_6600_ctor__rec__unique,axiom,
    ! [B: $tType,A: $tType,G: fun(A,A),F: fun(A,B),S: fun(A,B)] :
      ( ( G = id(A) )
     => ( ( aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),basic_BNF_xtor(A)) = aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),S),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),bNF_id_bnf(A)),G)),bNF_id_bnf(A))) )
       => ( F = basic_BNF_ctor_rec(fun(A,B),S) ) ) ) ).

% ctor_rec_unique
tff(fact_6601_Basic__BNF__LFPs_Octor__rec,axiom,
    ! [B: $tType,A: $tType,G: fun(A,A),F: fun(A,B),X: A] :
      ( ( G = id(A) )
     => ( aa(A,B,basic_BNF_ctor_rec(fun(A,B),F),aa(A,A,basic_BNF_xtor(A),X)) = aa(A,B,F,aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),bNF_id_bnf(A)),G)),bNF_id_bnf(A)),X)) ) ) ).

% Basic_BNF_LFPs.ctor_rec
tff(fact_6602_Basic__BNF__LFPs_OPair__def__alt,axiom,
    ! [A: $tType,B: $tType,X2: A,Xa3: B] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3) = aa(product_prod(A,B),product_prod(A,B),basic_BNF_xtor(product_prod(A,B)),aa(product_prod(A,B),product_prod(A,B),bNF_id_bnf(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa3))) ).

% Basic_BNF_LFPs.Pair_def_alt
tff(fact_6603_xtor__map__unique,axiom,
    ! [B: $tType,A: $tType,U: fun(A,B),F: fun(A,B)] :
      ( ( aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),U),basic_BNF_xtor(A)) = aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),basic_BNF_xtor(B)),F) )
     => ( U = F ) ) ).

% xtor_map_unique
tff(fact_6604_mk__less__def,axiom,
    ! [A: $tType,R4: fun(A,fun(A,bool)),X2: A,Xa3: A] :
      ( partial_mk_less(A,R4,X2,Xa3)
    <=> ( pp(aa(A,bool,aa(A,fun(A,bool),R4,X2),Xa3))
        & ~ pp(aa(A,bool,aa(A,fun(A,bool),R4,Xa3),X2)) ) ) ).

% mk_less_def
tff(fact_6605_img__ord__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C),Ord: fun(C,fun(C,B)),X2: A,Xa3: A] : partial_img_ord(A,C,B,F,Ord,X2,Xa3) = aa(C,B,aa(C,fun(C,B),Ord,aa(A,C,F,X2)),aa(A,C,F,Xa3)) ).

% img_ord_def
tff(fact_6606_and_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => comm_monoid(A,bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% and.comm_monoid_axioms
tff(fact_6607_Literal_Otransfer,axiom,
    pp(aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))))),bool,aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),fun(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))))),bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))),fun(bool,fun(bool,fun(bool,fun(literal,literal)))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(list(char),list(char)))),fun(bool,fun(bool,fun(literal,literal))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(list(char),list(char))),fun(bool,fun(literal,literal)),fequal(bool),bNF_rel_fun(bool,bool,fun(list(char),list(char)),fun(literal,literal),fequal(bool),bNF_rel_fun(list(char),literal,list(char),literal,pcr_literal,pcr_literal)))))))),aTP_Lamp_aoe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))))),literal2)) ).

% Literal.transfer
tff(fact_6608_comm__monoid_Ocomm__neutral,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,A3: A] :
      ( comm_monoid(A,F,Z)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),Z) = A3 ) ) ).

% comm_monoid.comm_neutral
tff(fact_6609_add_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => comm_monoid(A,plus_plus(A),zero_zero(A)) ) ).

% add.comm_monoid_axioms
tff(fact_6610_mult_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => comm_monoid(A,times_times(A),one_one(A)) ) ).

% mult.comm_monoid_axioms
tff(fact_6611_or_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se1065995026697491101ons_or(A),zero_zero(A)) ) ).

% or.comm_monoid_axioms
tff(fact_6612_gcd__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,gcd_gcd(nat),zero_zero(nat)) ).

% gcd_nat.comm_monoid_axioms
tff(fact_6613_xor_Ocomm__monoid__axioms,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => comm_monoid(A,bit_se5824344971392196577ns_xor(A),zero_zero(A)) ) ).

% xor.comm_monoid_axioms
tff(fact_6614_max__nat_Ocomm__monoid__axioms,axiom,
    comm_monoid(nat,ord_max(nat),zero_zero(nat)) ).

% max_nat.comm_monoid_axioms
tff(fact_6615_String_OLiteral__def,axiom,
    literal2 = aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))))),map_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))))),id(bool),map_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))),id(bool),map_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))),id(bool),map_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))),fun(bool,fun(bool,fun(bool,fun(literal,literal)))),id(bool),map_fun(bool,bool,fun(bool,fun(bool,fun(list(char),list(char)))),fun(bool,fun(bool,fun(literal,literal))),id(bool),map_fun(bool,bool,fun(bool,fun(list(char),list(char))),fun(bool,fun(literal,literal)),id(bool),map_fun(bool,bool,fun(list(char),list(char)),fun(literal,literal),id(bool),map_fun(literal,list(char),list(char),literal,explode,abs_literal)))))))),aTP_Lamp_aoe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))))) ).

% String.Literal_def
tff(fact_6616_sum__set__defs_I1_J,axiom,
    ! [A: $tType,B: $tType] : basic_setl(A,B) = aa(fun(B,set(A)),fun(sum_sum(A,B),set(A)),aa(fun(A,set(A)),fun(fun(B,set(A)),fun(sum_sum(A,B),set(A))),sum_case_sum(A,set(A),B),aTP_Lamp_nl(A,set(A))),aTP_Lamp_nr(B,set(A))) ).

% sum_set_defs(1)
tff(fact_6617_o__case__sum,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,H: fun(D,C),F: fun(A,D),G: fun(B,D)] : aa(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C),aa(fun(D,C),fun(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C)),comp(D,C,sum_sum(A,B)),H),aa(fun(B,D),fun(sum_sum(A,B),D),aa(fun(A,D),fun(fun(B,D),fun(sum_sum(A,B),D)),sum_case_sum(A,D,B),F),G)) = aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),H),F)),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),H),G)) ).

% o_case_sum
tff(fact_6618_disjE__realizer,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),X: sum_sum(A,B),R4: fun(C,bool),F: fun(A,C),G: fun(B,C)] :
      ( pp(aa(sum_sum(A,B),bool,aa(fun(B,bool),fun(sum_sum(A,B),bool),aa(fun(A,bool),fun(fun(B,bool),fun(sum_sum(A,B),bool)),sum_case_sum(A,bool,B),P),Q),X))
     => ( ! [P9: A] :
            ( pp(aa(A,bool,P,P9))
           => pp(aa(C,bool,R4,aa(A,C,F,P9))) )
       => ( ! [Q3: B] :
              ( pp(aa(B,bool,Q,Q3))
             => pp(aa(C,bool,R4,aa(B,C,G,Q3))) )
         => pp(aa(C,bool,R4,aa(sum_sum(A,B),C,aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),F),G),X))) ) ) ) ).

% disjE_realizer
tff(fact_6619_case__sum__step_I2_J,axiom,
    ! [E: $tType,A: $tType,C: $tType,B: $tType,F: fun(E,A),F4: fun(B,A),G4: fun(C,A),P2: sum_sum(B,C)] : aa(sum_sum(E,sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(sum_sum(E,sum_sum(B,C)),A),aa(fun(E,A),fun(fun(sum_sum(B,C),A),fun(sum_sum(E,sum_sum(B,C)),A)),sum_case_sum(E,A,sum_sum(B,C)),F),aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F4),G4)),aa(sum_sum(B,C),sum_sum(E,sum_sum(B,C)),sum_Inr(sum_sum(B,C),E),P2)) = aa(sum_sum(B,C),A,aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F4),G4),P2) ).

% case_sum_step(2)
tff(fact_6620_case__sum__step_I1_J,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F4: fun(B,A),G4: fun(C,A),G: fun(D,A),P2: sum_sum(B,C)] : aa(sum_sum(sum_sum(B,C),D),A,aa(fun(D,A),fun(sum_sum(sum_sum(B,C),D),A),aa(fun(sum_sum(B,C),A),fun(fun(D,A),fun(sum_sum(sum_sum(B,C),D),A)),sum_case_sum(sum_sum(B,C),A,D),aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F4),G4)),G),aa(sum_sum(B,C),sum_sum(sum_sum(B,C),D),sum_Inl(sum_sum(B,C),D),P2)) = aa(sum_sum(B,C),A,aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F4),G4),P2) ).

% case_sum_step(1)
tff(fact_6621_case__sum__if,axiom,
    ! [B: $tType,A: $tType,C: $tType,P2: bool,F: fun(B,A),G: fun(C,A),X: B,Y3: C] :
      ( ( pp(P2)
       => ( aa(sum_sum(B,C),A,aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F),G),aa(sum_sum(B,C),sum_sum(B,C),aa(sum_sum(B,C),fun(sum_sum(B,C),sum_sum(B,C)),aa(bool,fun(sum_sum(B,C),fun(sum_sum(B,C),sum_sum(B,C))),if(sum_sum(B,C)),P2),aa(B,sum_sum(B,C),sum_Inl(B,C),X)),aa(C,sum_sum(B,C),sum_Inr(C,B),Y3))) = aa(B,A,F,X) ) )
      & ( ~ pp(P2)
       => ( aa(sum_sum(B,C),A,aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),F),G),aa(sum_sum(B,C),sum_sum(B,C),aa(sum_sum(B,C),fun(sum_sum(B,C),sum_sum(B,C)),aa(bool,fun(sum_sum(B,C),fun(sum_sum(B,C),sum_sum(B,C))),if(sum_sum(B,C)),P2),aa(B,sum_sum(B,C),sum_Inl(B,C),X)),aa(C,sum_sum(B,C),sum_Inr(C,B),Y3))) = aa(C,A,G,Y3) ) ) ) ).

% case_sum_if
tff(fact_6622_case__sum__o__inj_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,B)] : aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),aa(fun(C,B),fun(sum_sum(A,C),B),aa(fun(A,B),fun(fun(C,B),fun(sum_sum(A,C),B)),sum_case_sum(A,B,C),F),G)),sum_Inr(C,A)) = G ).

% case_sum_o_inj(2)
tff(fact_6623_case__sum__o__inj_I1_J,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,B)] : aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),aa(fun(C,B),fun(sum_sum(A,C),B),aa(fun(A,B),fun(fun(C,B),fun(sum_sum(A,C),B)),sum_case_sum(A,B,C),F),G)),sum_Inl(A,C)) = F ).

% case_sum_o_inj(1)
tff(fact_6624_case__sum__expand__Inr,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B),X: sum_sum(A,C)] :
      ( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
     => ( aa(sum_sum(A,C),B,F,X) = aa(sum_sum(A,C),B,aa(fun(C,B),fun(sum_sum(A,C),B),aa(fun(A,B),fun(fun(C,B),fun(sum_sum(A,C),B)),sum_case_sum(A,B,C),G),aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),F),sum_Inr(C,A))),X) ) ) ).

% case_sum_expand_Inr
tff(fact_6625_case__sum__expand__Inr_H,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B),H: fun(C,B)] :
      ( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
     => ( ( H = aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),F),sum_Inr(C,A)) )
      <=> ( aa(fun(C,B),fun(sum_sum(A,C),B),aa(fun(A,B),fun(fun(C,B),fun(sum_sum(A,C),B)),sum_case_sum(A,B,C),G),H) = F ) ) ) ).

% case_sum_expand_Inr'
tff(fact_6626_case__sum__expand__Inr__pointfree,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B)] :
      ( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
     => ( aa(fun(C,B),fun(sum_sum(A,C),B),aa(fun(A,B),fun(fun(C,B),fun(sum_sum(A,C),B)),sum_case_sum(A,B,C),G),aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),F),sum_Inr(C,A))) = F ) ) ).

% case_sum_expand_Inr_pointfree
tff(fact_6627_sum__set__defs_I2_J,axiom,
    ! [C: $tType,D: $tType] : basic_setr(C,D) = aa(fun(D,set(D)),fun(sum_sum(C,D),set(D)),aa(fun(C,set(D)),fun(fun(D,set(D)),fun(sum_sum(C,D),set(D))),sum_case_sum(C,set(D),D),aTP_Lamp_aof(C,set(D))),aTP_Lamp_aog(D,set(D))) ).

% sum_set_defs(2)
tff(fact_6628_case__sum__KK,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: C,X2: sum_sum(A,B)] : aa(sum_sum(A,B),C,aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),aTP_Lamp_aoh(C,fun(A,C),A3)),aTP_Lamp_aoi(C,fun(B,C),A3)),X2) = A3 ).

% case_sum_KK
tff(fact_6629_surjective__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(sum_sum(A,B),C)] : aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),aTP_Lamp_aoj(fun(sum_sum(A,B),C),fun(A,C),F)),aTP_Lamp_aok(fun(sum_sum(A,B),C),fun(B,C),F)) = F ).

% surjective_sum
tff(fact_6630_map__option__o__case__sum,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(D,C),G: fun(A,option(D)),H: fun(B,option(D))] : aa(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C)),aa(fun(option(D),option(C)),fun(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C))),comp(option(D),option(C),sum_sum(A,B)),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),aa(fun(B,option(D)),fun(sum_sum(A,B),option(D)),aa(fun(A,option(D)),fun(fun(B,option(D)),fun(sum_sum(A,B),option(D))),sum_case_sum(A,option(D),B),G),H)) = aa(fun(B,option(C)),fun(sum_sum(A,B),option(C)),aa(fun(A,option(C)),fun(fun(B,option(C)),fun(sum_sum(A,B),option(C))),sum_case_sum(A,option(C),B),aa(fun(A,option(D)),fun(A,option(C)),aa(fun(option(D),option(C)),fun(fun(A,option(D)),fun(A,option(C))),comp(option(D),option(C),A),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),G)),aa(fun(B,option(D)),fun(B,option(C)),aa(fun(option(D),option(C)),fun(fun(B,option(D)),fun(B,option(C))),comp(option(D),option(C),B),aa(fun(D,C),fun(option(D),option(C)),map_option(D,C),F)),H)) ).

% map_option_o_case_sum
tff(fact_6631_sum_Ocase__distrib,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F1: fun(A,C),F22: fun(B,C),Sum: sum_sum(A,B)] : aa(C,D,H,aa(sum_sum(A,B),C,aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),F1),F22),Sum)) = aa(sum_sum(A,B),D,aa(fun(B,D),fun(sum_sum(A,B),D),aa(fun(A,D),fun(fun(B,D),fun(sum_sum(A,B),D)),sum_case_sum(A,D,B),aa(fun(A,C),fun(A,D),aTP_Lamp_aol(fun(C,D),fun(fun(A,C),fun(A,D)),H),F1)),aa(fun(B,C),fun(B,D),aTP_Lamp_aom(fun(C,D),fun(fun(B,C),fun(B,D)),H),F22)),Sum) ).

% sum.case_distrib
tff(fact_6632_sum__encode__def,axiom,
    ! [X: sum_sum(nat,nat)] : nat_sum_encode(X) = aa(sum_sum(nat,nat),nat,aa(fun(nat,nat),fun(sum_sum(nat,nat),nat),aa(fun(nat,nat),fun(fun(nat,nat),fun(sum_sum(nat,nat),nat)),sum_case_sum(nat,nat,nat),aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aTP_Lamp_aon(nat,nat)),X) ).

% sum_encode_def
tff(fact_6633_projl__def,axiom,
    ! [B: $tType,A: $tType,Sum: sum_sum(A,B)] : sum_projl(A,B,Sum) = aa(sum_sum(A,B),A,aa(fun(B,A),fun(sum_sum(A,B),A),aa(fun(A,A),fun(fun(B,A),fun(sum_sum(A,B),A)),sum_case_sum(A,A,B),aTP_Lamp_ak(A,A)),aTP_Lamp_aoo(B,A)),Sum) ).

% projl_def
tff(fact_6634_projr__def,axiom,
    ! [B: $tType,A: $tType,Sum: sum_sum(A,B)] : sum_projr(A,B,Sum) = aa(sum_sum(A,B),B,aa(fun(B,B),fun(sum_sum(A,B),B),aa(fun(A,B),fun(fun(B,B),fun(sum_sum(A,B),B)),sum_case_sum(A,B,B),aTP_Lamp_zn(A,B)),aTP_Lamp_lf(B,B)),Sum) ).

% projr_def
tff(fact_6635_old_Orec__sum__def,axiom,
    ! [T: $tType,B: $tType,A: $tType,X2: fun(A,T),Xa3: fun(B,T),Xb: sum_sum(A,B)] : sum_rec_sum(A,T,B,X2,Xa3,Xb) = the(T,sum_rec_set_sum(A,T,B,X2,Xa3,Xb)) ).

% old.rec_sum_def
tff(fact_6636_nat__of__num__code_I3_J,axiom,
    ! [N: num] : nat_of_num(aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(N)),nat_of_num(N))) ).

% nat_of_num_code(3)
tff(fact_6637_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_vw(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6638_nat__of__num__code_I1_J,axiom,
    nat_of_num(one2) = one_one(nat) ).

% nat_of_num_code(1)
tff(fact_6639_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] : nat_of_num(bit0(X)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X)),nat_of_num(X)) ).

% nat_of_num.simps(2)
tff(fact_6640_nat__of__num__neq__0,axiom,
    ! [X: num] : nat_of_num(X) != zero_zero(nat) ).

% nat_of_num_neq_0
tff(fact_6641_nat__of__num__pos,axiom,
    ! [X: num] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat_of_num(X))) ).

% nat_of_num_pos
tff(fact_6642_folding__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S2,F)
    <=> ! [X3: A,Y: A] :
          ( pp(aa(set(A),bool,member(A,X3),S2))
         => ( pp(aa(set(A),bool,member(A,Y),S2))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X3)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X3)),aa(A,fun(B,B),F,Y)) ) ) ) ) ).

% folding_on_def
tff(fact_6643_folding__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B)),X: A,Y3: A] :
      ( finite_folding_on(A,B,S2,F)
     => ( pp(aa(set(A),bool,member(A,X),S2))
       => ( pp(aa(set(A),bool,member(A,Y3),S2))
         => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y3)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y3)) ) ) ) ) ).

% folding_on.comp_fun_commute_on
tff(fact_6644_folding__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F: fun(A,fun(B,B))] :
      ( ! [X4: A,Y4: A] :
          ( pp(aa(set(A),bool,member(A,X4),S2))
         => ( pp(aa(set(A),bool,member(A,Y4),S2))
           => ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X4)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X4)),aa(A,fun(B,B),F,Y4)) ) ) )
     => finite_folding_on(A,B,S2,F) ) ).

% folding_on.intro
tff(fact_6645_nat__of__num__add,axiom,
    ! [X: num,Y3: num] : nat_of_num(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X)),nat_of_num(Y3)) ).

% nat_of_num_add
tff(fact_6646_nat__of__num__code_I2_J,axiom,
    ! [N: num] : nat_of_num(bit0(N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(N)),nat_of_num(N)) ).

% nat_of_num_code(2)
tff(fact_6647_nat__of__num_Osimps_I1_J,axiom,
    nat_of_num(one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_of_num.simps(1)
tff(fact_6648_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] : nat_of_num(aa(num,num,bit1,X)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(X)),nat_of_num(X))) ).

% nat_of_num.simps(3)
tff(fact_6649_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( nat_of_num(aa(nat,num,num_of_nat,N)) = N ) ) ).

% num_of_nat_inverse
tff(fact_6650_plus__num__def,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N) = aa(nat,num,num_of_nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_of_num(M)),nat_of_num(N))) ).

% plus_num_def
tff(fact_6651_snds__def,axiom,
    ! [B: $tType,A: $tType,X2: product_prod(A,B)] : basic_snds(A,B,X2) = aa(fun(B,bool),set(B),collect(B),basic_sndsp(A,B,X2)) ).

% snds_def
tff(fact_6652_setr__def,axiom,
    ! [B: $tType,A: $tType,X2: sum_sum(A,B)] : aa(sum_sum(A,B),set(B),basic_setr(A,B),X2) = aa(fun(B,bool),set(B),collect(B),basic_setrp(A,B,X2)) ).

% setr_def
tff(fact_6653_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_ro(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6654_setl__def,axiom,
    ! [B: $tType,A: $tType,X2: sum_sum(A,B)] : aa(sum_sum(A,B),set(A),basic_setl(A,B),X2) = aa(fun(A,bool),set(A),collect(A),basic_setlp(A,B,X2)) ).

% setl_def
tff(fact_6655_fsts__def,axiom,
    ! [B: $tType,A: $tType,X2: product_prod(A,B)] : aa(product_prod(A,B),set(A),basic_fsts(A,B),X2) = aa(fun(A,bool),set(A),collect(A),basic_fstsp(A,B,X2)) ).

% fsts_def
tff(fact_6656_prod_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups4802862169904069756st_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_list_set_axioms
tff(fact_6657_prod_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ( groups1962203154675924110t_prod(C,A) = groups_comm_monoid_G(A,C,times_times(A),one_one(A)) ) ) ).

% prod'_def
tff(fact_6658_sum_H__def,axiom,
    ! [C: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ( groups1027152243600224163dd_sum(C,A) = groups_comm_monoid_G(A,C,plus_plus(A),zero_zero(A)) ) ) ).

% sum'_def
tff(fact_6659_sum_Ocomm__monoid__list__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups4802862169904069756st_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_list_set_axioms
tff(fact_6660_Heap_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),nat),aa(fun(heap_Time_Heap(B),nat),fun(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),nat)),comp(heap_Time_Heap(B),nat,heap_Time_Heap(A)),heap_Time_size_Heap(B,F)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G)) = heap_Time_size_Heap(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).

% Heap.size_gen_o_map
tff(fact_6661_max_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => semilattice_order(A,ord_max(A),aTP_Lamp_lb(A,fun(A,bool)),aTP_Lamp_aji(A,fun(A,bool))) ) ).

% max.semilattice_order_axioms
tff(fact_6662_Heap_Omap__ident,axiom,
    ! [A: $tType,T2: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),aTP_Lamp_ak(A,A)),T2) = T2 ).

% Heap.map_ident
tff(fact_6663_Heap_Omap__id,axiom,
    ! [A: $tType,T2: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),heap_Time_Heap(A),aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),id(A)),T2) = T2 ).

% Heap.map_id
tff(fact_6664_Heap_Omap__id0,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(heap_Time_Heap(A),heap_Time_Heap(A)),heap_Time_map_Heap(A,A),id(A)) = id(heap_Time_Heap(A)) ).

% Heap.map_id0
tff(fact_6665_Heap_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: heap_Time_Heap(A)] : aa(heap_Time_Heap(B),heap_Time_Heap(C),aa(fun(B,C),fun(heap_Time_Heap(B),heap_Time_Heap(C)),heap_Time_map_Heap(B,C),G),aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),V)) = aa(heap_Time_Heap(A),heap_Time_Heap(C),aa(fun(A,C),fun(heap_Time_Heap(A),heap_Time_Heap(C)),heap_Time_map_Heap(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).

% Heap.map_comp
tff(fact_6666_Heap_Omap,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),map_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_map_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),F,id(product_prod(heap_ext(product_unit),nat))))),X)) ).

% Heap.map
tff(fact_6667_Heap_Oinj__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),top_top(set(heap_Time_Heap(A)))) ) ).

% Heap.inj_map
tff(fact_6668_gcd__nat_Osemilattice__order__axioms,axiom,
    semilattice_order(nat,gcd_gcd(nat),dvd_dvd(nat),aTP_Lamp_jg(nat,fun(nat,bool))) ).

% gcd_nat.semilattice_order_axioms
tff(fact_6669_sup_Osemilattice__order__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => semilattice_order(A,sup_sup(A),aTP_Lamp_ajv(A,fun(A,bool)),aTP_Lamp_ajw(A,fun(A,bool))) ) ).

% sup.semilattice_order_axioms
tff(fact_6670_Heap_Orec__o__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),F: fun(A,B)] : aa(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),C),aa(fun(heap_Time_Heap(B),C),fun(fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(A),C)),comp(heap_Time_Heap(B),C,heap_Time_Heap(A)),aa(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(B),C),heap_Time_rec_Heap(B,C),G)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F)) = aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_rec_Heap(A,C),aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_aop(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),G),F)) ).

% Heap.rec_o_map
tff(fact_6671_Heap_Oset,axiom,
    ! [A: $tType,X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),set(set(A)),image2(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A),aTP_Lamp_aoq(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A))),aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X),top_top(set(heap_ext(product_unit)))))) ).

% Heap.set
tff(fact_6672_Heap_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),Ya)))
           => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
       => ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),Ya) ) ) ) ).

% Heap.map_cong
tff(fact_6673_Heap_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [Z4: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X)))
         => ( aa(A,B,F,Z4) = aa(A,B,G,Z4) ) )
     => ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),X) ) ) ).

% Heap.map_cong0
tff(fact_6674_Heap_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: heap_Time_Heap(A),Xa: heap_Time_Heap(A),F: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z4: A,Za: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X)))
         => ( pp(aa(set(A),bool,member(A,Za),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),Xa)))
           => ( ( aa(A,B,F,Z4) = aa(A,B,Fa,Za) )
             => ( Z4 = Za ) ) ) )
     => ( ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),Fa),Xa) )
       => ( X = Xa ) ) ) ).

% Heap.inj_map_strong
tff(fact_6675_Heap_Oset__cases,axiom,
    ! [A: $tType,E2: A,A3: heap_Time_Heap(A)] :
      ( pp(aa(set(A),bool,member(A,E2),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),A3)))
     => ~ ! [Z4: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
            ( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Z4) )
           => ! [Xa4: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
                ( pp(aa(set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,member(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa4),aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Z4),top_top(set(heap_ext(product_unit))))))
               => ! [Xb4: product_prod(A,product_prod(heap_ext(product_unit),nat))] :
                    ( pp(aa(set(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool,member(product_prod(A,product_prod(heap_ext(product_unit),nat)),Xb4),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Xa4)))
                   => ~ pp(aa(set(A),bool,member(A,E2),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat)),Xb4))) ) ) ) ) ).

% Heap.set_cases
tff(fact_6676_Heap_Oset__intros,axiom,
    ! [A: $tType,Y3: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Ya: product_prod(A,product_prod(heap_ext(product_unit),nat)),Yb2: A] :
      ( pp(aa(set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,member(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y3),aa(set(heap_ext(product_unit)),set(option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),image2(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),X),top_top(set(heap_ext(product_unit))))))
     => ( pp(aa(set(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool,member(product_prod(A,product_prod(heap_ext(product_unit),nat)),Ya),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y3)))
       => ( pp(aa(set(A),bool,member(A,Yb2),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat)),Ya)))
         => pp(aa(set(A),bool,member(A,Yb2),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)))) ) ) ) ).

% Heap.set_intros
tff(fact_6677_Heap_Orec,axiom,
    ! [C: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_rec_Heap(A,C),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,F,X) ).

% Heap.rec
tff(fact_6678_Heap_Oset__map,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),V: heap_Time_Heap(A)] : aa(heap_Time_Heap(B),set(B),heap_Time_set_Heap(B),aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),V)) = aa(set(A),set(B),image2(A,B,F),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),V)) ).

% Heap.set_map
tff(fact_6679_Heap_Orel__Grp,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),bNF_Grp(A,B,A5,F)) = bNF_Grp(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(heap_Time_Heap(A),bool),set(heap_Time_Heap(A)),collect(heap_Time_Heap(A)),aTP_Lamp_aor(set(A),fun(heap_Time_Heap(A),bool),A5)),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F)) ).

% Heap.rel_Grp
tff(fact_6680_Heap_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(A),B2: heap_Time_Heap(B)] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),A3),B2))
    <=> ? [Z5: heap_Time_Heap(product_prod(A,B))] :
          ( pp(aa(set(heap_Time_Heap(product_prod(A,B))),bool,member(heap_Time_Heap(product_prod(A,B)),Z5),aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_aos(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4))))
          & ( aa(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),aa(fun(product_prod(A,B),A),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A)),heap_Time_map_Heap(product_prod(A,B),A),product_fst(A,B)),Z5) = A3 )
          & ( aa(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B),aa(fun(product_prod(A,B),B),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B)),heap_Time_map_Heap(product_prod(A,B),B),product_snd(A,B)),Z5) = B2 ) ) ) ).

% Heap.in_rel
tff(fact_6681_Heap_Orel__refl__strong,axiom,
    ! [A: $tType,X: heap_Time_Heap(A),Ra2: fun(A,fun(A,bool))] :
      ( ! [Z4: A] :
          ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra2,Z4),Z4)) )
     => pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),Ra2),X),X)) ) ).

% Heap.rel_refl_strong
tff(fact_6682_Heap_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: heap_Time_Heap(A),Y3: heap_Time_Heap(B),Ra2: fun(A,fun(B,bool))] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X),Y3))
     => ( ! [Z4: A,Yb: B] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X)))
           => ( pp(aa(set(B),bool,member(B,Yb),aa(heap_Time_Heap(B),set(B),heap_Time_set_Heap(B),Y3)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z4),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z4),Yb)) ) ) )
       => pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2),X),Y3)) ) ) ).

% Heap.rel_mono_strong
tff(fact_6683_Heap_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),Y3: heap_Time_Heap(B),Xa: heap_Time_Heap(B),R4: fun(A,fun(B,bool)),Ra2: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y3 = Xa )
       => ( ! [Z4: A,Yb: B] :
              ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),Ya)))
             => ( pp(aa(set(B),bool,member(B,Yb),aa(heap_Time_Heap(B),set(B),heap_Time_set_Heap(B),Xa)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R4,Z4),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra2,Z4),Yb)) ) ) )
         => ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X),Y3))
          <=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2),Ya),Xa)) ) ) ) ) ).

% Heap.rel_cong
tff(fact_6684_Heap_Orec__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(A),C),fun(heap_Time_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,D,bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),S2),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),S2)),heap_Time_rec_Heap(A,C)),heap_Time_rec_Heap(B,D))) ).

% Heap.rec_transfer
tff(fact_6685_Heap_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(A),B2: heap_Time_Heap(B)] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),A3),B2))
     => ~ ! [Xa4: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
            ( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Xa4) )
           => ! [Ya2: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
                ( ( B2 = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Ya2) )
               => ~ pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),Xa4),Ya2)) ) ) ) ).

% Heap.rel_cases
tff(fact_6686_Heap_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: heap_Time_Heap(A),Y3: heap_Time_Heap(B),Q: fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),X),Y3))
     => ( ! [A4: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B3: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
            ( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),A4),B3))
           => pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),Q,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),A4)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),B3))) )
       => pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),Q,X),Y3)) ) ) ).

% Heap.rel_induct
tff(fact_6687_Heap_Orel__inject,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Y3: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Y3)))
    <=> pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),X),Y3)) ) ).

% Heap.rel_inject
tff(fact_6688_Heap_Orel__intros,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Y3: fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))] :
      ( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),bool),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),X),Y3))
     => pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)),aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),Y3))) ) ).

% Heap.rel_intros
tff(fact_6689_Heap_Octr__transfer,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B)),bool,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A)),fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B)),bool),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap(B),bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)),heap_Time_Heap2(A)),heap_Time_Heap2(B))) ).

% Heap.ctr_transfer
tff(fact_6690_Heap_Orel__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(C),fun(heap_Time_Heap(B),bool),fun(heap_Time_Heap(D),bool),aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),Sa),bNF_rel_fun(heap_Time_Heap(B),heap_Time_Heap(D),bool,bool,aa(fun(B,fun(D,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(D),bool)),heap_Time_rel_Heap(B,D),Sc),fequal(bool)))),heap_Time_rel_Heap(A,B)),heap_Time_rel_Heap(C,D))) ).

% Heap.rel_transfer
tff(fact_6691_Heap_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R4: 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))),R4),Ra2))
     => pp(aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),bool,aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),bool),ord_less_eq(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),Ra2))) ) ).

% Heap.rel_mono
tff(fact_6692_Heap_Orel__compp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),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),R4),S2)) = aa(fun(heap_Time_Heap(B),fun(heap_Time_Heap(C),bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(fun(heap_Time_Heap(B),fun(heap_Time_Heap(C),bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool))),relcompp(heap_Time_Heap(A),heap_Time_Heap(B),heap_Time_Heap(C)),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)),aa(fun(B,fun(C,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(B,C),S2)) ).

% Heap.rel_compp
tff(fact_6693_Heap_Orel__conversep,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(B,A),conversep(A,B,R4)) = conversep(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ).

% Heap.rel_conversep
tff(fact_6694_Heap_Orel__flip,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),A3: heap_Time_Heap(B),B2: heap_Time_Heap(A)] :
      ( pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool),aa(fun(B,fun(A,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(B,A),conversep(A,B,R4)),A3),B2))
    <=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),B2),A3)) ) ).

% Heap.rel_flip
tff(fact_6695_Heap_Orel__eq,axiom,
    ! [A: $tType] : aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),fequal(A)) = fequal(heap_Time_Heap(A)) ).

% Heap.rel_eq
tff(fact_6696_Heap_Orel__refl,axiom,
    ! [B: $tType,Ra2: fun(B,fun(B,bool)),X: heap_Time_Heap(B)] :
      ( ! [X4: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra2,X4),X4))
     => pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(B),fun(heap_Time_Heap(B),bool),aa(fun(B,fun(B,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(B,B),Ra2),X),X)) ) ).

% Heap.rel_refl
tff(fact_6697_Heap_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X: heap_Time_Heap(A),Y3: heap_Time_Heap(B)] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(C),fun(heap_Time_Heap(B),bool),aa(fun(C,fun(B,bool)),fun(heap_Time_Heap(C),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(C,B),Sb),aa(heap_Time_Heap(A),heap_Time_Heap(C),aa(fun(A,C),fun(heap_Time_Heap(A),heap_Time_Heap(C)),heap_Time_map_Heap(A,C),I),X)),Y3))
    <=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_su(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X),Y3)) ) ).

% Heap.rel_map(1)
tff(fact_6698_Heap_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: heap_Time_Heap(A),G: fun(B,C),Y3: heap_Time_Heap(B)] :
      ( pp(aa(heap_Time_Heap(C),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool),aa(fun(A,fun(C,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(C),bool)),heap_Time_rel_Heap(A,C),Sa),X),aa(heap_Time_Heap(B),heap_Time_Heap(C),aa(fun(B,C),fun(heap_Time_Heap(B),heap_Time_Heap(C)),heap_Time_map_Heap(B,C),G),Y3)))
    <=> pp(aa(heap_Time_Heap(B),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_st(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y3)) ) ).

% Heap.rel_map(2)
tff(fact_6699_Heap_OQuotient,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
     => quotient(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),R4),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),Abs),aa(fun(B,A),fun(heap_Time_Heap(B),heap_Time_Heap(A)),heap_Time_map_Heap(B,A),Rep),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),T4)) ) ).

% Heap.Quotient
tff(fact_6700_Heap_Omap__transfer,axiom,
    ! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb2: fun(A,fun(E,bool)),Sd: fun(B,fun(F2,bool))] : pp(aa(fun(fun(E,F2),fun(heap_Time_Heap(E),heap_Time_Heap(F2))),bool,aa(fun(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B))),fun(fun(fun(E,F2),fun(heap_Time_Heap(E),heap_Time_Heap(F2))),bool),bNF_rel_fun(fun(A,B),fun(E,F2),fun(heap_Time_Heap(A),heap_Time_Heap(B)),fun(heap_Time_Heap(E),heap_Time_Heap(F2)),bNF_rel_fun(A,E,B,F2,Rb2,Sd),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(E),heap_Time_Heap(B),heap_Time_Heap(F2),aa(fun(A,fun(E,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(E),bool)),heap_Time_rel_Heap(A,E),Rb2),aa(fun(B,fun(F2,bool)),fun(heap_Time_Heap(B),fun(heap_Time_Heap(F2),bool)),heap_Time_rel_Heap(B,F2),Sd))),heap_Time_map_Heap(A,B)),heap_Time_map_Heap(E,F2))) ).

% Heap.map_transfer
tff(fact_6701_Heap_Orel__compp__Grp,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4) = aa(fun(heap_Time_Heap(product_prod(A,B)),fun(heap_Time_Heap(B),bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(product_prod(A,B)),bool)),fun(fun(heap_Time_Heap(product_prod(A,B)),fun(heap_Time_Heap(B),bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool))),relcompp(heap_Time_Heap(A),heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B)),conversep(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),bNF_Grp(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A),aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_aos(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),A),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(A)),heap_Time_map_Heap(product_prod(A,B),A),product_fst(A,B))))),bNF_Grp(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B),aa(fun(heap_Time_Heap(product_prod(A,B)),bool),set(heap_Time_Heap(product_prod(A,B))),collect(heap_Time_Heap(product_prod(A,B))),aTP_Lamp_aos(fun(A,fun(B,bool)),fun(heap_Time_Heap(product_prod(A,B)),bool),R4)),aa(fun(product_prod(A,B),B),fun(heap_Time_Heap(product_prod(A,B)),heap_Time_Heap(B)),heap_Time_map_Heap(product_prod(A,B),B),product_snd(A,B)))) ).

% Heap.rel_compp_Grp
tff(fact_6702_those_Osimps_I2_J,axiom,
    ! [A: $tType,X: option(A),Xs: list(option(A))] : those(A,aa(list(option(A)),list(option(A)),aa(option(A),fun(list(option(A)),list(option(A))),cons(option(A)),X),Xs)) = aa(option(A),option(list(A)),aa(fun(A,option(list(A))),fun(option(A),option(list(A))),aa(option(list(A)),fun(fun(A,option(list(A))),fun(option(A),option(list(A)))),case_option(option(list(A)),A),none(list(A))),aTP_Lamp_aot(list(option(A)),fun(A,option(list(A))),Xs)),X) ).

% those.simps(2)
tff(fact_6703_bdd__below__primitive__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ( condit1013018076250108175_below(A) = condit16957441358409770ng_bdd(A,aTP_Lamp_aou(A,fun(A,bool))) ) ) ).

% bdd_below_primitive_def
tff(fact_6704_Heap_Ocase__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R4: fun(A,fun(B,bool)),S2: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),D),fun(heap_Time_Heap(A),C),fun(heap_Time_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,D,bNF_rel_fun(heap_ext(product_unit),heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fequal(heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),fun(product_prod(B,product_prod(heap_ext(product_unit),nat)),bool)),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),bool)),rel_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),basic_rel_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),R4,fequal(product_prod(heap_ext(product_unit),nat))))),S2),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),S2)),heap_Time_case_Heap(A,C)),heap_Time_case_Heap(B,D))) ).

% Heap.case_transfer
tff(fact_6705_Heap_Opred__transfer,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(heap_Time_Heap(B),bool)),bool,aa(fun(fun(A,bool),fun(heap_Time_Heap(A),bool)),fun(fun(fun(B,bool),fun(heap_Time_Heap(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(heap_Time_Heap(A),bool),fun(heap_Time_Heap(B),bool),bNF_rel_fun(A,B,bool,bool,R4,fequal(bool)),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),bool,bool,aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),fequal(bool))),heap_Time_pred_Heap(A)),heap_Time_pred_Heap(B))) ).

% Heap.pred_transfer
tff(fact_6706_Heap_Opred__mono,axiom,
    ! [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(heap_Time_Heap(A),bool),bool,aa(fun(heap_Time_Heap(A),bool),fun(fun(heap_Time_Heap(A),bool),bool),ord_less_eq(fun(heap_Time_Heap(A),bool)),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P)),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa))) ) ).

% Heap.pred_mono
tff(fact_6707_Heap_Opred__True,axiom,
    ! [A: $tType,X2: heap_Time_Heap(A)] : pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aTP_Lamp_aq(A,bool)),X2)) ).

% Heap.pred_True
tff(fact_6708_Heap_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),Heap: heap_Time_Heap(A)] : aa(B,C,H,aa(heap_Time_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B),heap_Time_case_Heap(A,B),F),Heap)) = aa(heap_Time_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),fun(heap_Time_Heap(A),C),heap_Time_case_Heap(A,C),aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_aov(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),H),F)),Heap) ).

% Heap.case_distrib
tff(fact_6709_Heap_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
       => ( aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X) = aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),G),Ya) ) ) ) ).

% Heap.map_cong_pred
tff(fact_6710_Heap_Opred__cong,axiom,
    ! [A: $tType,X: heap_Time_Heap(A),Ya: heap_Time_Heap(A),P: fun(A,bool),Pa: fun(A,bool)] :
      ( ( X = Ya )
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),Ya)))
           => ( pp(aa(A,bool,P,Z4))
            <=> pp(aa(A,bool,Pa,Z4)) ) )
       => ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X))
        <=> pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa),Ya)) ) ) ) ).

% Heap.pred_cong
tff(fact_6711_Heap_Opred__mono__strong,axiom,
    ! [A: $tType,P: fun(A,bool),X: heap_Time_Heap(A),Pa: fun(A,bool)] :
      ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X))
     => ( ! [Z4: A] :
            ( pp(aa(set(A),bool,member(A,Z4),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X)))
           => ( pp(aa(A,bool,P,Z4))
             => pp(aa(A,bool,Pa,Z4)) ) )
       => pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),Pa),X)) ) ) ).

% Heap.pred_mono_strong
tff(fact_6712_Heap_Ocase,axiom,
    ! [B: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),X: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(heap_Time_Heap(A),B),heap_Time_case_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),X)) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B,F,X) ).

% Heap.case
tff(fact_6713_Heap_Opred__set,axiom,
    ! [A: $tType,P: fun(A,bool),X2: heap_Time_Heap(A)] :
      ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X2))
    <=> ! [Xa2: A] :
          ( pp(aa(set(A),bool,member(A,Xa2),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),X2)))
         => pp(aa(A,bool,P,Xa2)) ) ) ).

% Heap.pred_set
tff(fact_6714_Heap_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: heap_Time_Heap(A)] :
      ( pp(aa(heap_Time_Heap(B),bool,aa(fun(B,bool),fun(heap_Time_Heap(B),bool),heap_Time_pred_Heap(B),Q),aa(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,B),fun(heap_Time_Heap(A),heap_Time_Heap(B)),heap_Time_map_Heap(A,B),F),X)))
    <=> pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X)) ) ).

% Heap.pred_map
tff(fact_6715_Heap_Opred__inject,axiom,
    ! [A: $tType,P: fun(A,bool),A3: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),A3)))
    <=> pp(aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool,aa(fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),bool),basic_pred_fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aow(heap_ext(product_unit),bool)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),bool),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),bool),pred_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),basic_pred_prod(A,product_prod(heap_ext(product_unit),nat),P,top_top(fun(product_prod(heap_ext(product_unit),nat),bool))))),A3)) ) ).

% Heap.pred_inject
tff(fact_6716_bdd__below_Opreordering__bdd__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit622319405099724424ng_bdd(A,aTP_Lamp_aou(A,fun(A,bool)),aTP_Lamp_afd(A,fun(A,bool))) ) ).

% bdd_below.preordering_bdd_axioms
tff(fact_6717_pred__prod__inject,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: A,B2: B] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
    <=> ( pp(aa(A,bool,P12,A3))
        & pp(aa(B,bool,P23,B2)) ) ) ).

% pred_prod_inject
tff(fact_6718_prod_Opred__True,axiom,
    ! [B: $tType,A: $tType,X2: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aTP_Lamp_aq(A,bool),aTP_Lamp_aox(B,bool)),X2)) ).

% prod.pred_True
tff(fact_6719_pred__prod_Ointros,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,bool),A3: A,P23: fun(B,bool),B2: B] :
      ( pp(aa(A,bool,P12,A3))
     => ( pp(aa(B,bool,P23,B2))
       => pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ) ).

% pred_prod.intros
tff(fact_6720_pred__prod_Osimps,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),A3))
    <=> ? [A7: A,B7: B] :
          ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
          & pp(aa(A,bool,P12,A7))
          & pp(aa(B,bool,P23,B7)) ) ) ).

% pred_prod.simps
tff(fact_6721_pred__prod_Ocases,axiom,
    ! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),A3))
     => ~ ! [A4: A,B3: B] :
            ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
           => ( pp(aa(A,bool,P12,A4))
             => ~ pp(aa(B,bool,P23,B3)) ) ) ) ).

% pred_prod.cases
tff(fact_6722_pred__prod__split,axiom,
    ! [A: $tType,B: $tType,P: fun(bool,bool),Q: fun(A,bool),R4: fun(B,bool),Xy: product_prod(A,B)] :
      ( pp(aa(bool,bool,P,aa(product_prod(A,B),bool,basic_pred_prod(A,B,Q,R4),Xy)))
    <=> ! [X3: A,Y: B] :
          ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
         => pp(aa(bool,bool,P,fconj(aa(A,bool,Q,X3),aa(B,bool,R4,Y)))) ) ) ).

% pred_prod_split
tff(fact_6723_prod_Omap__cong__pred,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod(A,B),Ya: product_prod(A,B),F1: fun(A,C),G1: fun(A,C),F22: fun(B,D),G22: fun(B,D)] :
      ( ( X = Ya )
     => ( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aa(fun(A,C),fun(A,bool),aTP_Lamp_aoy(fun(A,C),fun(fun(A,C),fun(A,bool)),F1),G1),aa(fun(B,D),fun(B,bool),aTP_Lamp_aoz(fun(B,D),fun(fun(B,D),fun(B,bool)),F22),G22)),Ya))
       => ( aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),X) = aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,G1,G22),Ya) ) ) ) ).

% prod.map_cong_pred
tff(fact_6724_top__conj_I2_J,axiom,
    ! [A: $tType,P: bool,X: A] :
      ( ( pp(P)
        & pp(aa(A,bool,top_top(fun(A,bool)),X)) )
    <=> pp(P) ) ).

% top_conj(2)
tff(fact_6725_top__conj_I1_J,axiom,
    ! [A: $tType,X: A,P: bool] :
      ( ( pp(aa(A,bool,top_top(fun(A,bool)),X))
        & pp(P) )
    <=> pp(P) ) ).

% top_conj(1)
tff(fact_6726_prod_Opred__map,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Q12: fun(C,bool),Q23: fun(D,bool),F1: fun(A,C),F22: fun(B,D),X: product_prod(A,B)] :
      ( pp(aa(product_prod(C,D),bool,basic_pred_prod(C,D,Q12,Q23),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),X)))
    <=> pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aa(fun(A,C),fun(A,bool),aa(fun(C,bool),fun(fun(A,C),fun(A,bool)),comp(C,bool,A),Q12),F1),aa(fun(B,D),fun(B,bool),aa(fun(D,bool),fun(fun(B,D),fun(B,bool)),comp(D,bool,B),Q23),F22)),X)) ) ).

% prod.pred_map
tff(fact_6727_wfP__acyclicP,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] :
      ( wfP(A,R)
     => transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).

% wfP_acyclicP
tff(fact_6728_wfP__empty,axiom,
    ! [A: $tType] : wfP(A,aTP_Lamp_apa(A,fun(A,bool))) ).

% wfP_empty
tff(fact_6729_wfP__wf__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( wfP(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> wf(A,R) ) ).

% wfP_wf_eq
tff(fact_6730_wfP__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] :
      ( wfP(A,R)
    <=> 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),R))) ) ).

% wfP_def
tff(fact_6731_int__decode__def,axiom,
    ! [N: nat] : nat_int_decode(N) = aa(sum_sum(nat,nat),int,aa(fun(nat,int),fun(sum_sum(nat,nat),int),aa(fun(nat,int),fun(fun(nat,int),fun(sum_sum(nat,nat),int)),sum_case_sum(nat,int,nat),semiring_1_of_nat(int)),aTP_Lamp_apb(nat,int)),nat_sum_decode(N)) ).

% int_decode_def
tff(fact_6732_to__nat__def,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( to_nat(A) = fChoice(fun(A,nat),aTP_Lamp_apc(fun(A,nat),bool)) ) ) ).

% to_nat_def
tff(fact_6733_nat__to__rat__surj__def,axiom,
    ! [N: nat] : aa(nat,rat,nat_to_rat_surj,N) = aa(product_prod(nat,nat),rat,aa(fun(nat,fun(nat,rat)),fun(product_prod(nat,nat),rat),product_case_prod(nat,nat,rat),aTP_Lamp_apd(nat,fun(nat,rat))),aa(nat,product_prod(nat,nat),nat_prod_decode,N)) ).

% nat_to_rat_surj_def
tff(fact_6734_nth__item_Oelims,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [X: nat,Y3: set(old_node(A,product_unit))] :
          ( ( nth_item(A,X) = Y3 )
         => ( ( ( X = zero_zero(nat) )
             => ( Y3 != undefined(set(old_node(A,product_unit))) ) )
           => ~ ! [N2: nat] :
                  ( ( X = aa(nat,nat,suc,N2) )
                 => ( Y3 != aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_apg(nat,set(old_node(A,product_unit)))),aTP_Lamp_apk(nat,set(old_node(A,product_unit)))),nat_sum_decode(N2)) ) ) ) ) ) ).

% nth_item.elims
tff(fact_6735_nth__item_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ).

% nth_item.simps(1)
tff(fact_6736_nth__item_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [N: nat] : nth_item(A,aa(nat,nat,suc,N)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_apg(nat,set(old_node(A,product_unit)))),aTP_Lamp_apk(nat,set(old_node(A,product_unit)))),nat_sum_decode(N)) ) ).

% nth_item.simps(2)
tff(fact_6737_nth__item_Opelims,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [X: nat,Y3: set(old_node(A,product_unit))] :
          ( ( nth_item(A,X) = Y3 )
         => ( pp(aa(nat,bool,accp(nat,nth_item_rel),X))
           => ( ( ( X = zero_zero(nat) )
               => ( ( Y3 = undefined(set(old_node(A,product_unit))) )
                 => ~ pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat))) ) )
             => ~ ! [N2: nat] :
                    ( ( X = aa(nat,nat,suc,N2) )
                   => ( ( Y3 = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_apg(nat,set(old_node(A,product_unit)))),aTP_Lamp_apk(nat,set(old_node(A,product_unit)))),nat_sum_decode(N2)) )
                     => ~ pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% nth_item.pelims
tff(fact_6738_nth__item_Opsimps_I2_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N)))
         => ( nth_item(A,aa(nat,nat,suc,N)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_apg(nat,set(old_node(A,product_unit)))),aTP_Lamp_apk(nat,set(old_node(A,product_unit)))),nat_sum_decode(N)) ) ) ) ).

% nth_item.psimps(2)
tff(fact_6739_nth__item_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ( pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat)))
       => ( nth_item(A,zero_zero(nat)) = undefined(set(old_node(A,product_unit))) ) ) ) ).

% nth_item.psimps(1)
tff(fact_6740_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_Rats(A) = aa(set(nat),set(A),image2(nat,A,aa(fun(nat,rat),fun(nat,A),aa(fun(rat,A),fun(fun(nat,rat),fun(nat,A)),comp(rat,A,nat),field_char_0_of_rat(A)),nat_to_rat_surj)),top_top(set(nat))) ) ) ).

% Rats_eq_range_of_rat_o_nat_to_rat_surj
tff(fact_6741_trancl__def,axiom,
    ! [A: $tType,X2: set(product_prod(A,A))] : transitive_trancl(A,X2) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_tranclp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),X2)))) ).

% trancl_def
tff(fact_6742_tranclp__trancl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X2: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X2),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),X2),Xa3)),transitive_trancl(A,R))) ) ).

% tranclp_trancl_eq
tff(fact_6743_tranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( ! [A4: A,B3: B] :
            ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3)) )
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% tranclp_induct2
tff(fact_6744_less__nat__rel,axiom,
    ord_less(nat) = transitive_tranclp(nat,aTP_Lamp_kw(nat,fun(nat,bool))) ).

% less_nat_rel
tff(fact_6745_Rats__add,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),plus_plus(A),A3),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_add
tff(fact_6746_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,member(A,one_one(A)),field_char_0_Rats(A))) ) ).

% Rats_1
tff(fact_6747_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,member(A,zero_zero(A)),field_char_0_Rats(A))) ) ).

% Rats_0
tff(fact_6748_tranclp__power,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,P),X),Y3))
    <=> ? [N6: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N6))
          & 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))),N6),P),X),Y3)) ) ) ).

% tranclp_power
tff(fact_6749_tranclp__def,axiom,
    ! [A: $tType,X2: fun(A,fun(A,bool))] : transitive_tranclp(A,X2) = complete_lattice_lfp(fun(A,fun(A,bool)),aTP_Lamp_uf(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),X2)) ).

% tranclp_def
tff(fact_6750_Nitpick_Otranclp__unfold,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),A3: A,B2: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,R),A3),B2))
    <=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).

% Nitpick.tranclp_unfold
tff(fact_6751_semilattice__set_Oeq__fold_H,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A5: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( lattic1715443433743089157tice_F(A,F,A5) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_apl(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F),none(A),A5)) ) ) ).

% semilattice_set.eq_fold'
tff(fact_6752_reflp__refl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( reflp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
    <=> refl_on(A,top_top(set(A)),R) ) ).

% reflp_refl_eq
tff(fact_6753_Quotient__id__abs__transfer,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,R4,Abs,Rep,T4)
     => ( reflp(A,R4)
       => pp(aa(fun(A,B),bool,aa(fun(A,A),fun(fun(A,B),bool),bNF_rel_fun(A,A,A,B,fequal(A),T4),aTP_Lamp_ak(A,A)),Abs)) ) ) ).

% Quotient_id_abs_transfer
tff(fact_6754_Heap_Orel__reflp,axiom,
    ! [A: $tType,R4: fun(A,fun(A,bool))] :
      ( reflp(A,R4)
     => reflp(heap_Time_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),R4)) ) ).

% Heap.rel_reflp
tff(fact_6755_semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A5: set(A)] :
      ( lattic149705377957585745ce_set(A,F)
     => ( ~ pp(aa(set(A),bool,finite_finite2(A),A5))
       => ( lattic1715443433743089157tice_F(A,F,A5) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% semilattice_set.infinite
tff(fact_6756_option_Orel__reflp,axiom,
    ! [A: $tType,R4: fun(A,fun(A,bool))] :
      ( reflp(A,R4)
     => reflp(option(A),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),R4)) ) ).

% option.rel_reflp
tff(fact_6757_image2p__def,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,A),G: fun(D,B),R4: fun(C,fun(D,bool)),X2: A,Xa3: B] :
      ( bNF_Greatest_image2p(C,A,D,B,F,G,R4,X2,Xa3)
    <=> ? [X11: C,Y10: D] :
          ( pp(aa(D,bool,aa(C,fun(D,bool),R4,X11),Y10))
          & ( aa(C,A,F,X11) = X2 )
          & ( aa(D,B,G,Y10) = Xa3 ) ) ) ).

% image2p_def
tff(fact_6758_small__lazy_H_Ocases,axiom,
    ! [X: product_prod(int,int)] :
      ~ ! [D2: int,I2: int] : X != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D2),I2) ).

% small_lazy'.cases
tff(fact_6759_old_Orec__bool__def,axiom,
    ! [T: $tType,X2: T,Xa3: T,Xb: bool] : product_rec_bool(T,X2,Xa3,Xb) = the(T,product_rec_set_bool(T,X2,Xa3,Xb)) ).

% old.rec_bool_def
tff(fact_6760_lexord__asymmetric,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),A3: list(A),B2: list(A)] :
      ( asym(A,R4)
     => ( 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)),A3),B2)),lexord(A,R4)))
       => ~ 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)),B2),A3)),lexord(A,R4))) ) ) ).

% lexord_asymmetric
tff(fact_6761_old_Obool_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fFalse) = F22 ).

% old.bool.simps(6)
tff(fact_6762_old_Obool_Osimps_I5_J,axiom,
    ! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fTrue) = F1 ).

% old.bool.simps(5)
tff(fact_6763_asym_Ocases,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
     => ! [A17: A,B11: 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),A17),B11)),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),B11),A17)),A3)) ) ) ).

% asym.cases
tff(fact_6764_asym_Osimps,axiom,
    ! [A: $tType,A3: set(product_prod(A,A))] :
      ( asym(A,A3)
    <=> ? [R11: set(product_prod(A,A))] :
          ( ( A3 = R11 )
          & ! [X3: A,Xa2: 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),Xa2)),R11))
             => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X3)),R11)) ) ) ) ).

% asym.simps
tff(fact_6765_asym_Ointros,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( ! [A4: A,B3: 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),A4),B3)),R4))
         => ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B3),A4)),R4)) )
     => asym(A,R4) ) ).

% asym.intros
tff(fact_6766_asymD,axiom,
    ! [A: $tType,R4: set(product_prod(A,A)),X: A,Y3: A] :
      ( asym(A,R4)
     => ( pp(aa(set(product_prod(A,A)),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)),R4))
       => ~ pp(aa(set(product_prod(A,A)),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),X)),R4)) ) ) ).

% asymD
tff(fact_6767_asym__iff,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( asym(A,R4)
    <=> ! [X3: 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),X3),Y)),R4))
         => ~ pp(aa(set(product_prod(A,A)),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),X3)),R4)) ) ) ).

% asym_iff
tff(fact_6768_asymp__asym__eq,axiom,
    ! [A: $tType,R4: set(product_prod(A,A))] :
      ( asymp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R4))
    <=> asym(A,R4) ) ).

% asymp_asym_eq
tff(fact_6769_of__int__code_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),neg(K)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_code(1)
tff(fact_6770_less__eq__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),neg(L))) ).

% less_eq_int_code(3)
tff(fact_6771_less__eq__int__code_I7_J,axiom,
    ! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),zero_zero(int))) ).

% less_eq_int_code(7)
tff(fact_6772_less__int__code_I3_J,axiom,
    ! [L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),neg(L))) ).

% less_int_code(3)
tff(fact_6773_less__int__code_I7_J,axiom,
    ! [K: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),zero_zero(int))) ).

% less_int_code(7)
tff(fact_6774_nat__code_I1_J,axiom,
    ! [K: num] : aa(int,nat,nat2,neg(K)) = zero_zero(nat) ).

% nat_code(1)
tff(fact_6775_plus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% plus_int_code(6)
tff(fact_6776_asymp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => asymp(A,aTP_Lamp_afd(A,fun(A,bool))) ) ).

% asymp_greater
tff(fact_6777_less__eq__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),neg(L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),L),K)) ) ).

% less_eq_int_code(9)
tff(fact_6778_less__int__code_I9_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),neg(L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),L),K)) ) ).

% less_int_code(9)
tff(fact_6779_Int_Osub__code_I4_J,axiom,
    ! [N: num] : sub(one2,bit0(N)) = neg(bitM(N)) ).

% Int.sub_code(4)
tff(fact_6780_Int_Osub__code_I5_J,axiom,
    ! [N: num] : sub(one2,aa(num,num,bit1,N)) = neg(bit0(N)) ).

% Int.sub_code(5)
tff(fact_6781_Int_Osub__code_I1_J,axiom,
    sub(one2,one2) = zero_zero(int) ).

% Int.sub_code(1)
tff(fact_6782_Int_Osub__def,axiom,
    ! [M: num,N: num] : sub(M,N) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) ).

% Int.sub_def
tff(fact_6783_minus__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M)),neg(N)) = sub(N,M) ).

% minus_int_code(6)
tff(fact_6784_Int_Osub__code_I9_J,axiom,
    ! [M: num,N: num] : sub(bit0(M),aa(num,num,bit1,N)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),dup(sub(M,N))),one_one(int)) ).

% Int.sub_code(9)
tff(fact_6785_Int_Osub__code_I8_J,axiom,
    ! [M: num,N: num] : sub(aa(num,num,bit1,M),bit0(N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),dup(sub(M,N))),one_one(int)) ).

% Int.sub_code(8)
tff(fact_6786_Int_Odup__code_I1_J,axiom,
    dup(zero_zero(int)) = zero_zero(int) ).

% Int.dup_code(1)
tff(fact_6787_Int_Odup__def,axiom,
    ! [K: int] : dup(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),K) ).

% Int.dup_def
tff(fact_6788_Int_Odup__code_I3_J,axiom,
    ! [N: num] : dup(neg(N)) = neg(bit0(N)) ).

% Int.dup_code(3)
tff(fact_6789_Int_Osub__code_I6_J,axiom,
    ! [M: num,N: num] : sub(bit0(M),bit0(N)) = dup(sub(M,N)) ).

% Int.sub_code(6)
tff(fact_6790_Int_Osub__code_I7_J,axiom,
    ! [M: num,N: num] : sub(aa(num,num,bit1,M),aa(num,num,bit1,N)) = dup(sub(M,N)) ).

% Int.sub_code(7)
tff(fact_6791_Int_Osub__code_I2_J,axiom,
    ! [M: num] : sub(bit0(M),one2) = aa(num,int,pos,bitM(M)) ).

% Int.sub_code(2)
tff(fact_6792_Int_Osub__code_I3_J,axiom,
    ! [M: num] : sub(aa(num,num,bit1,M),one2) = aa(num,int,pos,bit0(M)) ).

% Int.sub_code(3)
tff(fact_6793_Int_ONeg__def,axiom,
    ! [N: num] : neg(N) = aa(int,int,uminus_uminus(int),aa(num,int,pos,N)) ).

% Int.Neg_def
tff(fact_6794_uminus__int__code_I2_J,axiom,
    ! [M: num] : aa(int,int,uminus_uminus(int),aa(num,int,pos,M)) = neg(M) ).

% uminus_int_code(2)
tff(fact_6795_uminus__int__code_I3_J,axiom,
    ! [M: num] : aa(int,int,uminus_uminus(int),neg(M)) = aa(num,int,pos,M) ).

% uminus_int_code(3)
tff(fact_6796_less__eq__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),neg(K)),aa(num,int,pos,L))) ).

% less_eq_int_code(8)
tff(fact_6797_less__eq__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),neg(L))) ).

% less_eq_int_code(6)
tff(fact_6798_less__int__code_I6_J,axiom,
    ! [K: num,L: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),neg(L))) ).

% less_int_code(6)
tff(fact_6799_less__int__code_I8_J,axiom,
    ! [K: num,L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),neg(K)),aa(num,int,pos,L))) ).

% less_int_code(8)
tff(fact_6800_Int_OPos__def,axiom,
    pos = numeral_numeral(int) ).

% Int.Pos_def
tff(fact_6801_one__int__code,axiom,
    one_one(int) = aa(num,int,pos,one2) ).

% one_int_code
tff(fact_6802_of__int__code_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,pos,K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_code(3)
tff(fact_6803_less__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),zero_zero(int))) ).

% less_int_code(4)
tff(fact_6804_less__int__code_I2_J,axiom,
    ! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(num,int,pos,L))) ).

% less_int_code(2)
tff(fact_6805_plus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% plus_int_code(3)
tff(fact_6806_less__eq__int__code_I4_J,axiom,
    ! [K: num] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),zero_zero(int))) ).

% less_eq_int_code(4)
tff(fact_6807_less__eq__int__code_I2_J,axiom,
    ! [L: num] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(num,int,pos,L))) ).

% less_eq_int_code(2)
tff(fact_6808_times__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).

% times_int_code(3)
tff(fact_6809_nat__code_I3_J,axiom,
    ! [K: num] : aa(int,nat,nat2,aa(num,int,pos,K)) = nat_of_num(K) ).

% nat_code(3)
tff(fact_6810_less__eq__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),K),L)) ) ).

% less_eq_int_code(5)
tff(fact_6811_less__int__code_I5_J,axiom,
    ! [K: num,L: num] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,pos,K)),aa(num,int,pos,L)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),K),L)) ) ).

% less_int_code(5)
tff(fact_6812_minus__int__code_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M)),aa(num,int,pos,N)) = sub(M,N) ).

% minus_int_code(3)
tff(fact_6813_Int_Odup__code_I2_J,axiom,
    ! [N: num] : dup(aa(num,int,pos,N)) = aa(num,int,pos,bit0(N)) ).

% Int.dup_code(2)
tff(fact_6814_minus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),neg(M)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% minus_int_code(5)
tff(fact_6815_minus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,pos,M)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% minus_int_code(4)
tff(fact_6816_times__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,pos,M)),neg(N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).

% times_int_code(4)
tff(fact_6817_times__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),aa(num,int,pos,N)) = neg(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).

% times_int_code(5)
tff(fact_6818_times__int__code_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),times_times(int),neg(M)),neg(N)) = aa(num,int,pos,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ).

% times_int_code(6)
tff(fact_6819_plus__int__code_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),neg(M)),aa(num,int,pos,N)) = sub(N,M) ).

% plus_int_code(5)
tff(fact_6820_plus__int__code_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,pos,M)),neg(N)) = sub(M,N) ).

% plus_int_code(4)
tff(fact_6821_rtranclp__imp__Sup__relpowp,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,P),X),Y3))
     => pp(aa(A,bool,aa(A,fun(A,bool),aa(set(fun(A,fun(A,bool))),fun(A,fun(A,bool)),complete_Sup_Sup(fun(A,fun(A,bool))),aa(set(nat),set(fun(A,fun(A,bool))),image2(nat,fun(A,fun(A,bool)),aTP_Lamp_apm(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),P)),top_top(set(nat)))),X),Y3)) ) ).

% rtranclp_imp_Sup_relpowp
tff(fact_6822_rtranclp__is__Sup__relpowp,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : transitive_rtranclp(A,P) = aa(set(fun(A,fun(A,bool))),fun(A,fun(A,bool)),complete_Sup_Sup(fun(A,fun(A,bool))),aa(set(nat),set(fun(A,fun(A,bool))),image2(nat,fun(A,fun(A,bool)),aTP_Lamp_apm(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),P)),top_top(set(nat)))) ).

% rtranclp_is_Sup_relpowp
tff(fact_6823_rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay))
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By)) ) ) ) ).

% rtranclp_induct2
tff(fact_6824_converse__rtranclpE2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Xa: A,Xb2: B,Za2: A,Zb: B] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)))
     => ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb2) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
       => ~ ! [A4: A,B3: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)))
             => ~ pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))) ) ) ) ).

% converse_rtranclpE2
tff(fact_6825_converse__rtranclp__induct2,axiom,
    ! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P: fun(A,fun(B,bool))] :
      ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Bx),By))
       => ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
              ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
             => ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),P,Aa2),Ba))
                 => pp(aa(B,bool,aa(A,fun(B,bool),P,A4),B3)) ) ) )
         => pp(aa(B,bool,aa(A,fun(B,bool),P,Ax),Ay)) ) ) ) ).

% converse_rtranclp_induct2
tff(fact_6826_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X2: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X2),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),X2),Xa3)),transitive_rtrancl(A,R))) ) ).

% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_6827_rtranclp__r__diff__Id,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] : transitive_rtranclp(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))),inf_inf(fun(A,fun(A,bool))),R),aTP_Lamp_adp(A,fun(A,bool)))) = transitive_rtranclp(A,R) ).

% rtranclp_r_diff_Id
tff(fact_6828_Enum_Ortranclp__rtrancl__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X: A,Y3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),X),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),X),Y3)),transitive_rtrancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).

% Enum.rtranclp_rtrancl_eq
tff(fact_6829_rtrancl__def,axiom,
    ! [A: $tType,X2: set(product_prod(A,A))] : transitive_rtrancl(A,X2) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_rtranclp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),X2)))) ).

% rtrancl_def
tff(fact_6830_rtranclp__def,axiom,
    ! [A: $tType,X2: fun(A,fun(A,bool))] : transitive_rtranclp(A,X2) = complete_lattice_lfp(fun(A,fun(A,bool)),aTP_Lamp_ug(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),X2)) ).

% rtranclp_def
tff(fact_6831_is__none__bind,axiom,
    ! [A: $tType,B: $tType,F: option(B),G: fun(B,option(A))] :
      ( is_none(A,aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),F),G))
    <=> ( is_none(B,F)
        | is_none(A,aa(B,option(A),G,aa(option(B),B,the2(B),F))) ) ) ).

% is_none_bind
tff(fact_6832_prod__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups1828464146339083142d_list(A,times_times(A),one_one(A)) ) ).

% prod_list.comm_monoid_list_axioms
tff(fact_6833_is__none__code_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_code(1)
tff(fact_6834_is__none__code_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_code(2)
tff(fact_6835_is__none__map__option,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),X: option(B)] :
      ( is_none(A,aa(option(B),option(A),aa(fun(B,A),fun(option(B),option(A)),map_option(B,A),F),X))
    <=> is_none(B,X) ) ).

% is_none_map_option
tff(fact_6836_is__none__simps_I2_J,axiom,
    ! [B: $tType,X: B] : ~ is_none(B,aa(B,option(B),some(B),X)) ).

% is_none_simps(2)
tff(fact_6837_Option_Ois__none__def,axiom,
    ! [A: $tType,X: option(A)] :
      ( is_none(A,X)
    <=> ( X = none(A) ) ) ).

% Option.is_none_def
tff(fact_6838_is__none__simps_I1_J,axiom,
    ! [A: $tType] : is_none(A,none(A)) ).

% is_none_simps(1)
tff(fact_6839_rel__optionI,axiom,
    ! [A: $tType,B: $tType,X: option(A),Y3: option(B),P: fun(A,fun(B,bool))] :
      ( ( is_none(A,X)
      <=> is_none(B,Y3) )
     => ( ( ~ is_none(A,X)
         => ( ~ is_none(B,Y3)
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y3))) ) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),P),X),Y3)) ) ) ).

% rel_optionI
tff(fact_6840_rel__option__unfold,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X: option(A),Y3: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),X),Y3))
    <=> ( ( is_none(A,X)
        <=> is_none(B,Y3) )
        & ( ~ is_none(A,X)
         => ( ~ is_none(B,Y3)
           => pp(aa(B,bool,aa(A,fun(B,bool),R4,aa(option(A),A,the2(A),X)),aa(option(B),B,the2(B),Y3))) ) ) ) ) ).

% rel_option_unfold
tff(fact_6841_the__map__option,axiom,
    ! [B: $tType,A: $tType,X: option(A),F: fun(A,B)] :
      ( ~ is_none(A,X)
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),aa(fun(A,B),fun(option(A),option(B)),map_option(A,B),F),X)) = aa(A,B,F,aa(option(A),A,the2(A),X)) ) ) ).

% the_map_option
tff(fact_6842_sum__list_Ocomm__monoid__list__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups1828464146339083142d_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.comm_monoid_list_axioms
tff(fact_6843_group_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F,Z,Inverse)
     => group_axioms(A,F,Z,Inverse) ) ).

% group.axioms(2)
tff(fact_6844_strict__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linorder(B)
        & linorder(A) )
     => ! [F: fun(A,B),G: fun(B,A)] :
          ( order_strict_mono(A,B,F)
         => ( ( aa(set(A),set(B),image2(A,B,F),top_top(set(A))) = top_top(set(B)) )
           => ( ! [X4: A] : aa(B,A,G,aa(A,B,F,X4)) = X4
             => order_strict_mono(B,A,G) ) ) ) ) ).

% strict_mono_inv
tff(fact_6845_strict__mono__imp__increasing,axiom,
    ! [F: fun(nat,nat),N: nat] :
      ( order_strict_mono(nat,nat,F)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,F,N))) ) ).

% strict_mono_imp_increasing
tff(fact_6846_strict__mono__imp__inj__on,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F: fun(A,B),A5: set(A)] :
          ( order_strict_mono(A,B,F)
         => inj_on(A,B,F,A5) ) ) ).

% strict_mono_imp_inj_on
tff(fact_6847_group__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group_axioms(A,F,Z,Inverse)
    <=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Z),A7) = A7
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A7)),A7) = Z ) ) ).

% group_axioms_def
tff(fact_6848_group__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Z),A4) = A4
     => ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Inverse,A4)),A4) = Z
       => group_axioms(A,F,Z,Inverse) ) ) ).

% group_axioms.intro
tff(fact_6849_group_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( semigroup(A,F)
     => ( group_axioms(A,F,Z,Inverse)
       => group(A,F,Z,Inverse) ) ) ).

% group.intro
tff(fact_6850_group__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F,Z,Inverse)
    <=> ( semigroup(A,F)
        & group_axioms(A,F,Z,Inverse) ) ) ).

% group_def
tff(fact_6851_mult_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => semigroup(A,times_times(A)) ) ).

% mult.semigroup_axioms
tff(fact_6852_gcd_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => semigroup(A,gcd_gcd(A)) ) ).

% gcd.semigroup_axioms
tff(fact_6853_semigroup_Oassoc,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A,C2: A] :
      ( semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A3),B2)),C2) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).

% semigroup.assoc
tff(fact_6854_semigroup_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( ! [A4: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A4),B3)),C3) = aa(A,A,aa(A,fun(A,A),F,A4),aa(A,A,aa(A,fun(A,A),F,B3),C3))
     => semigroup(A,F) ) ).

% semigroup.intro
tff(fact_6855_semigroup__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semigroup(A,F)
    <=> ! [A7: A,B7: A,C4: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,A7),B7)),C4) = aa(A,A,aa(A,fun(A,A),F,A7),aa(A,A,aa(A,fun(A,A),F,B7),C4)) ) ).

% semigroup_def
tff(fact_6856_group_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,Inverse: fun(A,A)] :
      ( group(A,F,Z,Inverse)
     => semigroup(A,F) ) ).

% group.axioms(1)
tff(fact_6857_lcm_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => semigroup(A,gcd_lcm(A)) ) ).

% lcm.semigroup_axioms
tff(fact_6858_add_Osemigroup__axioms,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => semigroup(A,plus_plus(A)) ) ).

% add.semigroup_axioms
tff(fact_6859_monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F,Z)
     => semigroup(A,F) ) ).

% monoid.axioms(1)
tff(fact_6860_monoid_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( semigroup(A,F)
     => ( monoid_axioms(A,F,Z)
       => monoid(A,F,Z) ) ) ).

% monoid.intro
tff(fact_6861_monoid__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F,Z)
    <=> ( semigroup(A,F)
        & monoid_axioms(A,F,Z) ) ) ).

% monoid_def
tff(fact_6862_monoid__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( monoid_axioms(A,F,Z)
    <=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Z),A7) = A7
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,A7),Z) = A7 ) ) ).

% monoid_axioms_def
tff(fact_6863_monoid__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Z),A4) = A4
     => ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),Z) = A4
       => monoid_axioms(A,F,Z) ) ) ).

% monoid_axioms.intro
tff(fact_6864_monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( monoid(A,F,Z)
     => monoid_axioms(A,F,Z) ) ).

% monoid.axioms(2)
tff(fact_6865_rec__natural__def,axiom,
    ! [T: $tType,X2: T,Xa3: fun(code_natural,fun(T,T)),Xb: code_natural] : aa(code_natural,T,code_rec_natural(T,X2,Xa3),Xb) = the(T,code_rec_set_natural(T,X2,Xa3,Xb)) ).

% rec_natural_def
tff(fact_6866_dsum__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: set(set(old_node(A,B))),B4: set(set(old_node(A,B))),C6: set(set(old_node(A,B))),D5: set(set(old_node(A,B)))] : pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dsum(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A5,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B4)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C6,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D5)))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_usum(A,B,A5,C6),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apo(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B4),D5)))) ).

% dsum_Sigma
tff(fact_6867_dsum__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),A5: set(set(old_node(A,B))),B4: set(set(old_node(A,B))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),C6: set(set(old_node(A,B))),D5: set(set(old_node(A,B)))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),R),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A5,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B4))))
     => ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),S),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C6,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D5))))
       => pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dsum(A,B,R,S)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_usum(A,B,A5,C6),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apo(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B4),D5)))) ) ) ).

% dsum_subset_Sigma
tff(fact_6868_case__natural__def,axiom,
    ! [T: $tType,X2: T,Xa3: fun(code_natural,T)] : code_case_natural(T,X2,Xa3) = code_rec_natural(T,X2,aTP_Lamp_app(fun(code_natural,T),fun(code_natural,fun(T,T)),Xa3)) ).

% case_natural_def
tff(fact_6869_dprod__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),A5: set(set(old_node(A,B))),B4: set(set(old_node(A,B))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),C6: set(set(old_node(A,B))),D5: set(set(old_node(A,B)))] :
      ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),R),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A5,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B4))))
     => ( pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),S),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C6,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D5))))
       => pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,R,S)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A5,C6),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apq(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B4),D5)))) ) ) ).

% dprod_subset_Sigma
tff(fact_6870_uprod__def,axiom,
    ! [B: $tType,A: $tType,A5: set(set(old_node(A,B))),B4: set(set(old_node(A,B)))] : old_uprod(A,B,A5,B4) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_aps(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B4)),A5)) ).

% uprod_def
tff(fact_6871_dprod__subset__Sigma2,axiom,
    ! [B: $tType,A: $tType,A5: set(set(old_node(A,B))),B4: fun(set(old_node(A,B)),set(set(old_node(A,B)))),C6: set(set(old_node(A,B))),D5: fun(set(old_node(A,B)),set(set(old_node(A,B))))] : pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A5,B4),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C6,D5))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A5,C6),old_Split(A,B,set(set(old_node(A,B))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B))))),aTP_Lamp_apt(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))))),B4),D5))))) ).

% dprod_subset_Sigma2
tff(fact_6872_dprod__Sigma,axiom,
    ! [B: $tType,A: $tType,A5: set(set(old_node(A,B))),B4: set(set(old_node(A,B))),C6: set(set(old_node(A,B))),D5: set(set(old_node(A,B)))] : pp(aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool,aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),bool),ord_less_eq(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),old_dprod(A,B,product_Sigma(set(old_node(A,B)),set(old_node(A,B)),A5,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B4)),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),C6,aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),D5)))),product_Sigma(set(old_node(A,B)),set(old_node(A,B)),old_uprod(A,B,A5,C6),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apq(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),B4),D5)))) ).

% dprod_Sigma
tff(fact_6873_Abs__int__inject,axiom,
    ! [X: set(product_prod(nat,nat)),Y3: set(product_prod(nat,nat))] :
      ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),X),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
     => ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y3),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
       => ( ( aa(set(product_prod(nat,nat)),int,abs_int,X) = aa(set(product_prod(nat,nat)),int,abs_int,Y3) )
        <=> ( X = Y3 ) ) ) ) ).

% Abs_int_inject
tff(fact_6874_Abs__int__induct,axiom,
    ! [P: fun(int,bool),X: int] :
      ( ! [Y4: set(product_prod(nat,nat))] :
          ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y4),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
         => pp(aa(int,bool,P,aa(set(product_prod(nat,nat)),int,abs_int,Y4))) )
     => pp(aa(int,bool,P,X)) ) ).

% Abs_int_induct
tff(fact_6875_Abs__int__cases,axiom,
    ! [X: int] :
      ~ ! [Y4: set(product_prod(nat,nat))] :
          ( ( X = aa(set(product_prod(nat,nat)),int,abs_int,Y4) )
         => ~ pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y4),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool)))) ) ).

% Abs_int_cases
tff(fact_6876_Abs__Integ__def,axiom,
    abs_Integ = quot_abs(product_prod(nat,nat),int,intrel,abs_int) ).

% Abs_Integ_def
tff(fact_6877_Abs__int__inverse,axiom,
    ! [Y3: set(product_prod(nat,nat))] :
      ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y3),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
     => ( aa(int,set(product_prod(nat,nat)),rep_int,aa(set(product_prod(nat,nat)),int,abs_int,Y3)) = Y3 ) ) ).

% Abs_int_inverse
tff(fact_6878_Rep__int__inject,axiom,
    ! [X: int,Y3: int] :
      ( ( aa(int,set(product_prod(nat,nat)),rep_int,X) = aa(int,set(product_prod(nat,nat)),rep_int,Y3) )
    <=> ( X = Y3 ) ) ).

% Rep_int_inject
tff(fact_6879_Rep__int__inverse,axiom,
    ! [X: int] : aa(set(product_prod(nat,nat)),int,abs_int,aa(int,set(product_prod(nat,nat)),rep_int,X)) = X ).

% Rep_int_inverse
tff(fact_6880_Rep__int__induct,axiom,
    ! [Y3: set(product_prod(nat,nat)),P: fun(set(product_prod(nat,nat)),bool)] :
      ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y3),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
     => ( ! [X4: int] : pp(aa(set(product_prod(nat,nat)),bool,P,aa(int,set(product_prod(nat,nat)),rep_int,X4)))
       => pp(aa(set(product_prod(nat,nat)),bool,P,Y3)) ) ) ).

% Rep_int_induct
tff(fact_6881_Rep__int__cases,axiom,
    ! [Y3: set(product_prod(nat,nat))] :
      ( pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),Y3),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))))
     => ~ ! [X4: int] : Y3 != aa(int,set(product_prod(nat,nat)),rep_int,X4) ) ).

% Rep_int_cases
tff(fact_6882_Rep__int,axiom,
    ! [X: int] : pp(aa(set(set(product_prod(nat,nat))),bool,member(set(product_prod(nat,nat)),aa(int,set(product_prod(nat,nat)),rep_int,X)),aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool)))) ).

% Rep_int
tff(fact_6883_type__definition__int,axiom,
    type_definition(int,set(product_prod(nat,nat)),rep_int,abs_int,aa(fun(set(product_prod(nat,nat)),bool),set(set(product_prod(nat,nat))),collect(set(product_prod(nat,nat))),aTP_Lamp_apu(set(product_prod(nat,nat)),bool))) ).

% type_definition_int
tff(fact_6884_Rep__Integ__def,axiom,
    rep_Integ = quot_rep(int,product_prod(nat,nat),rep_int) ).

% Rep_Integ_def
tff(fact_6885_less__eq__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),Xa),Xa))
     => ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
       => ( pp(aa(literal,bool,aa(literal,fun(literal,bool),ord_less_eq(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X)))
        <=> pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool))),Xa),X)) ) ) ) ).

% less_eq_literal.abs_eq
tff(fact_6886_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_aml(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_6887_plus__literal_Orsp,axiom,
    pp(aa(fun(list(char),fun(list(char),list(char))),bool,aa(fun(list(char),fun(list(char),list(char))),fun(fun(list(char),fun(list(char),list(char))),bool),bNF_rel_fun(list(char),list(char),fun(list(char),list(char)),fun(list(char),list(char)),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),bNF_rel_fun(list(char),list(char),list(char),list(char),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)))),append(char)),append(char))) ).

% plus_literal.rsp
tff(fact_6888_less__eq__literal_Orsp,axiom,
    pp(aa(fun(list(char),fun(list(char),bool)),bool,aa(fun(list(char),fun(list(char),bool)),fun(fun(list(char),fun(list(char),bool)),bool),bNF_rel_fun(list(char),list(char),fun(list(char),bool),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),bNF_rel_fun(list(char),list(char),bool,bool,bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),fequal(bool))),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool)))),lexordp_eq(char,aTP_Lamp_amo(char,fun(char,bool))))) ).

% less_eq_literal.rsp
tff(fact_6889_asciis__of__literal_Orsp,axiom,
    pp(aa(fun(list(char),list(code_integer)),bool,aa(fun(list(char),list(code_integer)),fun(fun(list(char),list(code_integer)),bool),bNF_rel_fun(list(char),list(char),list(code_integer),list(code_integer),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),fequal(list(code_integer))),map(char,code_integer,comm_s6883823935334413003f_char(code_integer))),map(char,code_integer,comm_s6883823935334413003f_char(code_integer)))) ).

% asciis_of_literal.rsp
tff(fact_6890_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_apw(fun(B,bool),fun(fun(A,B),bool),P)) ).

% rel_fun_eq_eq_onp
tff(fact_6891_size__literal_Orsp,axiom,
    pp(aa(fun(list(char),nat),bool,aa(fun(list(char),nat),fun(fun(list(char),nat),bool),bNF_rel_fun(list(char),list(char),nat,nat,bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),fequal(nat)),size_size(list(char))),size_size(list(char)))) ).

% size_literal.rsp
tff(fact_6892_Literal_Orsp,axiom,
    pp(aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),bool,aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),fun(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))),fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(bool,fun(list(char),list(char)))),fun(bool,fun(bool,fun(list(char),list(char)))),fequal(bool),bNF_rel_fun(bool,bool,fun(bool,fun(list(char),list(char))),fun(bool,fun(list(char),list(char))),fequal(bool),bNF_rel_fun(bool,bool,fun(list(char),list(char)),fun(list(char),list(char)),fequal(bool),bNF_rel_fun(list(char),list(char),list(char),list(char),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)))))))))),aTP_Lamp_aoe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))))),aTP_Lamp_aoe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))))) ).

% Literal.rsp
tff(fact_6893_less__literal_Orsp,axiom,
    pp(aa(fun(list(char),fun(list(char),bool)),bool,aa(fun(list(char),fun(list(char),bool)),fun(fun(list(char),fun(list(char),bool)),bool),bNF_rel_fun(list(char),list(char),fun(list(char),bool),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),bNF_rel_fun(list(char),list(char),bool,bool,bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),fequal(bool))),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool)))),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool))))) ).

% less_literal.rsp
tff(fact_6894_Heap_Orel__eq__onp,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),bNF_eq_onp(A,P)) = bNF_eq_onp(heap_Time_Heap(A),aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P)) ).

% Heap.rel_eq_onp
tff(fact_6895_Heap_Opred__rel,axiom,
    ! [A: $tType,P: fun(A,bool),X: heap_Time_Heap(A)] :
      ( pp(aa(heap_Time_Heap(A),bool,aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),P),X))
    <=> pp(aa(heap_Time_Heap(A),bool,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),bool)),heap_Time_rel_Heap(A,A),bNF_eq_onp(A,P)),X),X)) ) ).

% Heap.pred_rel
tff(fact_6896_plus__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),Xa),Xa))
     => ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
       => ( aa(literal,literal,aa(literal,fun(literal,literal),plus_plus(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X)) = aa(list(char),literal,abs_literal,aa(list(char),list(char),aa(list(char),fun(list(char),list(char)),append(char),Xa),X)) ) ) ) ).

% plus_literal.abs_eq
tff(fact_6897_literal_OAbs__literal__cases,axiom,
    ! [X: literal] :
      ~ ! [Y4: list(char)] :
          ( ( X = aa(list(char),literal,abs_literal,Y4) )
         => ~ pp(aa(set(list(char)),bool,member(list(char),Y4),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool)))) ) ).

% literal.Abs_literal_cases
tff(fact_6898_literal_OAbs__literal__induct,axiom,
    ! [P: fun(literal,bool),X: literal] :
      ( ! [Y4: list(char)] :
          ( pp(aa(set(list(char)),bool,member(list(char),Y4),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
         => pp(aa(literal,bool,P,aa(list(char),literal,abs_literal,Y4))) )
     => pp(aa(literal,bool,P,X)) ) ).

% literal.Abs_literal_induct
tff(fact_6899_literal_OAbs__literal__inject,axiom,
    ! [X: list(char),Y3: list(char)] :
      ( pp(aa(set(list(char)),bool,member(list(char),X),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
     => ( pp(aa(set(list(char)),bool,member(list(char),Y3),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
       => ( ( aa(list(char),literal,abs_literal,X) = aa(list(char),literal,abs_literal,Y3) )
        <=> ( X = Y3 ) ) ) ) ).

% literal.Abs_literal_inject
tff(fact_6900_size__literal_Oabs__eq,axiom,
    ! [X: list(char)] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
     => ( aa(literal,nat,size_size(literal),aa(list(char),literal,abs_literal,X)) = aa(list(char),nat,size_size(list(char)),X) ) ) ).

% size_literal.abs_eq
tff(fact_6901_literal_Oexplode,axiom,
    ! [X: literal] : pp(aa(set(list(char)),bool,member(list(char),aa(literal,list(char),explode,X)),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool)))) ).

% literal.explode
tff(fact_6902_literal_Oexplode__cases,axiom,
    ! [Y3: list(char)] :
      ( pp(aa(set(list(char)),bool,member(list(char),Y3),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
     => ~ ! [X4: literal] : Y3 != aa(literal,list(char),explode,X4) ) ).

% literal.explode_cases
tff(fact_6903_literal_Oexplode__induct,axiom,
    ! [Y3: list(char),P: fun(list(char),bool)] :
      ( pp(aa(set(list(char)),bool,member(list(char),Y3),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
     => ( ! [X4: literal] : pp(aa(list(char),bool,P,aa(literal,list(char),explode,X4)))
       => pp(aa(list(char),bool,P,Y3)) ) ) ).

% literal.explode_induct
tff(fact_6904_zero__literal_Orsp,axiom,
    pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),nil(char)),nil(char))) ).

% zero_literal.rsp
tff(fact_6905_eq__onp__def,axiom,
    ! [A: $tType,R4: fun(A,bool),X2: A,Xa3: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,R4),X2),Xa3))
    <=> ( pp(aa(A,bool,R4,X2))
        & ( X2 = Xa3 ) ) ) ).

% eq_onp_def
tff(fact_6906_eq__onp__True,axiom,
    ! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_aq(A,bool)) = fequal(A) ).

% eq_onp_True
tff(fact_6907_eq__onp__live__step,axiom,
    ! [A: $tType,X: bool,Y3: bool,P: fun(A,bool),A3: A] :
      ( ( pp(X)
      <=> pp(Y3) )
     => ( ( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,P),A3),A3))
          & pp(X) )
      <=> ( pp(aa(A,bool,P,A3))
          & pp(Y3) ) ) ) ).

% eq_onp_live_step
tff(fact_6908_Quotient__crel__typedef,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),Abs: fun(A,B),Rep: fun(B,A),T4: fun(A,fun(B,bool))] :
      ( quotient(A,B,bNF_eq_onp(A,P),Abs,Rep,T4)
     => ! [X2: A,Xa3: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X2),Xa3))
        <=> ( X2 = aa(B,A,Rep,Xa3) ) ) ) ).

% Quotient_crel_typedef
tff(fact_6909_String_OQuotient__literal,axiom,
    quotient(list(char),literal,bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),abs_literal,explode,cr_literal) ).

% String.Quotient_literal
tff(fact_6910_Literal_Oabs__eq,axiom,
    ! [X: list(char),Xg: bool,Xf: bool,Xe: bool,Xd: bool,Xc: bool,Xb2: bool,Xa: bool] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
     => ( aa(literal,literal,aa(bool,fun(literal,literal),aa(bool,fun(bool,fun(literal,literal)),aa(bool,fun(bool,fun(bool,fun(literal,literal))),aa(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))))),literal2,Xg),Xf),Xe),Xd),Xc),Xb2),Xa),aa(list(char),literal,abs_literal,X)) = aa(list(char),literal,abs_literal,aa(list(char),list(char),aa(char,fun(list(char),list(char)),cons(char),char2(Xg,Xf,Xe,Xd,Xc,Xb2,Xa,fFalse)),X)) ) ) ).

% Literal.abs_eq
tff(fact_6911_option_Opred__rel,axiom,
    ! [A: $tType,P: fun(A,bool),X: option(A)] :
      ( pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),pred_option(A),P),X))
    <=> pp(aa(option(A),bool,aa(option(A),fun(option(A),bool),aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),bNF_eq_onp(A,P)),X),X)) ) ).

% option.pred_rel
tff(fact_6912_option_Orel__eq__onp,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(option(A),fun(option(A),bool)),rel_option(A,A),bNF_eq_onp(A,P)) = bNF_eq_onp(option(A),aa(fun(A,bool),fun(option(A),bool),pred_option(A),P)) ).

% option.rel_eq_onp
tff(fact_6913_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_aml(D,bool)),P)) ).

% fun.rel_eq_onp
tff(fact_6914_typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: set(B),T4: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,S2)
     => ( ! [X4: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T4,X4),Xa4))
          <=> ( X4 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,aTP_Lamp_agi(set(B),fun(B,bool),S2)),Abs,Rep,T4) ) ) ).

% typedef_to_Quotient
tff(fact_6915_open__typedef__to__Quotient,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,bool),T4: fun(B,fun(A,bool))] :
      ( type_definition(A,B,Rep,Abs,aa(fun(B,bool),set(B),collect(B),P))
     => ( ! [X4: B,Xa4: A] :
            ( pp(aa(A,bool,aa(B,fun(A,bool),T4,X4),Xa4))
          <=> ( X4 = aa(A,B,Rep,Xa4) ) )
       => quotient(B,A,bNF_eq_onp(B,P),Abs,Rep,T4) ) ) ).

% open_typedef_to_Quotient
tff(fact_6916_Quotient__eq__onp__typedef,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Abs: fun(A,B),Rep: fun(B,A),Cr: fun(A,fun(B,bool))] :
      ( quotient(A,B,bNF_eq_onp(A,P),Abs,Rep,Cr)
     => type_definition(B,A,Rep,Abs,aa(fun(A,bool),set(A),collect(A),P)) ) ).

% Quotient_eq_onp_typedef
tff(fact_6917_rel__fun__eq__onp__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R4: fun(A,bool),S2: fun(B,fun(C,bool)),X2: 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,R4),S2),X2),Xa3))
    <=> ! [Xb3: A] :
          ( pp(aa(A,bool,R4,Xb3))
         => pp(aa(C,bool,aa(B,fun(C,bool),S2,aa(A,B,X2,Xb3)),aa(A,C,Xa3,Xb3))) ) ) ).

% rel_fun_eq_onp_rel
tff(fact_6918_literal_OAbs__literal__inverse,axiom,
    ! [Y3: list(char)] :
      ( pp(aa(set(list(char)),bool,member(list(char),Y3),aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))))
     => ( aa(literal,list(char),explode,aa(list(char),literal,abs_literal,Y3)) = Y3 ) ) ).

% literal.Abs_literal_inverse
tff(fact_6919_literal_Otype__definition__literal,axiom,
    type_definition(literal,list(char),explode,abs_literal,aa(fun(list(char),bool),set(list(char)),collect(list(char)),aTP_Lamp_apv(list(char),bool))) ).

% literal.type_definition_literal
tff(fact_6920_less__literal_Oabs__eq,axiom,
    ! [Xa: list(char),X: list(char)] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),Xa),Xa))
     => ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
       => ( pp(aa(literal,bool,aa(literal,fun(literal,bool),ord_less(literal),aa(list(char),literal,abs_literal,Xa)),aa(list(char),literal,abs_literal,X)))
        <=> pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),lexordp2(char,aTP_Lamp_amo(char,fun(char,bool))),Xa),X)) ) ) ) ).

% less_literal.abs_eq
tff(fact_6921_asciis__of__literal_Oabs__eq,axiom,
    ! [X: list(char)] :
      ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool)),X),X))
     => ( asciis_of_literal(aa(list(char),literal,abs_literal,X)) = aa(list(char),list(code_integer),map(char,code_integer,comm_s6883823935334413003f_char(code_integer)),X) ) ) ).

% asciis_of_literal.abs_eq
tff(fact_6922_literal__of__asciis_Orsp,axiom,
    pp(aa(fun(list(code_integer),list(char)),bool,aa(fun(list(code_integer),list(char)),fun(fun(list(code_integer),list(char)),bool),bNF_rel_fun(list(code_integer),list(code_integer),list(char),list(char),fequal(list(code_integer)),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool))),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer)))),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer))))) ).

% literal_of_asciis.rsp
tff(fact_6923_implode_Orsp,axiom,
    pp(aa(fun(list(char),list(char)),bool,aa(fun(list(char),list(char)),fun(fun(list(char),list(char)),bool),bNF_rel_fun(list(char),list(char),list(char),list(char),fequal(list(char)),bNF_eq_onp(list(char),aTP_Lamp_apv(list(char),bool))),map(char,char,ascii_of)),map(char,char,ascii_of))) ).

% implode.rsp
tff(fact_6924_literal__of__asciis_Otransfer,axiom,
    pp(aa(fun(list(code_integer),literal),bool,aa(fun(list(code_integer),list(char)),fun(fun(list(code_integer),literal),bool),bNF_rel_fun(list(code_integer),list(code_integer),list(char),literal,fequal(list(code_integer)),pcr_literal),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer)))),literal_of_asciis)) ).

% literal_of_asciis.transfer
tff(fact_6925_String_Oliteral__of__asciis__def,axiom,
    literal_of_asciis = aa(fun(list(code_integer),list(char)),fun(list(code_integer),literal),map_fun(list(code_integer),list(code_integer),list(char),literal,id(list(code_integer)),abs_literal),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer)))) ).

% String.literal_of_asciis_def
tff(fact_6926_literal__of__asciis_Orep__eq,axiom,
    ! [X: list(code_integer)] : aa(literal,list(char),explode,aa(list(code_integer),literal,literal_of_asciis,X)) = aa(list(code_integer),list(char),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer))),X) ).

% literal_of_asciis.rep_eq
tff(fact_6927_literal__of__asciis_Oabs__eq,axiom,
    ! [X: list(code_integer)] : aa(list(code_integer),literal,literal_of_asciis,X) = aa(list(char),literal,abs_literal,aa(list(code_integer),list(char),map(code_integer,char,aa(fun(code_integer,char),fun(code_integer,char),aa(fun(char,char),fun(fun(code_integer,char),fun(code_integer,char)),comp(char,char,code_integer),ascii_of),unique5772411509450598832har_of(code_integer))),X)) ).

% literal_of_asciis.abs_eq
tff(fact_6928_String_Oliteral__of__asciis__Cons,axiom,
    ! [K: code_integer,Ks: list(code_integer)] : aa(list(code_integer),literal,literal_of_asciis,aa(list(code_integer),list(code_integer),aa(code_integer,fun(list(code_integer),list(code_integer)),cons(code_integer),K),Ks)) = case_char(literal,aTP_Lamp_apx(list(code_integer),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal)))))))),Ks),aa(code_integer,char,unique5772411509450598832har_of(code_integer),K)) ).

% String.literal_of_asciis_Cons
tff(fact_6929_literal_Odomain,axiom,
    ! [X2: list(char)] :
      ( pp(aa(list(char),bool,aa(fun(list(char),fun(literal,bool)),fun(list(char),bool),domainp(list(char),literal),pcr_literal),X2))
    <=> ? [Y: list(char)] :
          ( pp(aa(list(char),bool,aa(list(char),fun(list(char),bool),list_all2(char,char,fequal(char)),X2),Y))
          & ! [Xa2: char] :
              ( pp(aa(set(char),bool,member(char,Xa2),aa(list(char),set(char),set2(char),Y)))
             => ~ digit7(Xa2) ) ) ) ).

% literal.domain
tff(fact_6930_fun_ODomainp__rel,axiom,
    ! [C: $tType,B: $tType,A: $tType,R4: 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),R4)) = aa(fun(A,bool),fun(fun(C,A),bool),basic_pred_fun(C,A,aTP_Lamp_amm(C,bool)),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R4)) ).

% fun.Domainp_rel
tff(fact_6931_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 )
     => ! [X2: 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)),X2))
        <=> ? [Y: A] :
              ( pp(aa(A,bool,aa(C,fun(A,bool),A5,X2),Y))
              & pp(aa(A,bool,P,Y)) ) ) ) ).

% pcr_Domainp
tff(fact_6932_option_ODomainp__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(option(A),fun(option(B),bool)),fun(option(A),bool),domainp(option(A),option(B)),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) = aa(fun(A,bool),fun(option(A),bool),pred_option(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R4)) ).

% option.Domainp_rel
tff(fact_6933_Domainp__Domain__eq,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),X2: A] :
      ( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),aa(set(product_prod(A,B)),set(A),domain(A,B),R))) ) ).

% Domainp_Domain_eq
tff(fact_6934_Domain__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),X2) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_as(set(product_prod(A,B)),fun(A,fun(B,bool))),X2))) ).

% Domain_def
tff(fact_6935_char_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F: fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),Char: char] : aa(A,B,H,case_char(A,F,Char)) = case_char(B,aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B)))))))),aTP_Lamp_apy(fun(A,B),fun(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B))))))))),H),F),Char) ).

% char.case_distrib
tff(fact_6936_Heap_ODomainp__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] : aa(fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),fun(heap_Time_Heap(A),bool),domainp(heap_Time_Heap(A),heap_Time_Heap(B)),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) = aa(fun(A,bool),fun(heap_Time_Heap(A),bool),heap_Time_pred_Heap(A),aa(fun(A,fun(B,bool)),fun(A,bool),domainp(A,B),R4)) ).

% Heap.Domainp_rel
tff(fact_6937_literal_Odomain__eq,axiom,
    ! [X2: list(char)] :
      ( pp(aa(list(char),bool,aa(fun(list(char),fun(literal,bool)),fun(list(char),bool),domainp(list(char),literal),pcr_literal),X2))
    <=> ! [Xa2: char] :
          ( pp(aa(set(char),bool,member(char,Xa2),aa(list(char),set(char),set2(char),X2)))
         => ~ digit7(Xa2) ) ) ).

% literal.domain_eq
tff(fact_6938_literal_Odomain__par__left__total,axiom,
    ! [P8: fun(list(char),bool)] :
      ( left_total(list(char),list(char),list_all2(char,char,fequal(char)))
     => ( pp(aa(fun(list(char),bool),bool,aa(fun(list(char),bool),fun(fun(list(char),bool),bool),bNF_rel_fun(list(char),list(char),bool,bool,list_all2(char,char,fequal(char)),fequal(bool)),P8),aTP_Lamp_apv(list(char),bool)))
       => ( aa(fun(list(char),fun(literal,bool)),fun(list(char),bool),domainp(list(char),literal),pcr_literal) = P8 ) ) ) ).

% literal.domain_par_left_total
tff(fact_6939_right__total__relcompp__transfer,axiom,
    ! [A: $tType,E: $tType,C: $tType,F2: $tType,B: $tType,D: $tType,B4: fun(A,fun(B,bool)),A5: fun(C,fun(D,bool)),C6: fun(E,fun(F2,bool))] :
      ( right_total(A,B,B4)
     => pp(aa(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,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(F2,bool)),fun(D,fun(F2,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(F2,bool)),fun(D,fun(F2,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(F2,bool)),fun(C,fun(E,bool)),fun(D,fun(F2,bool)),bNF_rel_fun(A,B,fun(E,bool),fun(F2,bool),B4,bNF_rel_fun(E,F2,bool,bool,C6,fequal(bool))),bNF_rel_fun(C,D,fun(E,bool),fun(F2,bool),A5,bNF_rel_fun(E,F2,bool,bool,C6,fequal(bool))))),aTP_Lamp_apz(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,F2))) ) ).

% right_total_relcompp_transfer
tff(fact_6940_rat_Odomain__par,axiom,
    ! [DR1: fun(int,bool),DR2: fun(int,bool),P24: fun(product_prod(int,int),bool)] :
      ( ( aa(fun(int,fun(int,bool)),fun(int,bool),domainp(int,int),fequal(int)) = DR1 )
     => ( ( aa(fun(int,fun(int,bool)),fun(int,bool),domainp(int,int),fequal(int)) = DR2 )
       => ( 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,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal(bool)),P24),aTP_Lamp_aqa(product_prod(int,int),bool)))
         => ( aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat) = aa(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool),aa(fun(product_prod(int,int),bool),fun(fun(product_prod(int,int),bool),fun(product_prod(int,int),bool)),inf_inf(fun(product_prod(int,int),bool)),basic_pred_prod(int,int,DR1,DR2)),P24) ) ) ) ) ).

% rat.domain_par
tff(fact_6941_digit7__def,axiom,
    ! [Char: char] :
      ( digit7(Char)
    <=> pp(case_char(bool,aTP_Lamp_aqb(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit7_def
tff(fact_6942_int_Odomain,axiom,
    ! [DR1: fun(nat,bool),DR2: fun(nat,bool)] :
      ( ( aa(fun(nat,fun(nat,bool)),fun(nat,bool),domainp(nat,nat),fequal(nat)) = DR1 )
     => ( ( aa(fun(nat,fun(nat,bool)),fun(nat,bool),domainp(nat,nat),fequal(nat)) = DR2 )
       => ( aa(fun(product_prod(nat,nat),fun(int,bool)),fun(product_prod(nat,nat),bool),domainp(product_prod(nat,nat),int),pcr_int) = basic_pred_prod(nat,nat,DR1,DR2) ) ) ) ).

% int.domain
tff(fact_6943_typedef__right__total,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T4: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X4: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X4),Xa4))
          <=> ( X4 = aa(B,A,Rep,Xa4) ) )
       => right_total(A,B,T4) ) ) ).

% typedef_right_total
tff(fact_6944_Heap_Oleft__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( left_total(A,B,R4)
     => left_total(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ) ).

% Heap.left_total_rel
tff(fact_6945_Heap_Oright__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( right_total(A,B,R4)
     => right_total(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ) ).

% Heap.right_total_rel
tff(fact_6946_int_Oright__total,axiom,
    right_total(product_prod(nat,nat),int,pcr_int) ).

% int.right_total
tff(fact_6947_option_Oleft__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( left_total(A,B,R4)
     => left_total(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ) ).

% option.left_total_rel
tff(fact_6948_option_Oright__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( right_total(A,B,R4)
     => right_total(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ) ).

% option.right_total_rel
tff(fact_6949_rat_Odomain,axiom,
    ! [X2: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat),X2))
    <=> ? [Y: product_prod(int,int)] :
          ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),X2),Y))
          & pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Y),Y)) ) ) ).

% rat.domain
tff(fact_6950_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_aqc(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),B4)),domainp(D,B))) ) ).

% right_total_Domainp_transfer
tff(fact_6951_literal_Odomain__par,axiom,
    ! [DR: fun(char,bool),P24: fun(list(char),bool)] :
      ( ( aa(fun(char,fun(char,bool)),fun(char,bool),domainp(char,char),fequal(char)) = DR )
     => ( pp(aa(fun(list(char),bool),bool,aa(fun(list(char),bool),fun(fun(list(char),bool),bool),bNF_rel_fun(list(char),list(char),bool,bool,list_all2(char,char,fequal(char)),fequal(bool)),P24),aTP_Lamp_apv(list(char),bool)))
       => ( aa(fun(list(char),fun(literal,bool)),fun(list(char),bool),domainp(list(char),literal),pcr_literal) = aa(fun(list(char),bool),fun(list(char),bool),aa(fun(list(char),bool),fun(fun(list(char),bool),fun(list(char),bool)),inf_inf(fun(list(char),bool)),list_all(char,DR)),P24) ) ) ) ).

% literal.domain_par
tff(fact_6952_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_aqd(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_6953_char_Odisc__eq__case,axiom,
    ! [Char: char] : pp(case_char(bool,aTP_Lamp_aqe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ).

% char.disc_eq_case
tff(fact_6954_int_Oleft__total,axiom,
    left_total(product_prod(nat,nat),int,pcr_int) ).

% int.left_total
tff(fact_6955_list_Opred__map,axiom,
    ! [B: $tType,A: $tType,Q: fun(B,bool),F: fun(A,B),X: list(A)] :
      ( pp(aa(list(B),bool,list_all(B,Q),aa(list(A),list(B),map(A,B,F),X)))
    <=> pp(aa(list(A),bool,list_all(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q),F)),X)) ) ).

% list.pred_map
tff(fact_6956_list_Omap__cong__pred,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( pp(aa(list(A),bool,list_all(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_agd(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
       => ( aa(list(A),list(B),map(A,B,F),X) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong_pred
tff(fact_6957_option_Obi__unique__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,R4)
     => bi_unique(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ) ).

% option.bi_unique_rel
tff(fact_6958_bi__unique__iff,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,R4)
    <=> ( ! [Z5: B] : uniq(A,aa(B,fun(A,bool),aTP_Lamp_un(fun(A,fun(B,bool)),fun(B,fun(A,bool)),R4),Z5))
        & ! [Z5: A] : uniq(B,aa(A,fun(B,bool),R4,Z5)) ) ) ).

% bi_unique_iff
tff(fact_6959_list_Opred__True,axiom,
    ! [A: $tType,X2: list(A)] : pp(aa(list(A),bool,list_all(A,aTP_Lamp_aq(A,bool)),X2)) ).

% list.pred_True
tff(fact_6960_Heap_Obi__unique__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( bi_unique(A,B,R4)
     => bi_unique(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ) ).

% Heap.bi_unique_rel
tff(fact_6961_typedef__bi__unique,axiom,
    ! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A5: set(A),T4: fun(A,fun(B,bool))] :
      ( type_definition(B,A,Rep,Abs,A5)
     => ( ! [X4: A,Xa4: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),T4,X4),Xa4))
          <=> ( X4 = aa(B,A,Rep,Xa4) ) )
       => bi_unique(A,B,T4) ) ) ).

% typedef_bi_unique
tff(fact_6962_rat_Odomain__par__left__total,axiom,
    ! [P8: fun(product_prod(int,int),bool)] :
      ( left_total(product_prod(int,int),product_prod(int,int),basic_rel_prod(int,int,int,int,fequal(int),fequal(int)))
     => ( 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,basic_rel_prod(int,int,int,int,fequal(int),fequal(int)),fequal(bool)),P8),aTP_Lamp_aqa(product_prod(int,int),bool)))
       => ( aa(fun(product_prod(int,int),fun(rat,bool)),fun(product_prod(int,int),bool),domainp(product_prod(int,int),rat),pcr_rat) = P8 ) ) ) ).

% rat.domain_par_left_total
tff(fact_6963_list__ex1__simps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( list_ex1(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
    <=> ( ( pp(aa(A,bool,P,X))
         => pp(aa(list(A),bool,list_all(A,aa(A,fun(A,bool),aTP_Lamp_aqf(fun(A,bool),fun(A,fun(A,bool)),P),X)),Xs)) )
        & ( ~ pp(aa(A,bool,P,X))
         => list_ex1(A,P,Xs) ) ) ) ).

% list_ex1_simps(2)
tff(fact_6964_Abs__rat__inject,axiom,
    ! [X: set(product_prod(int,int)),Y3: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),X),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
     => ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
       => ( ( aa(set(product_prod(int,int)),rat,abs_rat,X) = aa(set(product_prod(int,int)),rat,abs_rat,Y3) )
        <=> ( X = Y3 ) ) ) ) ).

% Abs_rat_inject
tff(fact_6965_Abs__rat__cases,axiom,
    ! [X: rat] :
      ~ ! [Y4: set(product_prod(int,int))] :
          ( ( X = aa(set(product_prod(int,int)),rat,abs_rat,Y4) )
         => ~ pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y4),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool)))) ) ).

% Abs_rat_cases
tff(fact_6966_Abs__rat__induct,axiom,
    ! [P: fun(rat,bool),X: rat] :
      ( ! [Y4: set(product_prod(int,int))] :
          ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y4),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
         => pp(aa(rat,bool,P,aa(set(product_prod(int,int)),rat,abs_rat,Y4))) )
     => pp(aa(rat,bool,P,X)) ) ).

% Abs_rat_induct
tff(fact_6967_Abs__rat__inverse,axiom,
    ! [Y3: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
     => ( aa(rat,set(product_prod(int,int)),rep_rat,aa(set(product_prod(int,int)),rat,abs_rat,Y3)) = Y3 ) ) ).

% Abs_rat_inverse
tff(fact_6968_type__definition__rat,axiom,
    type_definition(rat,set(product_prod(int,int)),rep_rat,abs_rat,aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))) ).

% type_definition_rat
tff(fact_6969_Rep__rat__induct,axiom,
    ! [Y3: set(product_prod(int,int)),P: fun(set(product_prod(int,int)),bool)] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
     => ( ! [X4: rat] : pp(aa(set(product_prod(int,int)),bool,P,aa(rat,set(product_prod(int,int)),rep_rat,X4)))
       => pp(aa(set(product_prod(int,int)),bool,P,Y3)) ) ) ).

% Rep_rat_induct
tff(fact_6970_Rep__rat__cases,axiom,
    ! [Y3: set(product_prod(int,int))] :
      ( pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),Y3),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool))))
     => ~ ! [X4: rat] : Y3 != aa(rat,set(product_prod(int,int)),rep_rat,X4) ) ).

% Rep_rat_cases
tff(fact_6971_Rep__rat,axiom,
    ! [X: rat] : pp(aa(set(set(product_prod(int,int))),bool,member(set(product_prod(int,int)),aa(rat,set(product_prod(int,int)),rep_rat,X)),aa(fun(set(product_prod(int,int)),bool),set(set(product_prod(int,int))),collect(set(product_prod(int,int))),aTP_Lamp_aqg(set(product_prod(int,int)),bool)))) ).

% Rep_rat
tff(fact_6972_quot__type_Orep__def,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(A,bool)),Abs: fun(set(A),B),Rep: fun(B,set(A)),A3: B] :
      ( quot_type(A,B,R4,Abs,Rep)
     => ( aa(B,A,quot_rep(B,A,Rep),A3) = fChoice(A,aa(B,fun(A,bool),aTP_Lamp_pp(fun(B,set(A)),fun(B,fun(A,bool)),Rep),A3)) ) ) ).

% quot_type.rep_def
tff(fact_6973_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_aqh(fun(C,fun(D,bool)),fun(fun(C,A),fun(set(A),set(C))),A5)),vimage(D,B))) ) ) ).

% vimage_right_total_transfer
tff(fact_6974_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_aqj(fun(A,fun(B,bool)),fun(fun(A,bool),set(A)),A5)),collect(B))) ) ).

% right_total_Collect_transfer
tff(fact_6975_Heap_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(heap_Time_Heap(B),set(B)),bool,aa(fun(heap_Time_Heap(A),set(A)),fun(fun(heap_Time_Heap(B),set(B)),bool),bNF_rel_fun(heap_Time_Heap(A),heap_Time_Heap(B),set(A),set(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4),bNF_rel_set(A,B,R4)),heap_Time_set_Heap(A)),heap_Time_set_Heap(B))) ).

% Heap.set_transfer
tff(fact_6976_option_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool))] : pp(aa(fun(option(B),set(B)),bool,aa(fun(option(A),set(A)),fun(fun(option(B),set(B)),bool),bNF_rel_fun(option(A),option(B),set(A),set(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4),bNF_rel_set(A,B,R4)),set_option(A)),set_option(B))) ).

% option.set_transfer
tff(fact_6977_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_6978_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_aqk(set(A),fun(fun(A,set(C)),set(C)))),aTP_Lamp_aql(set(B),fun(fun(B,set(D)),set(D))))) ).

% UNION_transfer
tff(fact_6979_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_aqm(fun(A,bool),fun(set(A),bool),P)) ).

% set_relator_eq_onp
tff(fact_6980_rel__set__def,axiom,
    ! [A: $tType,B: $tType,R4: fun(A,fun(B,bool)),X2: set(A),Xa3: set(B)] :
      ( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R4),X2),Xa3))
    <=> ( ! [Xb3: A] :
            ( pp(aa(set(A),bool,member(A,Xb3),X2))
           => ? [Xc2: B] :
                ( pp(aa(set(B),bool,member(B,Xc2),Xa3))
                & pp(aa(B,bool,aa(A,fun(B,bool),R4,Xb3),Xc2)) ) )
        & ! [Xb3: B] :
            ( pp(aa(set(B),bool,member(B,Xb3),Xa3))
           => ? [Xc2: A] :
                ( pp(aa(set(A),bool,member(A,Xc2),X2))
                & pp(aa(B,bool,aa(A,fun(B,bool),R4,Xc2),Xb3)) ) ) ) ) ).

% rel_set_def
tff(fact_6981_quot__type_Osome__collect,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(A,bool)),Abs: fun(set(A),B),Rep: fun(B,set(A)),R: A] :
      ( quot_type(A,B,R4,Abs,Rep)
     => ( pp(aa(A,bool,aa(A,fun(A,bool),R4,R),R))
       => ( aa(A,fun(A,bool),R4,fChoice(A,aa(A,fun(A,bool),aTP_Lamp_aqn(fun(A,fun(A,bool)),fun(A,fun(A,bool)),R4),R))) = aa(A,fun(A,bool),R4,R) ) ) ) ).

% quot_type.some_collect
tff(fact_6982_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_aqo(fun(A,fun(B,bool)),fun(set(set(A)),set(A)),A5)),complete_Inf_Inf(set(B)))) ) ) ).

% right_total_Inter_transfer
tff(fact_6983_fun_Oset__transfer,axiom,
    ! [A: $tType,B: $tType,D: $tType,R4: 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),R4),bNF_rel_set(A,B,R4)),aTP_Lamp_aqp(fun(D,A),set(A))),aTP_Lamp_aqq(fun(D,B),set(B)))) ).

% fun.set_transfer
tff(fact_6984_SUP__parametric,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( complete_Sup(C)
     => ! [R4: 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,R4),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,R4,fequal(C)),fequal(C))),aTP_Lamp_aqr(set(A),fun(fun(A,C),C))),aTP_Lamp_aqs(set(B),fun(fun(B,C),C)))) ) ).

% SUP_parametric
tff(fact_6985_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_aqt(set(A),fun(fun(A,C),C))),aTP_Lamp_aqu(set(B),fun(fun(B,C),C)))) ) ).

% INF_parametric
tff(fact_6986_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_aqv(fun(A,fun(B,bool)),fun(set(A),set(A)),A5)),uminus_uminus(set(B)))) ) ) ).

% right_total_Compl_transfer
tff(fact_6987_open__typedef__to__part__equivp,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),P: fun(B,bool)] :
      ( type_definition(A,B,Rep,Abs,aa(fun(B,bool),set(B),collect(B),P))
     => equiv_part_equivp(B,bNF_eq_onp(B,P)) ) ).

% open_typedef_to_part_equivp
tff(fact_6988_typedef__to__part__equivp,axiom,
    ! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),S2: set(B)] :
      ( type_definition(A,B,Rep,Abs,S2)
     => equiv_part_equivp(B,bNF_eq_onp(B,aTP_Lamp_agi(set(B),fun(B,bool),S2))) ) ).

% typedef_to_part_equivp
tff(fact_6989_part__equivp__typedef,axiom,
    ! [A: $tType,R4: fun(A,fun(A,bool))] :
      ( equiv_part_equivp(A,R4)
     => ? [D2: set(A)] : pp(aa(set(set(A)),bool,member(set(A),D2),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_aqw(fun(A,fun(A,bool)),fun(set(A),bool),R4)))) ) ).

% part_equivp_typedef
tff(fact_6990_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_6991_bounded__quasi__semilattice_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => bounde5338082073960204281axioms(A,F,Top,Bot,Normalize) ) ).

% bounded_quasi_semilattice.axioms(2)
tff(fact_6992_Heap_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( bi_total(A,B,R4)
     => bi_total(heap_Time_Heap(A),heap_Time_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Time_Heap(A),fun(heap_Time_Heap(B),bool)),heap_Time_rel_Heap(A,B),R4)) ) ).

% Heap.bi_total_rel
tff(fact_6993_bounded__quasi__semilattice__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Normalize: fun(A,A),Top: A,Bot: A] :
      ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),A4) = aa(A,A,Normalize,A4)
     => ( ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Normalize,A4)),B3) = aa(A,A,aa(A,fun(A,A),F,A4),B3)
       => ( ! [A4: A,B3: A] : aa(A,A,Normalize,aa(A,A,aa(A,fun(A,A),F,A4),B3)) = aa(A,A,aa(A,fun(A,A),F,A4),B3)
         => ( ( aa(A,A,Normalize,Top) = Top )
           => ( ( aa(A,A,Normalize,Bot) = Bot )
             => ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Top),A4) = aa(A,A,Normalize,A4)
               => ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,Bot),A4) = Bot
                 => bounde5338082073960204281axioms(A,F,Top,Bot,Normalize) ) ) ) ) ) ) ) ).

% bounded_quasi_semilattice_axioms.intro
tff(fact_6994_bounded__quasi__semilattice__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde5338082073960204281axioms(A,F,Top,Bot,Normalize)
    <=> ( ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,A7),A7) = aa(A,A,Normalize,A7)
        & ! [A7: A,B7: A] : aa(A,A,aa(A,fun(A,A),F,aa(A,A,Normalize,A7)),B7) = aa(A,A,aa(A,fun(A,A),F,A7),B7)
        & ! [A7: A,B7: A] : aa(A,A,Normalize,aa(A,A,aa(A,fun(A,A),F,A7),B7)) = aa(A,A,aa(A,fun(A,A),F,A7),B7)
        & ( aa(A,A,Normalize,Top) = Top )
        & ( aa(A,A,Normalize,Bot) = Bot )
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Top),A7) = aa(A,A,Normalize,A7)
        & ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,Bot),A7) = Bot ) ) ).

% bounded_quasi_semilattice_axioms_def
tff(fact_6995_int_Obi__total,axiom,
    bi_total(product_prod(nat,nat),int,pcr_int) ).

% int.bi_total
tff(fact_6996_option_Obi__total__rel,axiom,
    ! [B: $tType,A: $tType,R4: fun(A,fun(B,bool))] :
      ( bi_total(A,B,R4)
     => bi_total(option(A),option(B),aa(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),rel_option(A,B),R4)) ) ).

% option.bi_total_rel
tff(fact_6997_bounded__quasi__semilattice__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
    <=> ( abel_semigroup(A,F)
        & bounde5338082073960204281axioms(A,F,Top,Bot,Normalize) ) ) ).

% bounded_quasi_semilattice_def
tff(fact_6998_bounded__quasi__semilattice_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( abel_semigroup(A,F)
     => ( bounde5338082073960204281axioms(A,F,Top,Bot,Normalize)
       => bounde8507323023520639062attice(A,F,Top,Bot,Normalize) ) ) ).

% bounded_quasi_semilattice.intro
tff(fact_6999_abel__semigroup_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
     => semigroup(A,F) ) ).

% abel_semigroup.axioms(1)
tff(fact_7000_bounded__quasi__semilattice_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde8507323023520639062attice(A,F,Top,Bot,Normalize)
     => abel_semigroup(A,F) ) ).

% bounded_quasi_semilattice.axioms(1)
tff(fact_7001_lcm_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => abel_semigroup(A,gcd_lcm(A)) ) ).

% lcm.abel_semigroup_axioms
tff(fact_7002_abel__semigroup_Oleft__commute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),B2: A,A3: A,C2: A] :
      ( abel_semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,B2),aa(A,A,aa(A,fun(A,A),F,A3),C2)) = aa(A,A,aa(A,fun(A,A),F,A3),aa(A,A,aa(A,fun(A,A),F,B2),C2)) ) ) ).

% abel_semigroup.left_commute
tff(fact_7003_abel__semigroup_Ocommute,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A3: A,B2: A] :
      ( abel_semigroup(A,F)
     => ( aa(A,A,aa(A,fun(A,A),F,A3),B2) = aa(A,A,aa(A,fun(A,A),F,B2),A3) ) ) ).

% abel_semigroup.commute
tff(fact_7004_gcd_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => abel_semigroup(A,gcd_gcd(A)) ) ).

% gcd.abel_semigroup_axioms
tff(fact_7005_mult_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => abel_semigroup(A,times_times(A)) ) ).

% mult.abel_semigroup_axioms
tff(fact_7006_comm__monoid_Oaxioms_I1_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( comm_monoid(A,F,Z)
     => abel_semigroup(A,F) ) ).

% comm_monoid.axioms(1)
tff(fact_7007_add_Oabel__semigroup__axioms,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => abel_semigroup(A,plus_plus(A)) ) ).

% add.abel_semigroup_axioms
tff(fact_7008_comm__monoid__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( comm_monoid(A,F,Z)
    <=> ( abel_semigroup(A,F)
        & comm_monoid_axioms(A,F,Z) ) ) ).

% comm_monoid_def
tff(fact_7009_comm__monoid_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( abel_semigroup(A,F)
     => ( comm_monoid_axioms(A,F,Z)
       => comm_monoid(A,F,Z) ) ) ).

% comm_monoid.intro
tff(fact_7010_comm__monoid__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( ! [A4: A] : aa(A,A,aa(A,fun(A,A),F,A4),Z) = A4
     => comm_monoid_axioms(A,F,Z) ) ).

% comm_monoid_axioms.intro
tff(fact_7011_comm__monoid__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( comm_monoid_axioms(A,F,Z)
    <=> ! [A7: A] : aa(A,A,aa(A,fun(A,A),F,A7),Z) = A7 ) ).

% comm_monoid_axioms_def
tff(fact_7012_comm__monoid_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A] :
      ( comm_monoid(A,F,Z)
     => comm_monoid_axioms(A,F,Z) ) ).

% comm_monoid.axioms(2)
tff(fact_7013_abel__semigroup__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
    <=> ( semigroup(A,F)
        & abel_s757365448890700780axioms(A,F) ) ) ).

% abel_semigroup_def
tff(fact_7014_abel__semigroup_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( semigroup(A,F)
     => ( abel_s757365448890700780axioms(A,F)
       => abel_semigroup(A,F) ) ) ).

% abel_semigroup.intro
tff(fact_7015_abel__semigroup__axioms__def,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_s757365448890700780axioms(A,F)
    <=> ! [A7: A,B7: A] : aa(A,A,aa(A,fun(A,A),F,A7),B7) = aa(A,A,aa(A,fun(A,A),F,B7),A7) ) ).

% abel_semigroup_axioms_def
tff(fact_7016_abel__semigroup__axioms_Ointro,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( ! [A4: A,B3: A] : aa(A,A,aa(A,fun(A,A),F,A4),B3) = aa(A,A,aa(A,fun(A,A),F,B3),A4)
     => abel_s757365448890700780axioms(A,F) ) ).

% abel_semigroup_axioms.intro
tff(fact_7017_abel__semigroup_Oaxioms_I2_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A))] :
      ( abel_semigroup(A,F)
     => abel_s757365448890700780axioms(A,F) ) ).

% abel_semigroup.axioms(2)
tff(fact_7018_dual__order_Oordering__axioms,axiom,
    ! [A: $tType] :
      ( order(A)
     => ordering(A,aTP_Lamp_aqx(A,fun(A,bool)),aTP_Lamp_aqy(A,fun(A,bool))) ) ).

% dual_order.ordering_axioms
tff(fact_7019_frequently__cofinite,axiom,
    ! [A: $tType,P: fun(A,bool)] :
      ( frequently(A,P,cofinite(A))
    <=> ~ pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),P))) ) ).

% frequently_cofinite
tff(fact_7020_frequently__const,axiom,
    ! [A: $tType,F5: filter(A),P: bool] :
      ( ( F5 != bot_bot(filter(A)) )
     => ( frequently(A,aTP_Lamp_pl(bool,fun(A,bool),P),F5)
      <=> pp(P) ) ) ).

% frequently_const
tff(fact_7021_frequently__bex__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,fun(A,bool)),F5: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( frequently(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aek(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),F5)
      <=> ? [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),A5))
            & frequently(B,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X3),F5) ) ) ) ).

% frequently_bex_finite_distrib
tff(fact_7022_frequently__bex__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P: fun(B,fun(A,bool)),F5: filter(B)] :
      ( pp(aa(set(A),bool,finite_finite2(A),A5))
     => ( frequently(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_aek(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P),F5)
       => ? [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),A5))
            & frequently(B,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),X4),F5) ) ) ) ).

% frequently_bex_finite
tff(fact_7023_gcd__nat_Oordering__axioms,axiom,
    ordering(nat,dvd_dvd(nat),aTP_Lamp_jg(nat,fun(nat,bool))) ).

% gcd_nat.ordering_axioms
tff(fact_7024_frequently__imp__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( frequently(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
    <=> ( eventually(A,P,F5)
       => frequently(A,Q,F5) ) ) ).

% frequently_imp_iff
tff(fact_7025_frequently__rev__mp,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( frequently(A,P,F5)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
       => frequently(A,Q,F5) ) ) ).

% frequently_rev_mp
tff(fact_7026_not__frequently,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A)] :
      ( ~ frequently(A,P,F5)
    <=> eventually(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P),F5) ) ).

% not_frequently
tff(fact_7027_not__eventually,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A)] :
      ( ~ eventually(A,P,F5)
    <=> frequently(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P),F5) ) ).

% not_eventually
tff(fact_7028_frequently__def,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A)] :
      ( frequently(A,P,F5)
    <=> ~ eventually(A,aTP_Lamp_lv(fun(A,bool),fun(A,bool),P),F5) ) ).

% frequently_def
tff(fact_7029_frequently__mp,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_qm(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
     => ( frequently(A,P,F5)
       => frequently(A,Q,F5) ) ) ).

% frequently_mp
tff(fact_7030_eventually__frequently__const__simps_I5_J,axiom,
    ! [A: $tType,P: fun(A,bool),C6: bool,F5: filter(A)] :
      ( eventually(A,aa(bool,fun(A,bool),aTP_Lamp_aqz(fun(A,bool),fun(bool,fun(A,bool)),P),C6),F5)
    <=> ( frequently(A,P,F5)
       => pp(C6) ) ) ).

% eventually_frequently_const_simps(5)
tff(fact_7031_frequently__all,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),F5: filter(A)] :
      ( frequently(A,aTP_Lamp_aez(fun(A,fun(B,bool)),fun(A,bool),P),F5)
    <=> ! [Y8: fun(A,B)] : frequently(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_afr(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P),Y8),F5) ) ).

% frequently_all
tff(fact_7032_ordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( ordering(A,aTP_Lamp_aap(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_aap(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
     => ordering(A,Less_eq,Less) ) ).

% ordering_dualI
tff(fact_7033_eventually__frequently__const__simps_I2_J,axiom,
    ! [A: $tType,C6: bool,P: fun(A,bool),F5: filter(A)] :
      ( frequently(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ara(bool,fun(fun(A,bool),fun(A,bool)),C6),P),F5)
    <=> ( pp(C6)
        & frequently(A,P,F5) ) ) ).

% eventually_frequently_const_simps(2)
tff(fact_7034_eventually__frequently__const__simps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),C6: bool,F5: filter(A)] :
      ( frequently(A,aa(bool,fun(A,bool),aTP_Lamp_arb(fun(A,bool),fun(bool,fun(A,bool)),P),C6),F5)
    <=> ( frequently(A,P,F5)
        & pp(C6) ) ) ).

% eventually_frequently_const_simps(1)
tff(fact_7035_frequently__ex,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A)] :
      ( frequently(A,P,F5)
     => ? [X_13: A] : pp(aa(A,bool,P,X_13)) ) ).

% frequently_ex
tff(fact_7036_frequently__disj,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( frequently(A,P,F5)
     => ( frequently(A,Q,F5)
       => frequently(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ig(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5) ) ) ).

% frequently_disj
tff(fact_7037_frequently__elim1,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( frequently(A,P,F5)
     => ( ! [I2: A] :
            ( pp(aa(A,bool,P,I2))
           => pp(aa(A,bool,Q,I2)) )
       => frequently(A,Q,F5) ) ) ).

% frequently_elim1
tff(fact_7038_frequently__disj__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),F5: filter(A)] :
      ( frequently(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ig(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
    <=> ( frequently(A,P,F5)
        | frequently(A,Q,F5) ) ) ).

% frequently_disj_iff
tff(fact_7039_not__frequently__False,axiom,
    ! [A: $tType,F5: filter(A)] : ~ frequently(A,aTP_Lamp_ap(A,bool),F5) ).

% not_frequently_False
tff(fact_7040_frequently__const__iff,axiom,
    ! [A: $tType,P: bool,F5: filter(A)] :
      ( frequently(A,aTP_Lamp_pl(bool,fun(A,bool),P),F5)
    <=> ( pp(P)
        & ( F5 != bot_bot(filter(A)) ) ) ) ).

% frequently_const_iff
tff(fact_7041_eventually__frequentlyE,axiom,
    ! [A: $tType,P: fun(A,bool),F5: filter(A),Q: fun(A,bool)] :
      ( eventually(A,P,F5)
     => ( eventually(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_arc(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q),F5)
       => ( ( F5 != bot_bot(filter(A)) )
         => frequently(A,Q,F5) ) ) ) ).

% eventually_frequentlyE
tff(fact_7042_prod__list_Omonoid__list__axioms,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => groups_monoid_list(A,times_times(A),one_one(A)) ) ).

% prod_list.monoid_list_axioms
tff(fact_7043_case__sum__o__map__sum__id,axiom,
    ! [A: $tType,B: $tType,C: $tType,G: fun(B,A),F: fun(C,A),X: sum_sum(C,B)] : aa(sum_sum(C,B),A,aa(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A),aa(fun(sum_sum(A,B),A),fun(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A)),comp(sum_sum(A,B),A,sum_sum(C,B)),aa(fun(B,A),fun(sum_sum(A,B),A),aa(fun(A,A),fun(fun(B,A),fun(sum_sum(A,B),A)),sum_case_sum(A,A,B),id(A)),G)),sum_map_sum(C,A,B,B,F,id(B))),X) = aa(sum_sum(C,B),A,aa(fun(B,A),fun(sum_sum(C,B),A),aa(fun(C,A),fun(fun(B,A),fun(sum_sum(C,B),A)),sum_case_sum(C,A,B),aa(fun(C,C),fun(C,A),aa(fun(C,A),fun(fun(C,C),fun(C,A)),comp(C,A,C),F),id(C))),G),X) ).

% case_sum_o_map_sum_id
tff(fact_7044_map__sum_Oidentity,axiom,
    ! [B: $tType,A: $tType] : sum_map_sum(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)) = id(sum_sum(A,B)) ).

% map_sum.identity
tff(fact_7045_sum_Omap__ident,axiom,
    ! [B: $tType,A: $tType,T2: sum_sum(A,B)] : aa(sum_sum(A,B),sum_sum(A,B),sum_map_sum(A,A,B,B,aTP_Lamp_ak(A,A),aTP_Lamp_lf(B,B)),T2) = T2 ).

% sum.map_ident
tff(fact_7046_map__sum__if__distrib__then_I2_J,axiom,
    ! [H6: $tType,F2: $tType,G2: $tType,E: $tType,E2: bool,F: fun(E,F2),G: fun(G2,H6),X: G2,Y3: sum_sum(E,G2)] :
      ( ( pp(E2)
       => ( aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),aa(sum_sum(E,G2),sum_sum(E,G2),aa(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2)),aa(bool,fun(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2))),if(sum_sum(E,G2)),E2),aa(G2,sum_sum(E,G2),sum_Inr(G2,E),X)),Y3)) = aa(H6,sum_sum(F2,H6),sum_Inr(H6,F2),aa(G2,H6,G,X)) ) )
      & ( ~ pp(E2)
       => ( aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),aa(sum_sum(E,G2),sum_sum(E,G2),aa(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2)),aa(bool,fun(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2))),if(sum_sum(E,G2)),E2),aa(G2,sum_sum(E,G2),sum_Inr(G2,E),X)),Y3)) = aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),Y3) ) ) ) ).

% map_sum_if_distrib_then(2)
tff(fact_7047_map__sum__if__distrib__else_I2_J,axiom,
    ! [E: $tType,F2: $tType,H6: $tType,G2: $tType,E2: bool,F: fun(E,F2),G: fun(G2,H6),X: sum_sum(E,G2),Y3: G2] :
      ( ( pp(E2)
       => ( aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),aa(sum_sum(E,G2),sum_sum(E,G2),aa(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2)),aa(bool,fun(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2))),if(sum_sum(E,G2)),E2),X),aa(G2,sum_sum(E,G2),sum_Inr(G2,E),Y3))) = aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),X) ) )
      & ( ~ pp(E2)
       => ( aa(sum_sum(E,G2),sum_sum(F2,H6),sum_map_sum(E,F2,G2,H6,F,G),aa(sum_sum(E,G2),sum_sum(E,G2),aa(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2)),aa(bool,fun(sum_sum(E,G2),fun(sum_sum(E,G2),sum_sum(E,G2))),if(sum_sum(E,G2)),E2),X),aa(G2,sum_sum(E,G2),sum_Inr(G2,E),Y3))) = aa(H6,sum_sum(F2,H6),sum_Inr(H6,F2),aa(G2,H6,G,Y3)) ) ) ) ).

% map_sum_if_distrib_else(2)
tff(fact_7048_map__sum__if__distrib__then_I1_J,axiom,
    ! [D: $tType,B: $tType,C: $tType,A: $tType,E2: bool,F: fun(A,B),G: fun(C,D),X: A,Y3: sum_sum(A,C)] :
      ( ( pp(E2)
       => ( aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),aa(sum_sum(A,C),sum_sum(A,C),aa(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C)),aa(bool,fun(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C))),if(sum_sum(A,C)),E2),aa(A,sum_sum(A,C),sum_Inl(A,C),X)),Y3)) = aa(B,sum_sum(B,D),sum_Inl(B,D),aa(A,B,F,X)) ) )
      & ( ~ pp(E2)
       => ( aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),aa(sum_sum(A,C),sum_sum(A,C),aa(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C)),aa(bool,fun(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C))),if(sum_sum(A,C)),E2),aa(A,sum_sum(A,C),sum_Inl(A,C),X)),Y3)) = aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),Y3) ) ) ) ).

% map_sum_if_distrib_then(1)
tff(fact_7049_map__sum__if__distrib__else_I1_J,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,E2: bool,F: fun(A,B),G: fun(C,D),X: sum_sum(A,C),Y3: A] :
      ( ( pp(E2)
       => ( aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),aa(sum_sum(A,C),sum_sum(A,C),aa(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C)),aa(bool,fun(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C))),if(sum_sum(A,C)),E2),X),aa(A,sum_sum(A,C),sum_Inl(A,C),Y3))) = aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),X) ) )
      & ( ~ pp(E2)
       => ( aa(sum_sum(A,C),sum_sum(B,D),sum_map_sum(A,B,C,D,F,G),aa(sum_sum(A,C),sum_sum(A,C),aa(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C)),aa(bool,fun(sum_sum(A,C),fun(sum_sum(A,C),sum_sum(A,C))),if(sum_sum(A,C)),E2),X),aa(A,sum_sum(A,C),sum_Inl(A,C),Y3))) = aa(B,sum_sum(B,D),sum_Inl(B,D),aa(A,B,F,Y3)) ) ) ) ).

% map_sum_if_distrib_else(1)
tff(fact_7050_map__sum_Ocomp,axiom,
    ! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F2)),aa(fun(sum_sum(C,D),sum_sum(E,F2)),fun(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F2))),comp(sum_sum(C,D),sum_sum(E,F2),sum_sum(A,B)),sum_map_sum(C,E,D,F2,F,G)),sum_map_sum(A,C,B,D,H,I)) = sum_map_sum(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).

% map_sum.comp
tff(fact_7051_map__sum_Ocompositionality,axiom,
    ! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D),Sum: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F2),sum_map_sum(C,E,D,F2,F,G),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,H,I),Sum)) = aa(sum_sum(A,B),sum_sum(E,F2),sum_map_sum(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Sum) ).

% map_sum.compositionality
tff(fact_7052_sum_Omap__comp,axiom,
    ! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F2),F1: fun(A,C),F22: fun(B,D),V: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F2),sum_map_sum(C,E,D,F2,G1,G22),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),V)) = aa(sum_sum(A,B),sum_sum(E,F2),sum_map_sum(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G22),F22)),V) ).

% sum.map_comp
tff(fact_7053_map__sum__o__inj_I1_J,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(D,C)] : aa(fun(A,sum_sum(A,D)),fun(A,sum_sum(B,C)),aa(fun(sum_sum(A,D),sum_sum(B,C)),fun(fun(A,sum_sum(A,D)),fun(A,sum_sum(B,C))),comp(sum_sum(A,D),sum_sum(B,C),A),sum_map_sum(A,B,D,C,F,G)),sum_Inl(A,D)) = aa(fun(A,B),fun(A,sum_sum(B,C)),aa(fun(B,sum_sum(B,C)),fun(fun(A,B),fun(A,sum_sum(B,C))),comp(B,sum_sum(B,C),A),sum_Inl(B,C)),F) ).

% map_sum_o_inj(1)
tff(fact_7054_map__sum__o__inj_I2_J,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(A,B),G: fun(D,C)] : aa(fun(D,sum_sum(A,D)),fun(D,sum_sum(B,C)),aa(fun(sum_sum(A,D),sum_sum(B,C)),fun(fun(D,sum_sum(A,D)),fun(D,sum_sum(B,C))),comp(sum_sum(A,D),sum_sum(B,C),D),sum_map_sum(A,B,D,C,F,G)),sum_Inr(D,A)) = aa(fun(D,C),fun(D,sum_sum(B,C)),aa(fun(C,sum_sum(B,C)),fun(fun(D,C),fun(D,sum_sum(B,C))),comp(C,sum_sum(B,C),D),sum_Inr(C,B)),G) ).

% map_sum_o_inj(2)
tff(fact_7055_case__sum__o__map__sum,axiom,
    ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,C),G: fun(E,C),H1: fun(A,D),H22: fun(B,E)] : aa(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C),aa(fun(sum_sum(D,E),C),fun(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C)),comp(sum_sum(D,E),C,sum_sum(A,B)),aa(fun(E,C),fun(sum_sum(D,E),C),aa(fun(D,C),fun(fun(E,C),fun(sum_sum(D,E),C)),sum_case_sum(D,C,E),F),G)),sum_map_sum(A,D,B,E,H1,H22)) = aa(fun(B,C),fun(sum_sum(A,B),C),aa(fun(A,C),fun(fun(B,C),fun(sum_sum(A,B),C)),sum_case_sum(A,C,B),aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),F),H1)),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),G),H22)) ).

% case_sum_o_map_sum
tff(fact_7056_case__sum__map__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: fun(B,A),R: fun(C,A),F: fun(D,B),G: fun(E,C),X: sum_sum(D,E)] : aa(sum_sum(B,C),A,aa(fun(C,A),fun(sum_sum(B,C),A),aa(fun(B,A),fun(fun(C,A),fun(sum_sum(B,C),A)),sum_case_sum(B,A,C),L),R),aa(sum_sum(D,E),sum_sum(B,C),sum_map_sum(D,B,E,C,F,G),X)) = aa(sum_sum(D,E),A,aa(fun(E,A),fun(sum_sum(D,E),A),aa(fun(D,A),fun(fun(E,A),fun(sum_sum(D,E),A)),sum_case_sum(D,A,E),aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),L),F)),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),R),G)),X) ).

% case_sum_map_sum
tff(fact_7057_sum_Osize__gen__o__map,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),nat),aa(fun(sum_sum(C,D),nat),fun(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),nat)),comp(sum_sum(C,D),nat,sum_sum(A,B)),basic_BNF_size_sum(C,D,F,Fa)),sum_map_sum(A,C,B,D,G,Ga)) = basic_BNF_size_sum(A,B,aa(fun(A,C),fun(A,nat),aa(fun(C,nat),fun(fun(A,C),fun(A,nat)),comp(C,nat,A),F),G),aa(fun(B,D),fun(B,nat),aa(fun(D,nat),fun(fun(B,D),fun(B,nat)),comp(D,nat,B),Fa),Ga)) ).

% sum.size_gen_o_map
tff(fact_7058_sum__list_Omonoid__list__axioms,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => groups_monoid_list(A,plus_plus(A),zero_zero(A)) ) ).

% sum_list.monoid_list_axioms
tff(fact_7059_sum_Orel__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,A13: set(A),F1: fun(A,C),A24: set(B),F22: fun(B,D)] : bNF_rel_sum(A,C,B,D,bNF_Grp(A,C,A13,F1),bNF_Grp(B,D,A24,F22)) = bNF_Grp(sum_sum(A,B),sum_sum(C,D),aa(fun(sum_sum(A,B),bool),set(sum_sum(A,B)),collect(sum_sum(A,B)),aa(set(B),fun(sum_sum(A,B),bool),aTP_Lamp_ard(set(A),fun(set(B),fun(sum_sum(A,B),bool)),A13),A24)),sum_map_sum(A,C,B,D,F1,F22)) ).

% sum.rel_Grp
tff(fact_7060_sum_Oin__rel,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R12: fun(A,fun(C,bool)),R23: fun(B,fun(D,bool)),A3: sum_sum(A,B),B2: sum_sum(C,D)] :
      ( pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R12,R23),A3),B2))
    <=> ? [Z5: sum_sum(product_prod(A,C),product_prod(B,D))] :
          ( pp(aa(set(sum_sum(product_prod(A,C),product_prod(B,D))),bool,member(sum_sum(product_prod(A,C),product_prod(B,D)),Z5),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_are(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R12),R23))))
          & ( aa(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),sum_map_sum(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)),Z5) = A3 )
          & ( aa(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),sum_map_sum(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)),Z5) = B2 ) ) ) ).

% sum.in_rel
tff(fact_7061_sum_Orel__map_I1_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1b: fun(E,fun(C,bool)),S2b: fun(F2,fun(D,bool)),I1: fun(A,E),I22: fun(B,F2),X: sum_sum(A,B),Y3: sum_sum(C,D)] :
      ( pp(aa(sum_sum(C,D),bool,aa(sum_sum(E,F2),fun(sum_sum(C,D),bool),bNF_rel_sum(E,C,F2,D,S1b,S2b),aa(sum_sum(A,B),sum_sum(E,F2),sum_map_sum(A,E,B,F2,I1,I22),X)),Y3))
    <=> pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,aa(fun(A,E),fun(A,fun(C,bool)),aTP_Lamp_ali(fun(E,fun(C,bool)),fun(fun(A,E),fun(A,fun(C,bool))),S1b),I1),aa(fun(B,F2),fun(B,fun(D,bool)),aTP_Lamp_alj(fun(F2,fun(D,bool)),fun(fun(B,F2),fun(B,fun(D,bool))),S2b),I22)),X),Y3)) ) ).

% sum.rel_map(1)
tff(fact_7062_sum_Orel__map_I2_J,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,S1a: fun(A,fun(E,bool)),S2a: fun(B,fun(F2,bool)),X: sum_sum(A,B),G1: fun(C,E),G22: fun(D,F2),Y3: sum_sum(C,D)] :
      ( pp(aa(sum_sum(E,F2),bool,aa(sum_sum(A,B),fun(sum_sum(E,F2),bool),bNF_rel_sum(A,E,B,F2,S1a,S2a),X),aa(sum_sum(C,D),sum_sum(E,F2),sum_map_sum(C,E,D,F2,G1,G22),Y3)))
    <=> pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,aa(fun(C,E),fun(A,fun(C,bool)),aTP_Lamp_alg(fun(A,fun(E,bool)),fun(fun(C,E),fun(A,fun(C,bool))),S1a),G1),aa(fun(D,F2),fun(B,fun(D,bool)),aTP_Lamp_alh(fun(B,fun(F2,bool)),fun(fun(D,F2),fun(B,fun(D,bool))),S2a),G22)),X),Y3)) ) ).

% sum.rel_map(2)
tff(fact_7063_case__sum__transfer,axiom,
    ! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,R4: fun(A,fun(C,bool)),T4: fun(B,fun(D,bool)),S2: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(F2,D),fun(sum_sum(C,F2),D))),bool,aa(fun(fun(A,B),fun(fun(E,B),fun(sum_sum(A,E),B))),fun(fun(fun(C,D),fun(fun(F2,D),fun(sum_sum(C,F2),D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,B),fun(sum_sum(A,E),B)),fun(fun(F2,D),fun(sum_sum(C,F2),D)),bNF_rel_fun(A,C,B,D,R4,T4),bNF_rel_fun(fun(E,B),fun(F2,D),fun(sum_sum(A,E),B),fun(sum_sum(C,F2),D),bNF_rel_fun(E,F2,B,D,S2,T4),bNF_rel_fun(sum_sum(A,E),sum_sum(C,F2),B,D,bNF_rel_sum(A,C,E,F2,R4,S2),T4))),sum_case_sum(A,B,E)),sum_case_sum(C,D,F2))) ).

% case_sum_transfer
tff(fact_7064_Inr__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,T4: fun(A,fun(B,bool)),S2: fun(C,fun(D,bool))] : pp(aa(fun(B,sum_sum(D,B)),bool,aa(fun(A,sum_sum(C,A)),fun(fun(B,sum_sum(D,B)),bool),bNF_rel_fun(A,B,sum_sum(C,A),sum_sum(D,B),T4,bNF_rel_sum(C,D,A,B,S2,T4)),sum_Inr(A,C)),sum_Inr(B,D))) ).

% Inr_transfer
tff(fact_7065_Inl__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,S2: fun(A,fun(B,bool)),T4: fun(C,fun(D,bool))] : pp(aa(fun(B,sum_sum(B,D)),bool,aa(fun(A,sum_sum(A,C)),fun(fun(B,sum_sum(B,D)),bool),bNF_rel_fun(A,B,sum_sum(A,C),sum_sum(B,D),S2,bNF_rel_sum(A,B,C,D,S2,T4)),sum_Inl(A,C)),sum_Inl(B,D))) ).

% Inl_transfer
tff(fact_7066_sum_Orel__compp__Grp,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,R12: fun(A,fun(C,bool)),R23: fun(B,fun(D,bool))] : bNF_rel_sum(A,C,B,D,R12,R23) = aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),bool)),fun(sum_sum(A,B),fun(sum_sum(C,D),bool)),aa(fun(sum_sum(A,B),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),fun(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),bool)),fun(sum_sum(A,B),fun(sum_sum(C,D),bool))),relcompp(sum_sum(A,B),sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D)),conversep(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_are(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R12),R23)),sum_map_sum(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D))))),bNF_Grp(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_are(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R12),R23)),sum_map_sum(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)))) ).

% sum.rel_compp_Grp
tff(fact_7067_typerep_Osize__neq,axiom,
    ! [X: typerep] : aa(typerep,nat,size_size(typerep),X) != zero_zero(nat) ).

% typerep.size_neq
tff(fact_7068_dual__order_Opreordering__axioms,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => preordering(A,aTP_Lamp_aou(A,fun(A,bool)),aTP_Lamp_afd(A,fun(A,bool))) ) ).

% dual_order.preordering_axioms
tff(fact_7069_preordering__dualI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool))] :
      ( preordering(A,aTP_Lamp_aap(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less_eq),aTP_Lamp_aap(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Less))
     => preordering(A,Less_eq,Less) ) ).

% preordering_dualI
tff(fact_7070_gcd__nat_Opreordering__axioms,axiom,
    preordering(nat,dvd_dvd(nat),aTP_Lamp_jg(nat,fun(nat,bool))) ).

% gcd_nat.preordering_axioms
tff(fact_7071_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X22: list(typerep)] : aa(typerep,nat,size_size(typerep),typerep2(X1,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(typerep),nat,size_list(typerep,size_size(typerep)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% typerep.size(2)
tff(fact_7072_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_arf(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ) ).

% admissible_disj
tff(fact_7073_admissible__True,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool))] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_aq(A,bool)) ).

% admissible_True
tff(fact_7074_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_av(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)) ) ) ).

% admissible_conj
tff(fact_7075_admissible__const,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,bool)),T2: bool] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_pl(bool,fun(A,bool),T2)) ).

% admissible_const
tff(fact_7076_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,bool)),P: fun(B,fun(A,bool))] :
      ( ! [Y4: A] : comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),Y4))
     => comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_arg(fun(B,fun(A,bool)),fun(B,bool),P)) ) ).

% admissible_all
tff(fact_7077_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))] :
      ( ! [Y4: A] :
          ( pp(aa(set(A),bool,member(A,Y4),A5))
         => comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,bool),aTP_Lamp_tw(fun(B,fun(A,bool)),fun(A,fun(B,bool)),P),Y4)) )
     => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_agb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),A5),P)) ) ).

% admissible_ball
tff(fact_7078_typerep_Osize__gen,axiom,
    ! [X1: literal,X22: list(typerep)] : aa(typerep,nat,size_typerep,typerep2(X1,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(typerep),nat,size_list(typerep,size_typerep),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% typerep.size_gen
tff(fact_7079_typerep_Orec,axiom,
    ! [A: $tType,F: fun(literal,fun(list(product_prod(typerep,A)),A)),X1: literal,X22: list(typerep)] : rec_typerep(A,F,typerep2(X1,X22)) = aa(list(product_prod(typerep,A)),A,aa(literal,fun(list(product_prod(typerep,A)),A),F,X1),aa(list(typerep),list(product_prod(typerep,A)),map(typerep,product_prod(typerep,A),aTP_Lamp_arh(fun(literal,fun(list(product_prod(typerep,A)),A)),fun(typerep,product_prod(typerep,A)),F)),X22)) ).

% typerep.rec
tff(fact_7080_comm__monoid__set_Ozero__middle,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_ari(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Z),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_arj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% comm_monoid_set.zero_middle
tff(fact_7081_comm__monoid__set_Oinsert_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),P2: fun(B,A),I: B] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),P2))))
       => ( ( pp(aa(set(B),bool,member(B,I),I5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),I5) ) )
          & ( ~ pp(aa(set(B),bool,member(B,I),I5))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),aa(set(B),set(B),insert2(B,I),I5)) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,P2,I)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),I5)) ) ) ) ) ) ).

% comm_monoid_set.insert'
tff(fact_7082_comm__monoid__set_OUNION__disjoint,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),A5: fun(B,set(C)),G: fun(C,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),I5))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),I5))
             => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),A5,X4))) )
         => ( ! [X4: B] :
                ( pp(aa(set(B),bool,member(B,X4),I5))
               => ! [Xa4: B] :
                    ( pp(aa(set(B),bool,member(B,Xa4),I5))
                   => ( ( X4 != Xa4 )
                     => ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A5,X4)),aa(B,set(C),A5,Xa4)) = bot_bot(set(C)) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),G),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),groups_comm_monoid_F(A,B,F,Z),aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_arl(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F),Z),A5),G)),I5) ) ) ) ) ) ).

% comm_monoid_set.UNION_disjoint
tff(fact_7083_comm__monoid__set_OIf__cases,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_qe(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A5) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),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),groups_comm_monoid_F(A,B,F,Z),G),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))))) ) ) ) ).

% comm_monoid_set.If_cases
tff(fact_7084_comm__monoid__set_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),G: fun(B,A),B4: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),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),groups_comm_monoid_F(A,B,F,Z),aa(set(B),fun(B,A),aa(fun(B,A),fun(set(B),fun(B,A)),aTP_Lamp_arm(A,fun(fun(B,A),fun(set(B),fun(B,A))),Z),G),B4)),A5) ) ) ) ).

% comm_monoid_set.inter_restrict
tff(fact_7085_comm__monoid__set_Oub__add__nat,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,M: nat,N: nat,G: fun(nat,A),P2: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( 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),N),one_one(nat))))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% comm_monoid_set.ub_add_nat
tff(fact_7086_comm__monoid__add__class_Osum__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ( groups7311177749621191930dd_sum(B,A) = groups_comm_monoid_F(A,B,plus_plus(A),zero_zero(A)) ) ) ).

% comm_monoid_add_class.sum_def
tff(fact_7087_comm__monoid__set_Onat__group,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),K: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_arn(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),F),Z),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ) ).

% comm_monoid_set.nat_group
tff(fact_7088_comm__monoid__set_OatLeastLessThan__rev,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat,M: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_aro(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev
tff(fact_7089_sum_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => groups778175481326437816id_set(A,plus_plus(A),zero_zero(A)) ) ).

% sum.comm_monoid_set_axioms
tff(fact_7090_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aTP_Lamp_arp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.shift_bounds_nat_ivl
tff(fact_7091_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aTP_Lamp_arp(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.shift_bounds_cl_nat_ivl
tff(fact_7092_comm__monoid__set_OatLeastAtMost__rev,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat,M: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_arq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ) ).

% comm_monoid_set.atLeastAtMost_rev
tff(fact_7093_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat,M: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_arq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ) ).

% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_7094_comm__monoid__mult__class_Oprod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ( groups7121269368397514597t_prod(B,A) = groups_comm_monoid_F(A,B,times_times(A),one_one(A)) ) ) ).

% comm_monoid_mult_class.prod_def
tff(fact_7095_comm__monoid__set_Onested__swap,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,A3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_arr(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_art(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F),Z),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% comm_monoid_set.nested_swap
tff(fact_7096_comm__monoid__set_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),X: fun(B,A),Y3: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),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),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),Y3))))
         => 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)),aa(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool)))),aTP_Lamp_aru(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))))),F),Z),I5),X),Y3)))) ) ) ) ).

% comm_monoid_set.finite_Collect_op
tff(fact_7097_comm__monoid__set_Odelta,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z: A,S2: set(B),A3: B,B2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S2))
       => ( ( pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arv(A,fun(B,fun(fun(B,A),fun(B,A))),Z),A3),B2)),S2) = aa(B,A,B2,A3) ) )
          & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arv(A,fun(B,fun(fun(B,A),fun(B,A))),Z),A3),B2)),S2) = Z ) ) ) ) ) ).

% comm_monoid_set.delta
tff(fact_7098_comm__monoid__set_Odelta_H,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z: A,S2: set(B),A3: B,B2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S2))
       => ( ( pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arw(A,fun(B,fun(fun(B,A),fun(B,A))),Z),A3),B2)),S2) = aa(B,A,B2,A3) ) )
          & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arw(A,fun(B,fun(fun(B,A),fun(B,A))),Z),A3),B2)),S2) = Z ) ) ) ) ) ).

% comm_monoid_set.delta'
tff(fact_7099_comm__monoid__set_Ointer__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),G: fun(B,A),P: fun(B,bool)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),aa(fun(B,bool),set(B),collect(B),aa(fun(B,bool),fun(B,bool),aTP_Lamp_hw(set(B),fun(fun(B,bool),fun(B,bool)),A5),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,bool),fun(B,A),aa(fun(B,A),fun(fun(B,bool),fun(B,A)),aTP_Lamp_arx(A,fun(fun(B,A),fun(fun(B,bool),fun(B,A))),Z),G),P)),A5) ) ) ) ).

% comm_monoid_set.inter_filter
tff(fact_7100_comm__monoid__set_Oswap__restrict,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),B4: set(C),G: fun(B,fun(C,A)),R4: fun(B,fun(C,bool))] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( 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),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,fun(C,bool)),fun(B,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)),aa(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),aa(A,fun(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)))),aTP_Lamp_ary(fun(A,fun(A,A)),fun(A,fun(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))))),F),Z),B4),G),R4)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),aa(fun(B,fun(C,bool)),fun(C,A),aa(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)),aa(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),aa(A,fun(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)))),aTP_Lamp_asa(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))))),F),Z),A5),G),R4)),B4) ) ) ) ) ).

% comm_monoid_set.swap_restrict
tff(fact_7101_comm__monoid__set_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),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),aa(fun(B,A),fun(B,bool),aTP_Lamp_asb(A,fun(fun(B,A),fun(B,bool)),Z),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),A5) ) ) ) ).

% comm_monoid_set.setdiff_irrelevant
tff(fact_7102_comm__monoid__set_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,S5: set(B),T3: set(C),H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S5))
       => ( pp(aa(set(C),bool,finite_finite2(C),T3))
         => ( bij_betw(B,C,H,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),S5),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),minus_minus(set(C)),T4),T3))
           => ( ! [A4: B] :
                  ( pp(aa(set(B),bool,member(B,A4),S5))
                 => ( aa(C,A,G,aa(B,C,H,A4)) = Z ) )
             => ( ! [B3: C] :
                    ( pp(aa(set(C),bool,member(C,B3),T3))
                   => ( aa(C,A,G,B3) = Z ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(C,A),fun(B,A),aTP_Lamp_asc(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),G),T4) ) ) ) ) ) ) ) ).

% comm_monoid_set.reindex_bij_betw_not_neutral
tff(fact_7103_prod_Ocomm__monoid__set__axioms,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => groups778175481326437816id_set(A,times_times(A),one_one(A)) ) ).

% prod.comm_monoid_set_axioms
tff(fact_7104_comm__monoid__set_Oreindex__bij__betw,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,H: fun(B,C),S2: set(B),T4: set(C),G: fun(C,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( bij_betw(B,C,H,S2,T4)
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(C,A),fun(B,A),aTP_Lamp_asc(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),G),T4) ) ) ) ).

% comm_monoid_set.reindex_bij_betw
tff(fact_7105_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.shift_bounds_Suc_ivl
tff(fact_7106_comm__monoid__set_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,fun(C,A)),B4: set(C),A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(set(C),fun(B,A),aa(fun(B,fun(C,A)),fun(set(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A))),aTP_Lamp_asd(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A)))),F),Z),G),B4)),A5) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),aa(set(B),fun(C,A),aa(fun(B,fun(C,A)),fun(set(B),fun(C,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(B),fun(C,A))),aTP_Lamp_ase(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(B),fun(C,A)))),F),Z),G),A5)),B4) ) ) ).

% comm_monoid_set.swap
tff(fact_7107_comm__monoid__set_Odistrib,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,A),H: fun(B,A),A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),F),G),H)),A5) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),A5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),H),A5)) ) ) ).

% comm_monoid_set.distrib
tff(fact_7108_comm__monoid__set_Oneutral__const,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(A,fun(B,A),aTP_Lamp_cu(A,fun(B,A)),Z)),A5) = Z ) ) ).

% comm_monoid_set.neutral_const
tff(fact_7109_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.shift_bounds_cl_Suc_ivl
tff(fact_7110_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc
tff(fact_7111_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc
tff(fact_7112_comm__monoid__set_OatMost__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ) ).

% comm_monoid_set.atMost_Suc_shift
tff(fact_7113_comm__monoid__set_OlessThan__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% comm_monoid_set.lessThan_Suc_shift
tff(fact_7114_comm__monoid__set_Onat__diff__reindex,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aTP_Lamp_asg(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.nat_diff_reindex
tff(fact_7115_comm__monoid__set_OatLeast1__atMost__eq,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.atLeast1_atMost_eq
tff(fact_7116_comm__monoid__set_OatMost__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% comm_monoid_set.atMost_shift
tff(fact_7117_comm__monoid__set_Onested__swap_H,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,A3: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ash(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z),A3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_art(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),F),Z),A3),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.nested_swap'
tff(fact_7118_comm__monoid__set_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,A),I5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),G),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_asi(A,fun(fun(B,A),fun(set(B),fun(B,bool))),Z),G),I5))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),G),I5) ) ) ).

% comm_monoid_set.non_neutral'
tff(fact_7119_comm__monoid__set_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),G: fun(B,A),H: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),I5))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),F),G),H)),I5) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),H),I5)) ) ) ) ).

% comm_monoid_set.distrib_triv'
tff(fact_7120_comm__monoid__set_Odistrib_H,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),G: fun(B,A),H: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),G))))
       => ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),H))))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),F),G),H)),I5) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),G),I5)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),H),I5)) ) ) ) ) ).

% comm_monoid_set.distrib'
tff(fact_7121_comm__monoid__set_OG__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,fun(A,A)),Z: A,I5: set(B),P2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( ( pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),P2))))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),P2),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),P2))) ) )
        & ( ~ pp(aa(set(B),bool,finite_finite2(B),aa(fun(B,bool),set(B),collect(B),aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Z),I5),P2))))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_G(A,B,F,Z),P2),I5) = Z ) ) ) ) ).

% comm_monoid_set.G_def
tff(fact_7122_comm__monoid__set_OSuc__reindex__ivl,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,M: nat,N: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(A,A,aa(A,fun(A,A),F,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aTP_Lamp_zy(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% comm_monoid_set.Suc_reindex_ivl
tff(fact_7123_comm__monoid__set_OSigma,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),B4: fun(B,set(C)),G: fun(B,fun(C,A))] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( ! [X4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => pp(aa(set(C),bool,finite_finite2(C),aa(B,set(C),B4,X4))) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,fun(C,A)),fun(B,A),aa(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A))),aTP_Lamp_asj(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)))),F),Z),B4),G)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups_comm_monoid_F(A,product_prod(B,C),F,Z),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A5,B4)) ) ) ) ) ).

% comm_monoid_set.Sigma
tff(fact_7124_comm__monoid__set_Ocartesian__product,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,fun(C,A)),B4: set(C),A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(set(C),fun(B,A),aa(fun(B,fun(C,A)),fun(set(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A))),aTP_Lamp_asd(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A)))),F),Z),G),B4)),A5) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups_comm_monoid_F(A,product_prod(B,C),F,Z),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A5,aTP_Lamp_xm(set(C),fun(B,set(C)),B4))) ) ) ).

% comm_monoid_set.cartesian_product
tff(fact_7125_comm__monoid__set_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,S2: set(B),T4: set(C),G: fun(B,C),H: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S2))
       => ( pp(aa(set(C),bool,finite_finite2(C),T4))
         => ( 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,G),S2)),T4))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aa(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),aa(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A)))),aTP_Lamp_ask(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))))),F),Z),S2),G),H)),T4) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),H),S2) ) ) ) ) ) ).

% comm_monoid_set.group
tff(fact_7126_comm__monoid__set_Odelta__remove,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,S2: set(B),A3: B,B2: fun(B,A),C2: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S2))
       => ( ( pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),F,aa(B,A,B2,A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B)))))) ) )
          & ( ~ pp(aa(set(B),bool,member(B,A3),S2))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A3),B2),C2)),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),S2),aa(set(B),set(B),insert2(B,A3),bot_bot(set(B))))) ) ) ) ) ) ).

% comm_monoid_set.delta_remove
tff(fact_7127_comm__monoid__set_Oimage__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,A),A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( inj_on(B,A,G,A5)
       => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups_comm_monoid_F(A,A,F,Z),aTP_Lamp_ak(A,A)),aa(set(B),set(A),image2(B,A,G),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),A5) ) ) ) ).

% comm_monoid_set.image_eq
tff(fact_7128_comm__monoid__set_Oimage__gen,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,S2: set(B),H: fun(B,A),G: fun(B,C)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),S2))
       => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),H),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aa(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A)))),aTP_Lamp_asm(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))))),F),Z),S2),H),G)),aa(set(B),set(C),image2(B,C,G),S2)) ) ) ) ).

% comm_monoid_set.image_gen
tff(fact_7129_comm__monoid__set_Oin__pairs,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,A),fun(nat,A),aTP_Lamp_asn(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.in_pairs
tff(fact_7130_comm__monoid__set_Oin__pairs__0,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,A),fun(nat,A),aTP_Lamp_asn(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),F),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% comm_monoid_set.in_pairs_0
tff(fact_7131_comm__monoid__set_OUnion__disjoint,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,C6: set(set(B)),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( ! [X4: set(B)] :
            ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
           => pp(aa(set(B),bool,finite_finite2(B),X4)) )
       => ( ! [X4: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X4),C6))
             => ! [Xa4: set(B)] :
                  ( pp(aa(set(set(B)),bool,member(set(B),Xa4),C6))
                 => ( ( X4 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),Xa4) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C6)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),groups_comm_monoid_F(A,set(B),F,Z)),groups_comm_monoid_F(A,B,F,Z)),G),C6) ) ) ) ) ).

% comm_monoid_set.Union_disjoint
tff(fact_7132_comm__monoid__set_Oreindex,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,H: fun(B,C),A5: set(B),G: fun(C,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( inj_on(B,C,H,A5)
       => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ).

% comm_monoid_set.reindex
tff(fact_7133_comm__monoid__set_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),H: fun(B,C),G: fun(C,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( ! [X4: B,Y4: B] :
              ( pp(aa(set(B),bool,member(B,X4),A5))
             => ( pp(aa(set(B),bool,member(B,Y4),A5))
               => ( ( X4 != Y4 )
                 => ( ( aa(B,C,H,X4) = aa(B,C,H,Y4) )
                   => ( aa(C,A,G,aa(B,C,H,X4)) = Z ) ) ) ) )
         => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),G),aa(set(B),set(C),image2(B,C,H),A5)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A5) ) ) ) ) ).

% comm_monoid_set.reindex_nontrivial
tff(fact_7134_comm__monoid__set_OUnion__comp,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,B4: set(set(B)),G: fun(B,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( ! [X4: set(B)] :
            ( pp(aa(set(set(B)),bool,member(set(B),X4),B4))
           => pp(aa(set(B),bool,finite_finite2(B),X4)) )
       => ( ! [A12: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),A12),B4))
             => ! [A23: set(B)] :
                  ( pp(aa(set(set(B)),bool,member(set(B),A23),B4))
                 => ( ( A12 != A23 )
                   => ! [X4: B] :
                        ( pp(aa(set(B),bool,member(B,X4),A12))
                       => ( pp(aa(set(B),bool,member(B,X4),A23))
                         => ( aa(B,A,G,X4) = Z ) ) ) ) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B4)) = 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)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(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)),groups_comm_monoid_F(A,set(B),F,Z)),groups_comm_monoid_F(A,B,F,Z)),G),B4) ) ) ) ) ).

% comm_monoid_set.Union_comp
tff(fact_7135_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,M: nat,N: nat,G: fun(nat,A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_0
tff(fact_7136_comm__monoid__set_OPlus,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z: A,A5: set(B),B4: set(C),G: fun(sum_sum(B,C),A)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( pp(aa(set(B),bool,finite_finite2(B),A5))
       => ( pp(aa(set(C),bool,finite_finite2(C),B4))
         => ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups_comm_monoid_F(A,sum_sum(B,C),F,Z),G),sum_Plus(B,C,A5,B4)) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A5)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B4)) ) ) ) ) ).

% comm_monoid_set.Plus
tff(fact_7137_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),M))) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_0
tff(fact_7138_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.atLeastLessThan_shift_bounds
tff(fact_7139_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,K: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.atLeastAtMost_shift_bounds
tff(fact_7140_comm__monoid__set_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,fun(A,A)),Z: A,H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( groups778175481326437816id_set(A,F,Z)
         => ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ) ).

% comm_monoid_set.atLeastLessThan_reindex
tff(fact_7141_comm__monoid__set_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,fun(A,A)),Z: A,H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( groups778175481326437816id_set(A,F,Z)
         => ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ) ).

% comm_monoid_set.atLeastAtMost_reindex
tff(fact_7142_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ) ).

% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_7143_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ) ).

% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_7144_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_7145_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_7146_comm__monoid__set_Oeq__fold,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(B,A),A5: set(B)] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z),G),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),F),G),Z,A5) ) ) ).

% comm_monoid_set.eq_fold
tff(fact_7147_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_lessThan_pred_shift
tff(fact_7148_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_dx(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),G),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_atMost_pred_shift
tff(fact_7149_comm__monoid__set_Otriangle__reindex,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ge(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_asp(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ) ).

% comm_monoid_set.triangle_reindex
tff(fact_7150_comm__monoid__set_Otriangle__reindex__eq,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(nat,fun(nat,A)),N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),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_ec(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_asp(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% comm_monoid_set.triangle_reindex_eq
tff(fact_7151_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(int,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_int_atMost_int_shift
tff(fact_7152_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),Z: A,G: fun(int,A),M: nat,N: nat] :
      ( groups778175481326437816id_set(A,F,Z)
     => ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% comm_monoid_set.atLeast_int_lessThan_int_shift
tff(fact_7153_digit6__def,axiom,
    ! [Char: char] :
      ( digit6(Char)
    <=> pp(case_char(bool,aTP_Lamp_asq(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit6_def
tff(fact_7154_digit5__def,axiom,
    ! [Char: char] :
      ( digit5(Char)
    <=> pp(case_char(bool,aTP_Lamp_asr(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit5_def
tff(fact_7155_digit0__def,axiom,
    ! [Char: char] :
      ( digit0(Char)
    <=> pp(case_char(bool,aTP_Lamp_ass(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit0_def
tff(fact_7156_digit1__def,axiom,
    ! [Char: char] :
      ( digit1(Char)
    <=> pp(case_char(bool,aTP_Lamp_ast(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit1_def
tff(fact_7157_digit4__def,axiom,
    ! [Char: char] :
      ( digit4(Char)
    <=> pp(case_char(bool,aTP_Lamp_asu(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit4_def
tff(fact_7158_digit3__def,axiom,
    ! [Char: char] :
      ( digit3(Char)
    <=> pp(case_char(bool,aTP_Lamp_asv(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit3_def
tff(fact_7159_digit2__def,axiom,
    ! [Char: char] :
      ( digit2(Char)
    <=> pp(case_char(bool,aTP_Lamp_asw(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Char)) ) ).

% digit2_def
tff(fact_7160_accp__eq__acc,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X2: A] :
      ( pp(aa(A,bool,accp(A,R),X2))
    <=> pp(aa(set(A),bool,member(A,X2),acc(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).

% accp_eq_acc
tff(fact_7161_acc__downwards,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A)),B2: A] :
      ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
     => ( pp(aa(set(product_prod(A,A)),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)),transitive_rtrancl(A,R)))
       => pp(aa(set(A),bool,member(A,B2),acc(A,R))) ) ) ).

% acc_downwards
tff(fact_7162_acc__downwards__aux,axiom,
    ! [A: $tType,B2: A,A3: A,R: 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),B2),A3)),transitive_rtrancl(A,R)))
     => ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
       => pp(aa(set(A),bool,member(A,B2),acc(A,R))) ) ) ).

% acc_downwards_aux
tff(fact_7163_acc_Ocases,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
     => ! [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),A3)),R))
         => pp(aa(set(A),bool,member(A,Y5),acc(A,R))) ) ) ).

% acc.cases
tff(fact_7164_acc_Osimps,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
    <=> ? [X3: A] :
          ( ( A3 = X3 )
          & ! [Xa2: 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),Xa2),X3)),R))
             => pp(aa(set(A),bool,member(A,Xa2),acc(A,R))) ) ) ) ).

% acc.simps
tff(fact_7165_acc_Ointros,axiom,
    ! [A: $tType,X: A,R: set(product_prod(A,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),X)),R))
         => pp(aa(set(A),bool,member(A,Y4),acc(A,R))) )
     => pp(aa(set(A),bool,member(A,X),acc(A,R))) ) ).

% acc.intros
tff(fact_7166_acc__induct,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),acc(A,R)))
           => ( ! [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)),R))
                 => pp(aa(A,bool,P,Y5)) )
             => pp(aa(A,bool,P,X4)) ) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% acc_induct
tff(fact_7167_acc__downward,axiom,
    ! [A: $tType,B2: A,R: set(product_prod(A,A)),A3: A] :
      ( pp(aa(set(A),bool,member(A,B2),acc(A,R)))
     => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
       => pp(aa(set(A),bool,member(A,A3),acc(A,R))) ) ) ).

% acc_downward
tff(fact_7168_not__acc__down,axiom,
    ! [A: $tType,X: A,R4: set(product_prod(A,A))] :
      ( ~ pp(aa(set(A),bool,member(A,X),acc(A,R4)))
     => ~ ! [Z4: 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),Z4),X)),R4))
           => pp(aa(set(A),bool,member(A,Z4),acc(A,R4))) ) ) ).

% not_acc_down
tff(fact_7169_acc__induct__rule,axiom,
    ! [A: $tType,A3: A,R: set(product_prod(A,A)),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
     => ( ! [X4: A] :
            ( pp(aa(set(A),bool,member(A,X4),acc(A,R)))
           => ( ! [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)),R))
                 => pp(aa(A,bool,P,Y5)) )
             => pp(aa(A,bool,P,X4)) ) )
       => pp(aa(A,bool,P,A3)) ) ) ).

% acc_induct_rule
tff(fact_7170_acc__subset__induct,axiom,
    ! [A: $tType,D5: set(A),R4: set(product_prod(A,A)),X: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D5),acc(A,R4)))
     => ( ! [X4: A,Z4: A] :
            ( pp(aa(set(A),bool,member(A,X4),D5))
           => ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X4)),R4))
             => pp(aa(set(A),bool,member(A,Z4),D5)) ) )
       => ( pp(aa(set(A),bool,member(A,X),D5))
         => ( ! [X4: A] :
                ( pp(aa(set(A),bool,member(A,X4),D5))
               => ( ! [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),Z6),X4)),R4))
                     => pp(aa(A,bool,P,Z6)) )
                 => pp(aa(A,bool,P,X4)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% acc_subset_induct
tff(fact_7171_accp__acc__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),X2: A] :
      ( pp(aa(A,bool,accp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X2))
    <=> pp(aa(set(A),bool,member(A,X2),acc(A,R))) ) ).

% accp_acc_eq
tff(fact_7172_acc__def,axiom,
    ! [A: $tType,X2: set(product_prod(A,A))] : acc(A,X2) = aa(fun(A,bool),set(A),collect(A),accp(A,aTP_Lamp_sm(set(product_prod(A,A)),fun(A,fun(A,bool)),X2))) ).

% acc_def
tff(fact_7173_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y3: code_natural] :
      ( ( log(X,Xa) = Y3 )
     => ( 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),Xa)))
       => ~ ( ( ( ( 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),Xa),X)) )
               => ( Y3 = 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),Xa),X)) )
               => ( Y3 = aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),one_one(code_natural)),log(X,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),divide_divide(code_natural),Xa),X))) ) ) )
           => ~ 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),Xa))) ) ) ) ).

% log.pelims
tff(fact_7174_map__tailrec__rev_Opelims,axiom,
    ! [A: $tType,B: $tType,X: fun(A,B),Xa: list(A),Xb2: list(B),Y3: list(B)] :
      ( ( map_tailrec_rev(A,B,X,Xa,Xb2) = Y3 )
     => ( 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)),Xa),Xb2))))
       => ( ( ( Xa = nil(A) )
           => ( ( Y3 = Xb2 )
             => ~ 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)),Xb2)))) ) )
         => ~ ! [A4: A,As2: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As2) )
               => ( ( Y3 = map_tailrec_rev(A,B,X,As2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,X,A4)),Xb2)) )
                 => ~ 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),A4),As2)),Xb2)))) ) ) ) ) ) ).

% map_tailrec_rev.pelims
tff(fact_7175_log_Ocases,axiom,
    ! [X: product_prod(code_natural,code_natural)] :
      ~ ! [B3: code_natural,I2: code_natural] : X != aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),B3),I2) ).

% log.cases
tff(fact_7176_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))] :
      ~ ! [F3: fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),D2: code_natural,I2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(code_natural,fun(product_unit,code_term)),option(product_prod(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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I2)) ).

% full_exhaustive_natural'.cases
tff(fact_7177_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))] :
          ~ ! [F3: fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(bool,list(code_term)))),I2: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(product_prod(fun(A,B),fun(product_unit,code_term)),option(product_prod(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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I2),D2)) ) ).

% full_exhaustive_fun'.cases
tff(fact_7178_exhaustive__natural_H_Ocases,axiom,
    ! [X: product_prod(fun(code_natural,option(product_prod(bool,list(code_term)))),product_prod(code_natural,code_natural))] :
      ~ ! [F3: fun(code_natural,option(product_prod(bool,list(code_term)))),D2: code_natural,I2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(code_natural,option(product_prod(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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),D2),I2)) ).

% exhaustive_natural'.cases
tff(fact_7179_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))] :
          ~ ! [F3: fun(fun(A,B),option(product_prod(bool,list(code_term)))),I2: code_natural,D2: code_natural] : X != aa(product_prod(code_natural,code_natural),product_prod(fun(fun(A,B),option(product_prod(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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),I2),D2)) ) ).

% exhaustive_fun'.cases
tff(fact_7180_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(code_natural,A),product_prod(code_natural,code_natural))] :
      ~ ! [F3: fun(code_natural,A),N2: code_natural,M4: 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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N2),M4)) ).

% Lazy_Sequence.iterate_upto.cases
tff(fact_7181_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))))
     => ( ! [F3: fun(code_natural,A),N2: code_natural,M4: 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)),F3),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N2),M4))))
           => ( ! [X2: product_unit] :
                  ( ~ pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),M4),N2))
                 => 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,F3),aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),N2),one_one(code_natural))),M4)) )
             => 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,F3),N2),M4)) ) )
       => 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_7182_sum_Opred__map,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Q12: fun(C,bool),Q23: fun(D,bool),F1: fun(A,C),F22: fun(B,D),X: sum_sum(A,B)] :
      ( basic_pred_sum(C,D,Q12,Q23,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),X))
    <=> basic_pred_sum(A,B,aa(fun(A,C),fun(A,bool),aa(fun(C,bool),fun(fun(A,C),fun(A,bool)),comp(C,bool,A),Q12),F1),aa(fun(B,D),fun(B,bool),aa(fun(D,bool),fun(fun(B,D),fun(B,bool)),comp(D,bool,B),Q23),F22),X) ) ).

% sum.pred_map
tff(fact_7183_sum_Omap__cong__pred,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,X: sum_sum(A,B),Ya: sum_sum(A,B),F1: fun(A,C),G1: fun(A,C),F22: fun(B,D),G22: fun(B,D)] :
      ( ( X = Ya )
     => ( basic_pred_sum(A,B,aa(fun(A,C),fun(A,bool),aTP_Lamp_aoy(fun(A,C),fun(fun(A,C),fun(A,bool)),F1),G1),aa(fun(B,D),fun(B,bool),aTP_Lamp_aoz(fun(B,D),fun(fun(B,D),fun(B,bool)),F22),G22),Ya)
       => ( aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),X) = aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,G1,G22),Ya) ) ) ) ).

% sum.map_cong_pred
tff(fact_7184_sum_Opred__True,axiom,
    ! [B: $tType,A: $tType,X2: sum_sum(A,B)] : basic_pred_sum(A,B,aTP_Lamp_aq(A,bool),aTP_Lamp_aox(B,bool),X2) ).

% sum.pred_True
tff(fact_7185_Predicate_Oiterate__upto_Opsimps,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: 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)),F),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),N),M))))
     => ( iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_asx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ) ) ).

% Predicate.iterate_upto.psimps
tff(fact_7186_Predicate_Oiterate__upto_Opelims,axiom,
    ! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb2: code_natural,Y3: pred(A)] :
      ( ( iterate_upto(A,X,Xa,Xb2) = Y3 )
     => ( 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)),X),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb2))))
       => ~ ( ( Y3 = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_asx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb2)) )
           => ~ 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)),X),aa(code_natural,product_prod(code_natural,code_natural),aa(code_natural,fun(code_natural,product_prod(code_natural,code_natural)),product_Pair(code_natural,code_natural),Xa),Xb2)))) ) ) ) ).

% Predicate.iterate_upto.pelims
tff(fact_7187_bot__set__code,axiom,
    ! [A: $tType] : bot_bot(pred(A)) = seq2(A,aTP_Lamp_asy(product_unit,seq(A))) ).

% bot_set_code
tff(fact_7188_Predicate_Oiterate__upto_Osimps,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : iterate_upto(A,F,N,M) = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_asx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),F),N),M)) ).

% Predicate.iterate_upto.simps
tff(fact_7189_Predicate_Oiterate__upto_Oelims,axiom,
    ! [A: $tType,X: fun(code_natural,A),Xa: code_natural,Xb2: code_natural,Y3: pred(A)] :
      ( ( iterate_upto(A,X,Xa,Xb2) = Y3 )
     => ( Y3 = seq2(A,aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_asx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),X),Xa),Xb2)) ) ) ).

% Predicate.iterate_upto.elims
tff(fact_7190_Predicate_Osingle__code,axiom,
    ! [A: $tType,X: A] : aa(A,pred(A),single(A),X) = seq2(A,aTP_Lamp_asz(A,fun(product_unit,seq(A)),X)) ).

% Predicate.single_code
tff(fact_7191_of__pred__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A))] : aa(pred(A),set(A),set_of_pred(A),seq2(A,F)) = case_seq(set(A),A,bot_bot(set(A)),aTP_Lamp_ata(A,fun(pred(A),set(A))),aTP_Lamp_atb(pred(A),fun(seq(A),set(A))),aa(product_unit,seq(A),F,product_Unity)) ).

% of_pred_code
tff(fact_7192_seq_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F22: fun(A,fun(pred(A),B)),F32: fun(pred(A),fun(seq(A),B)),Seq: seq(A)] : aa(B,C,H,case_seq(B,A,F1,F22,F32,Seq)) = case_seq(C,A,aa(B,C,H,F1),aa(fun(A,fun(pred(A),B)),fun(A,fun(pred(A),C)),aTP_Lamp_atc(fun(B,C),fun(fun(A,fun(pred(A),B)),fun(A,fun(pred(A),C))),H),F22),aa(fun(pred(A),fun(seq(A),B)),fun(pred(A),fun(seq(A),C)),aTP_Lamp_atd(fun(B,C),fun(fun(pred(A),fun(seq(A),B)),fun(pred(A),fun(seq(A),C))),H),F32),Seq) ).

% seq.case_distrib
tff(fact_7193_set__of__seq__def,axiom,
    ! [A: $tType] : set_of_seq(A) = aa(fun(seq(A),pred(A)),fun(seq(A),set(A)),aa(fun(pred(A),set(A)),fun(fun(seq(A),pred(A)),fun(seq(A),set(A))),comp(pred(A),set(A),seq(A)),set_of_pred(A)),pred_of_seq(A)) ).

% set_of_seq_def
tff(fact_7194_eval__SUP,axiom,
    ! [A: $tType,B: $tType,F: fun(B,pred(A)),A5: set(B)] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),image2(B,pred(A),F),A5))) = 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),aa(fun(B,pred(A)),fun(B,fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(fun(B,pred(A)),fun(B,fun(A,bool))),comp(pred(A),fun(A,bool),B),eval(A)),F)),A5)) ).

% eval_SUP
tff(fact_7195_eval__INF,axiom,
    ! [A: $tType,B: $tType,F: fun(B,pred(A)),A5: set(B)] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Inf_Inf(pred(A)),aa(set(B),set(pred(A)),image2(B,pred(A),F),A5))) = 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),aa(fun(B,pred(A)),fun(B,fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(fun(B,pred(A)),fun(B,fun(A,bool))),comp(pred(A),fun(A,bool),B),eval(A)),F)),A5)) ).

% eval_INF
tff(fact_7196_set__of__pred__def,axiom,
    ! [A: $tType] : set_of_pred(A) = aa(fun(pred(A),fun(A,bool)),fun(pred(A),set(A)),aa(fun(fun(A,bool),set(A)),fun(fun(pred(A),fun(A,bool)),fun(pred(A),set(A))),comp(fun(A,bool),set(A),pred(A)),collect(A)),eval(A)) ).

% set_of_pred_def
tff(fact_7197_eval__map,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: pred(B)] : aa(pred(A),fun(A,bool),eval(A),aa(pred(B),pred(A),map2(B,A,F),P)) = 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_ate(fun(B,A),fun(B,fun(A,bool)),F)),aa(fun(B,bool),set(B),collect(B),aa(pred(B),fun(B,bool),eval(B),P)))) ).

% eval_map
tff(fact_7198_less__eq__pred__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A)),Q: pred(A)] :
      ( pp(aa(pred(A),bool,aa(pred(A),fun(pred(A),bool),ord_less_eq(pred(A)),seq2(A,F)),Q))
    <=> pp(case_seq(bool,A,fTrue,aTP_Lamp_atf(pred(A),fun(A,fun(pred(A),bool)),Q),aTP_Lamp_atg(pred(A),fun(pred(A),fun(seq(A),bool)),Q),aa(product_unit,seq(A),F,product_Unity))) ) ).

% less_eq_pred_code
tff(fact_7199_Predicate_Omap_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Pred: pred(A)] : aa(pred(B),pred(C),map2(B,C,F),aa(pred(A),pred(B),map2(A,B,G),Pred)) = aa(pred(A),pred(C),map2(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Pred) ).

% Predicate.map.compositionality
tff(fact_7200_Predicate_Omap_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(pred(A),pred(B)),fun(pred(A),pred(C)),aa(fun(pred(B),pred(C)),fun(fun(pred(A),pred(B)),fun(pred(A),pred(C))),comp(pred(B),pred(C),pred(A)),map2(B,C,F)),map2(A,B,G)) = map2(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).

% Predicate.map.comp
tff(fact_7201_Predicate_Omap_Oidentity,axiom,
    ! [A: $tType] : map2(A,A,aTP_Lamp_ak(A,A)) = id(pred(A)) ).

% Predicate.map.identity
tff(fact_7202_eval__bind,axiom,
    ! [A: $tType,B: $tType,P: pred(B),F: fun(B,pred(A))] : aa(pred(A),fun(A,bool),eval(A),bind3(B,A,P,F)) = aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),image2(B,pred(A),F),aa(fun(B,bool),set(B),collect(B),aa(pred(B),fun(B,bool),eval(B),P))))) ).

% eval_bind
tff(fact_7203_Predicate_Othe__def,axiom,
    ! [A: $tType,A5: pred(A)] : the3(A,A5) = the(A,aa(pred(A),fun(A,bool),eval(A),A5)) ).

% Predicate.the_def
tff(fact_7204_Predicate_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: pred(C),Q: fun(C,pred(B)),R4: fun(B,pred(A))] : bind3(B,A,bind3(C,B,P,Q),R4) = bind3(C,A,P,aa(fun(B,pred(A)),fun(C,pred(A)),aTP_Lamp_ath(fun(C,pred(B)),fun(fun(B,pred(A)),fun(C,pred(A))),Q),R4)) ).

% Predicate.bind_bind
tff(fact_7205_Sup__bind,axiom,
    ! [A: $tType,B: $tType,A5: set(pred(B)),F: fun(B,pred(A))] : bind3(B,A,aa(set(pred(B)),pred(B),complete_Sup_Sup(pred(B)),A5),F) = aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(pred(B)),set(pred(A)),image2(pred(B),pred(A),aTP_Lamp_ati(fun(B,pred(A)),fun(pred(B),pred(A)),F)),A5)) ).

% Sup_bind
tff(fact_7206_Predicate_Omap__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),P: pred(A)] : aa(pred(A),pred(B),map2(A,B,F),P) = bind3(A,B,P,aa(fun(A,B),fun(A,pred(B)),aa(fun(B,pred(B)),fun(fun(A,B),fun(A,pred(B))),comp(B,pred(B),A),single(B)),F)) ).

% Predicate.map_def
tff(fact_7207_Predicate_Obind__def,axiom,
    ! [B: $tType,A: $tType,P: pred(A),F: fun(A,pred(B))] : bind3(A,B,P,F) = aa(set(pred(B)),pred(B),complete_Sup_Sup(pred(B)),aa(set(A),set(pred(B)),image2(A,pred(B),F),aa(fun(A,bool),set(A),collect(A),aa(pred(A),fun(A,bool),eval(A),P)))) ).

% Predicate.bind_def
tff(fact_7208_the__eqI,axiom,
    ! [A: $tType,P: pred(A),X: A] :
      ( ( the(A,aa(pred(A),fun(A,bool),eval(A),P)) = X )
     => ( the3(A,P) = X ) ) ).

% the_eqI
tff(fact_7209_bind__code,axiom,
    ! [A: $tType,B: $tType,G: fun(product_unit,seq(B)),F: fun(B,pred(A))] : bind3(B,A,seq2(B,G),F) = seq2(A,aa(fun(B,pred(A)),fun(product_unit,seq(A)),aTP_Lamp_atj(fun(product_unit,seq(B)),fun(fun(B,pred(A)),fun(product_unit,seq(A))),G),F)) ).

% bind_code
tff(fact_7210_sup__code,axiom,
    ! [A: $tType,F: fun(product_unit,seq(A)),G: fun(product_unit,seq(A))] : aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),seq2(A,F)),seq2(A,G)) = seq2(A,aa(fun(product_unit,seq(A)),fun(product_unit,seq(A)),aTP_Lamp_atm(fun(product_unit,seq(A)),fun(fun(product_unit,seq(A)),fun(product_unit,seq(A))),F),G)) ).

% sup_code
tff(fact_7211_Random__Pred_Ounion__def,axiom,
    ! [A: $tType,R12: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),R23: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),X2: product_prod(code_natural,code_natural)] : random_union(A,R12,R23,X2) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_ato(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),R23)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R12,X2)) ).

% Random_Pred.union_def
tff(fact_7212_eval__def,axiom,
    ! [A: $tType,Pred: pred(A)] : aa(pred(A),fun(A,bool),eval(A),Pred) = case_pred(A,fun(A,bool),aTP_Lamp_atp(fun(A,bool),fun(A,bool)),Pred) ).

% eval_def
tff(fact_7213_pred_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F: fun(fun(A,bool),B),Pred: pred(A)] : aa(B,C,H,case_pred(A,B,F,Pred)) = case_pred(A,C,aa(fun(fun(A,bool),B),fun(fun(A,bool),C),aTP_Lamp_atq(fun(B,C),fun(fun(fun(A,bool),B),fun(fun(A,bool),C)),H),F),Pred) ).

% pred.case_distrib
tff(fact_7214_Random__Pred_Oiterate__upto__def,axiom,
    ! [A: $tType,F: fun(code_natural,A),N: code_natural,M: code_natural] : random_iterate_upto(A,F,N,M) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),iterate_upto(A,F,N,M)) ).

% Random_Pred.iterate_upto_def
tff(fact_7215_Random__Pred_Oempty__def,axiom,
    ! [A: $tType] : random_empty(A) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),bot_bot(pred(A))) ).

% Random_Pred.empty_def
tff(fact_7216_Random__Pred_ORandom__def,axiom,
    ! [A: $tType,G: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural)))] : random_Random(A,G) = product_scomp(product_prod(code_natural,code_natural),product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),G,aa(fun(product_prod(A,fun(product_unit,code_term)),pred(A)),fun(product_prod(A,fun(product_unit,code_term)),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(fun(product_prod(A,fun(product_unit,code_term)),pred(A)),fun(product_prod(A,fun(product_unit,code_term)),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),comp(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_prod(A,fun(product_unit,code_term))),product_Pair(pred(A),product_prod(code_natural,code_natural))),aa(fun(product_prod(A,fun(product_unit,code_term)),A),fun(product_prod(A,fun(product_unit,code_term)),pred(A)),aa(fun(A,pred(A)),fun(fun(product_prod(A,fun(product_unit,code_term)),A),fun(product_prod(A,fun(product_unit,code_term)),pred(A))),comp(A,pred(A),product_prod(A,fun(product_unit,code_term))),single(A)),product_fst(A,fun(product_unit,code_term))))) ).

% Random_Pred.Random_def
tff(fact_7217_Random__Pred_Obind__def,axiom,
    ! [B: $tType,A: $tType,R4: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),F: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),X2: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),random_bind(A,B,R4,F),X2) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_att(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),F)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R4,X2)) ).

% Random_Pred.bind_def
tff(fact_7218_Random__Pred_Osingle__def,axiom,
    ! [A: $tType,X: A] : aa(A,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),random_single(A),X) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(A,pred(A),single(A),X)) ).

% Random_Pred.single_def
tff(fact_7219_Random__Pred_Onot__randompred__def,axiom,
    ! [P: fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),X2: product_prod(code_natural,code_natural)] : random6974930770145893639ompred(P,X2) = aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(fun(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_case_prod(pred(product_unit),product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_atu(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))))),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),P,X2)) ).

% Random_Pred.not_randompred_def
tff(fact_7220_Random__Pred_Omap__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),P: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))] : random_map(A,B,F,P) = random_bind(A,B,P,aa(fun(A,B),fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aa(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(fun(A,B),fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),comp(B,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),A),random_single(B)),F)) ).

% Random_Pred.map_def
tff(fact_7221_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_sx(product_prod(int,int),product_prod(int,int)),Uu) = aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),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_7222_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_apk(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_aph(nat,set(old_node(A,product_unit)))),aTP_Lamp_apj(nat,set(old_node(A,product_unit)))),nat_sum_decode(Uu)) ) ).

% ATP.lambda_2
tff(fact_7223_ATP_Olambda__3,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_apg(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit))),aa(fun(nat,set(old_node(A,product_unit))),fun(fun(nat,set(old_node(A,product_unit))),fun(sum_sum(nat,nat),set(old_node(A,product_unit)))),sum_case_sum(nat,set(old_node(A,product_unit)),nat),aTP_Lamp_ape(nat,set(old_node(A,product_unit)))),aTP_Lamp_apf(nat,set(old_node(A,product_unit)))),nat_sum_decode(Uu)) ) ).

% ATP.lambda_3
tff(fact_7224_ATP_Olambda__4,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_mz(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(set(A)),nat,finite_card(set(A)),Uu)),one_one(nat)))),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),Uu)))) ).

% ATP.lambda_4
tff(fact_7225_ATP_Olambda__5,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_rs(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_5
tff(fact_7226_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_at(A,bool),Uu))
        <=> ( pp(aa(set(A),bool,member(A,Uu),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).

% ATP.lambda_6
tff(fact_7227_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_tc(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_7
tff(fact_7228_ATP_Olambda__8,axiom,
    ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),product_prod(int,int),aTP_Lamp_sy(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_8
tff(fact_7229_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa(nat,int,aTP_Lamp_apb(nat,int),Uu) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Uu))),one_one(int)) ).

% ATP.lambda_9
tff(fact_7230_ATP_Olambda__10,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_wt(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_10
tff(fact_7231_ATP_Olambda__11,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),product_prod(nat,list(A)),aTP_Lamp_akk(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_11
tff(fact_7232_ATP_Olambda__12,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( pp(aa(nat,bool,aTP_Lamp_my(nat,bool),Uu))
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_12
tff(fact_7233_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_dx(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_13
tff(fact_7234_ATP_Olambda__14,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_kz(int,int),Uu) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),Uu) ).

% ATP.lambda_14
tff(fact_7235_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_fu(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_15
tff(fact_7236_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_dm(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_16
tff(fact_7237_ATP_Olambda__17,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_aqa(product_prod(int,int),bool),Uu))
    <=> pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,Uu),Uu)) ) ).

% ATP.lambda_17
tff(fact_7238_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ar(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_18
tff(fact_7239_ATP_Olambda__19,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_alw(A,bool),Uu))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uu),one_one(A))) ) ) ).

% ATP.lambda_19
tff(fact_7240_ATP_Olambda__20,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_tb(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_20
tff(fact_7241_ATP_Olambda__21,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_vn(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_21
tff(fact_7242_ATP_Olambda__22,axiom,
    ! [D: $tType,Uu: D] : aa(D,set(D),aTP_Lamp_aog(D,set(D)),Uu) = aa(set(D),set(D),insert2(D,Uu),bot_bot(set(D))) ).

% ATP.lambda_22
tff(fact_7243_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_nl(A,set(A)),Uu) = aa(set(A),set(A),insert2(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_23
tff(fact_7244_ATP_Olambda__24,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: fun(A,nat)] :
          ( pp(aa(fun(A,nat),bool,aTP_Lamp_apc(fun(A,nat),bool),Uu))
        <=> inj_on(A,nat,Uu,top_top(set(A))) ) ) ).

% ATP.lambda_24
tff(fact_7245_ATP_Olambda__25,axiom,
    ! [B: $tType,D: $tType,Uu: fun(D,B)] : aa(fun(D,B),set(B),aTP_Lamp_aqq(fun(D,B),set(B)),Uu) = aa(set(D),set(B),image2(D,B,Uu),top_top(set(D))) ).

% ATP.lambda_25
tff(fact_7246_ATP_Olambda__26,axiom,
    ! [A: $tType,D: $tType,Uu: fun(D,A)] : aa(fun(D,A),set(A),aTP_Lamp_aqp(fun(D,A),set(A)),Uu) = aa(set(D),set(A),image2(D,A,Uu),top_top(set(D))) ).

% ATP.lambda_26
tff(fact_7247_ATP_Olambda__27,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_anw(option(B),bool),Uu))
    <=> ( Uu = none(B) ) ) ).

% ATP.lambda_27
tff(fact_7248_ATP_Olambda__28,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_anv(option(A),bool),Uu))
    <=> ( Uu = none(A) ) ) ).

% ATP.lambda_28
tff(fact_7249_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_acm(list(B),bool),Uu))
    <=> ( Uu = nil(B) ) ) ).

% ATP.lambda_29
tff(fact_7250_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_acl(list(A),bool),Uu))
    <=> ( Uu = nil(A) ) ) ).

% ATP.lambda_30
tff(fact_7251_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_apj(nat,set(old_node(A,product_unit))),Uu) = aa(product_prod(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,fun(nat,set(old_node(A,product_unit)))),fun(product_prod(nat,nat),set(old_node(A,product_unit))),product_case_prod(nat,nat,set(old_node(A,product_unit))),aTP_Lamp_api(nat,fun(nat,set(old_node(A,product_unit))))),aa(nat,product_prod(nat,nat),nat_prod_decode,Uu)) ) ).

% ATP.lambda_31
tff(fact_7252_ATP_Olambda__32,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_ta(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_32
tff(fact_7253_ATP_Olambda__33,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_aon(nat,nat),Uu) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)) ).

% ATP.lambda_33
tff(fact_7254_ATP_Olambda__34,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_abw(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_34
tff(fact_7255_ATP_Olambda__35,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_rr(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert2(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).

% ATP.lambda_35
tff(fact_7256_ATP_Olambda__36,axiom,
    ! [B: $tType,Uu: option(B)] :
      ( pp(aa(option(B),bool,aTP_Lamp_any(option(B),bool),Uu))
    <=> ( Uu != none(B) ) ) ).

% ATP.lambda_36
tff(fact_7257_ATP_Olambda__37,axiom,
    ! [A: $tType,Uu: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_anx(option(A),bool),Uu))
    <=> ( Uu != none(A) ) ) ).

% ATP.lambda_37
tff(fact_7258_ATP_Olambda__38,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_abx(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_38
tff(fact_7259_ATP_Olambda__39,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_abl(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_39
tff(fact_7260_ATP_Olambda__40,axiom,
    ! [A: $tType,Uu: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A),aTP_Lamp_aoq(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(A)),Uu) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(set(A)),image2(product_prod(A,product_prod(heap_ext(product_unit),nat)),set(A),basic_fsts(A,product_prod(heap_ext(product_unit),nat))),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),set(product_prod(A,product_prod(heap_ext(product_unit),nat))),set_option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu))) ).

% ATP.lambda_40
tff(fact_7261_ATP_Olambda__41,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_ajk(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_ajj(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_41
tff(fact_7262_ATP_Olambda__42,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_acz(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_acy(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).

% ATP.lambda_42
tff(fact_7263_ATP_Olambda__43,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_aai(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_aah(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_43
tff(fact_7264_ATP_Olambda__44,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_rb(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_ic(nat,fun(nat,bool),Uu)) ).

% ATP.lambda_44
tff(fact_7265_ATP_Olambda__45,axiom,
    ! [A: $tType,Uu: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_agm(fun(A,bool),bool),Uu))
    <=> pp(aa(set(A),bool,finite_finite2(A),aa(fun(A,bool),set(A),collect(A),aTP_Lamp_lv(fun(A,bool),fun(A,bool),Uu)))) ) ).

% ATP.lambda_45
tff(fact_7266_ATP_Olambda__46,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_wn(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_wm(fun(A,B),fun(A,fun(B,bool)),Uu))) ).

% ATP.lambda_46
tff(fact_7267_ATP_Olambda__47,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_abv(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_47
tff(fact_7268_ATP_Olambda__48,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_abu(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_48
tff(fact_7269_ATP_Olambda__49,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ca(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_49
tff(fact_7270_ATP_Olambda__50,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_bo(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_50
tff(fact_7271_ATP_Olambda__51,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jp(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_51
tff(fact_7272_ATP_Olambda__52,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_jq(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_52
tff(fact_7273_ATP_Olambda__53,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bm(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_53
tff(fact_7274_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_aph(nat,set(old_node(A,product_unit))),Uu) = aa(A,set(old_node(A,product_unit)),old_Leaf(A,product_unit),from_nat(A,Uu)) ) ).

% ATP.lambda_54
tff(fact_7275_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_apf(nat,set(old_node(A,product_unit))),Uu) = old_In1(A,product_unit,nth_item(A,Uu)) ) ).

% ATP.lambda_55
tff(fact_7276_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_ape(nat,set(old_node(A,product_unit))),Uu) = old_In0(A,product_unit,nth_item(A,Uu)) ) ).

% ATP.lambda_56
tff(fact_7277_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ahl(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_57
tff(fact_7278_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_ahj(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).

% ATP.lambda_58
tff(fact_7279_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_se(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_59
tff(fact_7280_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_sd(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_60
tff(fact_7281_ATP_Olambda__61,axiom,
    ! [Uu: int] : aa(int,nat,aTP_Lamp_nn(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_61
tff(fact_7282_ATP_Olambda__62,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_vf(A,bool),Uu))
        <=> ? [N6: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N6) )
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N6)) ) ) ) ).

% ATP.lambda_62
tff(fact_7283_ATP_Olambda__63,axiom,
    ! [Uu: list(char)] :
      ( pp(aa(list(char),bool,aTP_Lamp_apv(list(char),bool),Uu))
    <=> ! [X3: char] :
          ( pp(aa(set(char),bool,member(char,X3),aa(list(char),set(char),set2(char),Uu)))
         => ~ digit7(X3) ) ) ).

% ATP.lambda_63
tff(fact_7284_ATP_Olambda__64,axiom,
    ! [Uu: set(product_prod(int,int))] :
      ( pp(aa(set(product_prod(int,int)),bool,aTP_Lamp_aqg(set(product_prod(int,int)),bool),Uu))
    <=> ? [X3: product_prod(int,int)] :
          ( pp(aa(product_prod(int,int),bool,aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X3),X3))
          & ( Uu = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(product_prod(int,int),fun(product_prod(int,int),bool),ratrel,X3)) ) ) ) ).

% ATP.lambda_64
tff(fact_7285_ATP_Olambda__65,axiom,
    ! [Uu: set(product_prod(nat,nat))] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aTP_Lamp_apu(set(product_prod(nat,nat)),bool),Uu))
    <=> ? [X3: product_prod(nat,nat)] :
          ( pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X3),X3))
          & ( Uu = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),intrel,X3)) ) ) ) ).

% ATP.lambda_65
tff(fact_7286_ATP_Olambda__66,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_ws(product_prod(A,A),bool),Uu))
    <=> ? [X3: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),X3) ) ).

% ATP.lambda_66
tff(fact_7287_ATP_Olambda__67,axiom,
    ! [A: $tType,B: $tType,Uu: product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))] :
      ( pp(aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool,aTP_Lamp_amw(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),bool),Uu))
    <=> ? [F6: fun(nat,sum_sum(B,nat)),X3: sum_sum(A,nat),K3: nat] :
          ( ( Uu = aa(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat))),product_Pair(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),F6),X3) )
          & ( aa(nat,sum_sum(B,nat),F6,K3) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ) ).

% ATP.lambda_67
tff(fact_7288_ATP_Olambda__68,axiom,
    ! [B: $tType,Uu: nat] : aa(nat,sum_sum(B,nat),aTP_Lamp_ana(nat,sum_sum(B,nat)),Uu) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ).

% ATP.lambda_68
tff(fact_7289_ATP_Olambda__69,axiom,
    ! [A: $tType,Uu: nat] : aa(nat,sum_sum(A,nat),aTP_Lamp_amx(nat,sum_sum(A,nat)),Uu) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ).

% ATP.lambda_69
tff(fact_7290_ATP_Olambda__70,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_cd(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_cb(nat,fun(num,option(num)),Uua),aTP_Lamp_cc(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_70
tff(fact_7291_ATP_Olambda__71,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_do(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))),zero_zero(A)) ) ).

% ATP.lambda_71
tff(fact_7292_ATP_Olambda__72,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_aam(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = 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))),aa(bool,fun(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B)))),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_72
tff(fact_7293_ATP_Olambda__73,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_cm(code_integer,fun(code_integer,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),aa(bool,fun(int,fun(int,int)),if(int),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),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),bit0(one2))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).

% ATP.lambda_73
tff(fact_7294_ATP_Olambda__74,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_cf(code_integer,fun(code_integer,num)),Uu),Uua) = aa(num,num,aa(num,fun(num,num),aa(bool,fun(num,fun(num,num)),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),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,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),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,code_num_of_integer,Uu))),one2)) ).

% ATP.lambda_74
tff(fact_7295_ATP_Olambda__75,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ci(code_integer,fun(code_integer,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),aa(bool,fun(nat,fun(nat,nat)),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_75
tff(fact_7296_ATP_Olambda__76,axiom,
    ! [Uu: pred(product_unit),Uua: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_atu(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aa(bool,fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),if(product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aa(product_unit,bool,aa(pred(product_unit),fun(product_unit,bool),eval(product_unit),Uu),product_Unity)),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),bot_bot(pred(product_unit))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)),Uua)) ).

% ATP.lambda_76
tff(fact_7297_ATP_Olambda__77,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kl(int,fun(int,product_prod(int,int))),Uu),Uua) = aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),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_77
tff(fact_7298_ATP_Olambda__78,axiom,
    ! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_tf(int,fun(int,product_prod(int,int))),Uu),Uua) = aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),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_78
tff(fact_7299_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dn(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))),zero_zero(A)) ) ).

% ATP.lambda_79
tff(fact_7300_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_tr(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_80
tff(fact_7301_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_ti(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_81
tff(fact_7302_ATP_Olambda__82,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_acq(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_acp(A,fun(list(A),bool))),Uua),Uu)) ) ).

% ATP.lambda_82
tff(fact_7303_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_ce(nat,fun(num,option(num))),Uu),Uua) = aa(nat,option(num),aa(fun(nat,option(num)),fun(nat,option(num)),aa(option(num),fun(fun(nat,option(num)),fun(nat,option(num))),case_nat(option(num)),none(num)),aTP_Lamp_cd(num,fun(nat,option(num)),Uua)),Uu) ).

% ATP.lambda_83
tff(fact_7304_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_cb(nat,fun(num,option(num)),Uu),Uua) = aa(option(num),option(num),aa(fun(num,option(num)),fun(option(num),option(num)),aa(option(num),fun(fun(num,option(num)),fun(option(num),option(num))),case_option(option(num),num),none(num)),aTP_Lamp_bo(num,option(num))),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_84
tff(fact_7305_ATP_Olambda__85,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_kp(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_85
tff(fact_7306_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_wj(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_86
tff(fact_7307_ATP_Olambda__87,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aac(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aab(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_87
tff(fact_7308_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gm(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_gl(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_88
tff(fact_7309_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ee(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_ed(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_89
tff(fact_7310_ATP_Olambda__90,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_sw(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_90
tff(fact_7311_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ev(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_91
tff(fact_7312_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ew(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),bit0(one2))),Uua)) ) ).

% ATP.lambda_92
tff(fact_7313_ATP_Olambda__93,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_sz(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_93
tff(fact_7314_ATP_Olambda__94,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_anl(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B,Uu)),old_In1(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_94
tff(fact_7315_ATP_Olambda__95,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_ank(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B,Uu)),old_In0(A,B,Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_95
tff(fact_7316_ATP_Olambda__96,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_aid(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_1: fun(A,A2)] : bNF_Wellorder_embedS(A,A2,Uu,Uua,X_1) ) ) ).

% ATP.lambda_96
tff(fact_7317_ATP_Olambda__97,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_aih(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_1: fun(A,A2)] : pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,Uu,Uua),X_1)) ) ) ).

% ATP.lambda_97
tff(fact_7318_ATP_Olambda__98,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_aii(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
    <=> ( order_well_order_on(A,field2(A,Uu),Uu)
        & order_well_order_on(A2,field2(A2,Uua),Uua)
        & ? [X_1: fun(A,A2)] : bNF_Wellorder_iso(A,A2,Uu,Uua,X_1) ) ) ).

% ATP.lambda_98
tff(fact_7319_ATP_Olambda__99,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_km(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_99
tff(fact_7320_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fa(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_100
tff(fact_7321_ATP_Olambda__101,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_amb(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_101
tff(fact_7322_ATP_Olambda__102,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_uz(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),insert2(set(A),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_102
tff(fact_7323_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_eb(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_103
tff(fact_7324_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dp(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),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_104
tff(fact_7325_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_el(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_105
tff(fact_7326_ATP_Olambda__106,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jg(nat,fun(nat,bool)),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uu),Uua))
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_106
tff(fact_7327_ATP_Olambda__107,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_na(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_107
tff(fact_7328_ATP_Olambda__108,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_rc(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_108
tff(fact_7329_ATP_Olambda__109,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_eo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_109
tff(fact_7330_ATP_Olambda__110,axiom,
    ! [Uu: set(int),Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_ady(set(int),fun(int,bool),Uu),Uua))
    <=> ( pp(aa(set(int),bool,member(int,Uua),Uu))
        & ! [X3: int] :
            ( pp(aa(set(int),bool,member(int,X3),Uu))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X3),Uua)) ) ) ) ).

% ATP.lambda_110
tff(fact_7331_ATP_Olambda__111,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_afe(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
        & ! [A7: A,B7: A,C4: 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),A7),B7)),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),B7),C4)),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),A7),B7)),Uu)) ) ) ) ).

% ATP.lambda_111
tff(fact_7332_ATP_Olambda__112,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_ald(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_112
tff(fact_7333_ATP_Olambda__113,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ami(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_113
tff(fact_7334_ATP_Olambda__114,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_ip(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_114
tff(fact_7335_ATP_Olambda__115,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ss(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_115
tff(fact_7336_ATP_Olambda__116,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dz(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_116
tff(fact_7337_ATP_Olambda__117,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dy(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_117
tff(fact_7338_ATP_Olambda__118,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_apr(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),insert2(set(old_node(A,B)),old_Scons(A,B,Uu,Uua)),bot_bot(set(set(old_node(A,B))))) ).

% ATP.lambda_118
tff(fact_7339_ATP_Olambda__119,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_rm(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_119
tff(fact_7340_ATP_Olambda__120,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_yb(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).

% ATP.lambda_120
tff(fact_7341_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_id(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_121
tff(fact_7342_ATP_Olambda__122,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_aod(fun(A,fun(B,bool)),fun(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(product_prod(A,B)),set(product_prod(A,B)),insert2(product_prod(A,B),Uua),bot_bot(set(product_prod(A,B))))),aa(fun(product_prod(A,B),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_122
tff(fact_7343_ATP_Olambda__123,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_sr(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_123
tff(fact_7344_ATP_Olambda__124,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_ue(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A9: set(A),A7: A] :
            ( ( Uua = aa(set(A),set(A),insert2(A,A7),A9) )
            & pp(aa(set(A),bool,Uu,A9)) ) ) ) ).

% ATP.lambda_124
tff(fact_7345_ATP_Olambda__125,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_aiq(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_125
tff(fact_7346_ATP_Olambda__126,axiom,
    ! [D: $tType,A: $tType,Uu: set(A),Uua: fun(D,A)] :
      ( pp(aa(fun(D,A),bool,aTP_Lamp_akw(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_126
tff(fact_7347_ATP_Olambda__127,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_vt(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_127
tff(fact_7348_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( pp(aa(list(A),bool,aTP_Lamp_ajh(set(A),fun(list(A),bool),Uu),Uua))
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_128
tff(fact_7349_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_tm(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_129
tff(fact_7350_ATP_Olambda__130,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_to(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_130
tff(fact_7351_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ga(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_131
tff(fact_7352_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ek(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_132
tff(fact_7353_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_133
tff(fact_7354_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gx(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_134
tff(fact_7355_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gv(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_135
tff(fact_7356_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hj(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_136
tff(fact_7357_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ei(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_137
tff(fact_7358_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_eh(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_138
tff(fact_7359_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dv(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_139
tff(fact_7360_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aln(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y: A] :
                ( pp(aa(A,bool,Uu,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y)) ) ) ) ) ).

% ATP.lambda_140
tff(fact_7361_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_afb(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y: A] :
                ( pp(aa(A,bool,Uu,Y))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Uua)) ) ) ) ) ).

% ATP.lambda_141
tff(fact_7362_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gr(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_142
tff(fact_7363_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_go(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_143
tff(fact_7364_ATP_Olambda__144,axiom,
    ! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_amy(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu),Uua))
    <=> ( aa(nat,sum_sum(B,nat),Uu,Uua) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ).

% ATP.lambda_144
tff(fact_7365_ATP_Olambda__145,axiom,
    ! [A: $tType,Uu: fun(nat,sum_sum(A,nat)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_amv(fun(nat,sum_sum(A,nat)),fun(nat,bool),Uu),Uua))
    <=> ( aa(nat,sum_sum(A,nat),Uu,Uua) = aa(nat,sum_sum(A,nat),sum_Inr(nat,A),zero_zero(nat)) ) ) ).

% ATP.lambda_145
tff(fact_7366_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wf(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_146
tff(fact_7367_ATP_Olambda__147,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,product_prod(nat,A),aTP_Lamp_akl(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_147
tff(fact_7368_ATP_Olambda__148,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_aao(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_148
tff(fact_7369_ATP_Olambda__149,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_pa(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),insert2(A,aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_149
tff(fact_7370_ATP_Olambda__150,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_hy(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = zero_zero(A) ) ) ) ).

% ATP.lambda_150
tff(fact_7371_ATP_Olambda__151,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_hz(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_151
tff(fact_7372_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: list(fun(A,nat)),Uua: A] : aa(A,list(nat),aTP_Lamp_akq(list(fun(A,nat)),fun(A,list(nat)),Uu),Uua) = aa(list(fun(A,nat)),list(nat),map(fun(A,nat),nat,aTP_Lamp_akp(A,fun(fun(A,nat),nat),Uua)),Uu) ).

% ATP.lambda_152
tff(fact_7373_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_abh(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_wk(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_153
tff(fact_7374_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_kt(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).

% ATP.lambda_154
tff(fact_7375_ATP_Olambda__155,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_bl(num,fun(num,int),Uu),Uua) = bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_155
tff(fact_7376_ATP_Olambda__156,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_agt(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_156
tff(fact_7377_ATP_Olambda__157,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: heap_Time_Heap(product_prod(A,B))] :
      ( pp(aa(heap_Time_Heap(product_prod(A,B)),bool,aTP_Lamp_aos(fun(A,fun(B,bool)),fun(heap_Time_Heap(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(heap_Time_Heap(product_prod(A,B)),set(product_prod(A,B)),heap_Time_set_Heap(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_157
tff(fact_7378_ATP_Olambda__158,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(product_prod(A,B))] :
      ( pp(aa(option(product_prod(A,B)),bool,aTP_Lamp_anu(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(option(product_prod(A,B)),set(product_prod(A,B)),set_option(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).

% ATP.lambda_158
tff(fact_7379_ATP_Olambda__159,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_acj(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_159
tff(fact_7380_ATP_Olambda__160,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(set(A))] : aa(set(set(A)),set(A),aTP_Lamp_aqo(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_160
tff(fact_7381_ATP_Olambda__161,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_aqv(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_161
tff(fact_7382_ATP_Olambda__162,axiom,
    ! [Uu: char,Uua: char] :
      ( pp(aa(char,bool,aa(char,fun(char,bool),aTP_Lamp_amo(char,fun(char,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),Uu)),aa(char,nat,comm_s6883823935334413003f_char(nat),Uua))) ) ).

% ATP.lambda_162
tff(fact_7383_ATP_Olambda__163,axiom,
    ! [A: $tType,Uu: list(option(A)),Uua: A] : aa(A,option(list(A)),aTP_Lamp_aot(list(option(A)),fun(A,option(list(A))),Uu),Uua) = aa(option(list(A)),option(list(A)),aa(fun(list(A),list(A)),fun(option(list(A)),option(list(A))),map_option(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Uua)),those(A,Uu)) ).

% ATP.lambda_163
tff(fact_7384_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_jw(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_164
tff(fact_7385_ATP_Olambda__165,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_ly(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_165
tff(fact_7386_ATP_Olambda__166,axiom,
    ! [A: $tType,Uu: pred(A),Uua: seq(A)] : aa(seq(A),set(A),aa(pred(A),fun(seq(A),set(A)),aTP_Lamp_atb(pred(A),fun(seq(A),set(A))),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(pred(A),set(A),set_of_pred(A),Uu)),aa(seq(A),set(A),set_of_seq(A),Uua)) ).

% ATP.lambda_166
tff(fact_7387_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( countable(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,set(old_node(A,product_unit)),aa(nat,fun(nat,set(old_node(A,product_unit))),aTP_Lamp_api(nat,fun(nat,set(old_node(A,product_unit)))),Uu),Uua) = old_Scons(A,product_unit,nth_item(A,Uu),nth_item(A,Uua)) ) ).

% ATP.lambda_167
tff(fact_7388_ATP_Olambda__168,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_abk(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_168
tff(fact_7389_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,rat,aa(nat,fun(nat,rat),aTP_Lamp_apd(nat,fun(nat,rat)),Uu),Uua) = aa(int,rat,aa(int,fun(int,rat),fract,nat_int_decode(Uu)),nat_int_decode(Uua)) ).

% ATP.lambda_169
tff(fact_7390_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_adt(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_170
tff(fact_7391_ATP_Olambda__171,axiom,
    ! [A: $tType,Uu: set(A),Uua: heap_Time_Heap(A)] :
      ( pp(aa(heap_Time_Heap(A),bool,aTP_Lamp_aor(set(A),fun(heap_Time_Heap(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(heap_Time_Heap(A),set(A),heap_Time_set_Heap(A),Uua)),Uu)) ) ).

% ATP.lambda_171
tff(fact_7392_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: set(A),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_ant(set(A),fun(option(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(option(A),set(A),set_option(A),Uua)),Uu)) ) ).

% ATP.lambda_172
tff(fact_7393_ATP_Olambda__173,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_akv(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_173
tff(fact_7394_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_et(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_174
tff(fact_7395_ATP_Olambda__175,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_aas(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_175
tff(fact_7396_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_abr(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_176
tff(fact_7397_ATP_Olambda__177,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_vr(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_177
tff(fact_7398_ATP_Olambda__178,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(set(product_prod(A,A))),aTP_Lamp_ake(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(set(product_prod(A,A)))),Uu),Uua) = aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),image(set(product_prod(A,A)),set(product_prod(A,A)),converse(set(product_prod(A,A)),set(product_prod(A,A)),bNF_We4044943003108391690rdLess(A,A))),aa(set(set(product_prod(A,A))),set(set(product_prod(A,A))),insert2(set(product_prod(A,A)),Uu),bot_bot(set(set(product_prod(A,A)))))) ).

% ATP.lambda_178
tff(fact_7399_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_bn(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_179
tff(fact_7400_ATP_Olambda__180,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_be(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_180
tff(fact_7401_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( field_char_0(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(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_181
tff(fact_7402_ATP_Olambda__182,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_sl(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),insert2(A,Uu)),Uua)) ).

% ATP.lambda_182
tff(fact_7403_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ey(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_183
tff(fact_7404_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: fun(literal,fun(list(product_prod(typerep,A)),A)),Uua: typerep] : aa(typerep,product_prod(typerep,A),aTP_Lamp_arh(fun(literal,fun(list(product_prod(typerep,A)),A)),fun(typerep,product_prod(typerep,A)),Uu),Uua) = aa(A,product_prod(typerep,A),aa(typerep,fun(A,product_prod(typerep,A)),product_Pair(typerep,A),Uua),rec_typerep(A,Uu,Uua)) ).

% ATP.lambda_184
tff(fact_7405_ATP_Olambda__185,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_ah(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_185
tff(fact_7406_ATP_Olambda__186,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_ag(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_186
tff(fact_7407_ATP_Olambda__187,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_us(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),insert2(B,Uua),bot_bot(set(B)))) ).

% ATP.lambda_187
tff(fact_7408_ATP_Olambda__188,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_yt(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),insert2(B,Uua),bot_bot(set(B)))) ).

% ATP.lambda_188
tff(fact_7409_ATP_Olambda__189,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_lt(set(A),fun(A,bool),Uu),Uua))
    <=> ( Uu = aa(set(A),set(A),insert2(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_189
tff(fact_7410_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahc(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_190
tff(fact_7411_ATP_Olambda__191,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_zw(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_191
tff(fact_7412_ATP_Olambda__192,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_wl(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_192
tff(fact_7413_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_jh(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_193
tff(fact_7414_ATP_Olambda__194,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_adg(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_194
tff(fact_7415_ATP_Olambda__195,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_bu(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_195
tff(fact_7416_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_tn(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_196
tff(fact_7417_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aau(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_197
tff(fact_7418_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gd(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_198
tff(fact_7419_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hd(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_199
tff(fact_7420_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fy(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_200
tff(fact_7421_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_ahw(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),nat_list_decode(Uua)) ).

% ATP.lambda_201
tff(fact_7422_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_acp(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_202
tff(fact_7423_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_abf(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_203
tff(fact_7424_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: A,Uua: pred(A)] : aa(pred(A),set(A),aa(A,fun(pred(A),set(A)),aTP_Lamp_ata(A,fun(pred(A),set(A))),Uu),Uua) = aa(set(A),set(A),insert2(A,Uu),aa(pred(A),set(A),set_of_pred(A),Uua)) ).

% ATP.lambda_204
tff(fact_7425_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_acd(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_205
tff(fact_7426_ATP_Olambda__206,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adk(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = last(A,Uu) ) ) ).

% ATP.lambda_206
tff(fact_7427_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kw(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_207
tff(fact_7428_ATP_Olambda__208,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_if(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_208
tff(fact_7429_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ie(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_209
tff(fact_7430_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qz(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_210
tff(fact_7431_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ajv(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_211
tff(fact_7432_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afp(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_212
tff(fact_7433_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aou(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_213
tff(fact_7434_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lb(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_214
tff(fact_7435_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aqx(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_215
tff(fact_7436_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_dq(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_216
tff(fact_7437_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qb(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_217
tff(fact_7438_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mc(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_218
tff(fact_7439_ATP_Olambda__219,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_219
tff(fact_7440_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_220
tff(fact_7441_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ra(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_221
tff(fact_7442_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_agc(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_222
tff(fact_7443_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ajw(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_223
tff(fact_7444_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afq(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_224
tff(fact_7445_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afd(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_225
tff(fact_7446_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aji(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_226
tff(fact_7447_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aqy(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_227
tff(fact_7448_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_gi(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_228
tff(fact_7449_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mq(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_229
tff(fact_7450_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_lz(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).

% ATP.lambda_230
tff(fact_7451_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mb(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_231
tff(fact_7452_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mk(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).

% ATP.lambda_232
tff(fact_7453_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_qr(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_233
tff(fact_7454_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_no(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_234
tff(fact_7455_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_pe(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).

% ATP.lambda_235
tff(fact_7456_ATP_Olambda__236,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_236
tff(fact_7457_ATP_Olambda__237,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_ot(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_237
tff(fact_7458_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_vy(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_238
tff(fact_7459_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ma(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_239
tff(fact_7460_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ic(nat,fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),Uu)) ) ).

% ATP.lambda_240
tff(fact_7461_ATP_Olambda__241,axiom,
    ! [Uu: int,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_im(int,fun(int,bool),Uu),Uua))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),dvd_dvd(int),Uua),Uu)) ) ).

% ATP.lambda_241
tff(fact_7462_ATP_Olambda__242,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_bb(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),Uu)) ) ) ).

% ATP.lambda_242
tff(fact_7463_ATP_Olambda__243,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_xo(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).

% ATP.lambda_243
tff(fact_7464_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_jv(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_244
tff(fact_7465_ATP_Olambda__245,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_fx(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_245
tff(fact_7466_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ach(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_246
tff(fact_7467_ATP_Olambda__247,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_fw(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_247
tff(fact_7468_ATP_Olambda__248,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_du(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_248
tff(fact_7469_ATP_Olambda__249,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,pred(A)),Uua: pred(B)] : aa(pred(B),pred(A),aTP_Lamp_ati(fun(B,pred(A)),fun(pred(B),pred(A)),Uu),Uua) = bind3(B,A,Uua,Uu) ).

% ATP.lambda_249
tff(fact_7470_ATP_Olambda__250,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_wk(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_250
tff(fact_7471_ATP_Olambda__251,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_alr(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,member(nat,Uua),Uu)) ) ).

% ATP.lambda_251
tff(fact_7472_ATP_Olambda__252,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_agi(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,member(B,Uua),Uu)) ) ).

% ATP.lambda_252
tff(fact_7473_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alq(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).

% ATP.lambda_253
tff(fact_7474_ATP_Olambda__254,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aa(set(A),fun(A,bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_254
tff(fact_7475_ATP_Olambda__255,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_tl(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_255
tff(fact_7476_ATP_Olambda__256,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: nat] : aa(nat,fun(A,fun(A,bool)),aTP_Lamp_apm(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),Uu),Uua) = 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))),Uua),Uu) ).

% ATP.lambda_256
tff(fact_7477_ATP_Olambda__257,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_amh(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_257
tff(fact_7478_ATP_Olambda__258,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_abi(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_258
tff(fact_7479_ATP_Olambda__259,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_sc(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),Uua),Uu) ).

% ATP.lambda_259
tff(fact_7480_ATP_Olambda__260,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_kn(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_260
tff(fact_7481_ATP_Olambda__261,axiom,
    ! [A: $tType,Uu: A,Uua: product_unit] : aa(product_unit,seq(A),aTP_Lamp_asz(A,fun(product_unit,seq(A)),Uu),Uua) = insert(A,Uu,bot_bot(pred(A))) ).

% ATP.lambda_261
tff(fact_7482_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_aaq(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_262
tff(fact_7483_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_aar(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_263
tff(fact_7484_ATP_Olambda__264,axiom,
    ! [A: $tType,Uu: code_natural,Uua: A] :
      ( pp(aa(A,bool,aa(code_natural,fun(A,bool),aTP_Lamp_ame(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_264
tff(fact_7485_ATP_Olambda__265,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aja(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_265
tff(fact_7486_ATP_Olambda__266,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_xy(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_266
tff(fact_7487_ATP_Olambda__267,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_abp(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_267
tff(fact_7488_ATP_Olambda__268,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_abq(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_268
tff(fact_7489_ATP_Olambda__269,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: product_prod(B,A)] : aa(product_prod(B,A),nat,aTP_Lamp_ql(fun(A,nat),fun(product_prod(B,A),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(B,A),A,product_snd(B,A),Uua)) ).

% ATP.lambda_269
tff(fact_7490_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,nat),Uua: product_prod(A,B)] : aa(product_prod(A,B),nat,aTP_Lamp_qk(fun(A,nat),fun(product_prod(A,B),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_270
tff(fact_7491_ATP_Olambda__271,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(sum_sum(A,B),C),Uua: B] : aa(B,C,aTP_Lamp_aok(fun(sum_sum(A,B),C),fun(B,C),Uu),Uua) = aa(sum_sum(A,B),C,Uu,aa(B,sum_sum(A,B),sum_Inr(B,A),Uua)) ).

% ATP.lambda_271
tff(fact_7492_ATP_Olambda__272,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(sum_sum(A,B),C),Uua: A] : aa(A,C,aTP_Lamp_aoj(fun(sum_sum(A,B),C),fun(A,C),Uu),Uua) = aa(sum_sum(A,B),C,Uu,aa(A,sum_sum(A,B),sum_Inl(A,B),Uua)) ).

% ATP.lambda_272
tff(fact_7493_ATP_Olambda__273,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_agy(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_273
tff(fact_7494_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_274
tff(fact_7495_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_275
tff(fact_7496_ATP_Olambda__276,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_zy(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_276
tff(fact_7497_ATP_Olambda__277,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_aij(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).

% ATP.lambda_277
tff(fact_7498_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_rz(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).

% ATP.lambda_278
tff(fact_7499_ATP_Olambda__279,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,nat)] : aa(fun(A,nat),nat,aTP_Lamp_akp(A,fun(fun(A,nat),nat),Uu),Uua) = aa(A,nat,Uua,Uu) ).

% ATP.lambda_279
tff(fact_7500_ATP_Olambda__280,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Sup(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_nb(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_280
tff(fact_7501_ATP_Olambda__281,axiom,
    ! [B: $tType,A: $tType] :
      ( complete_Inf(B)
     => ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_nc(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).

% ATP.lambda_281
tff(fact_7502_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_am(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).

% ATP.lambda_282
tff(fact_7503_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_rl(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_min(A),Uu)),Uua)) ) ).

% ATP.lambda_283
tff(fact_7504_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_rh(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),ord_max(A),Uu)),Uua)) ) ).

% ATP.lambda_284
tff(fact_7505_ATP_Olambda__285,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_rj(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),sup_sup(A),Uu)),Uua)) ) ).

% ATP.lambda_285
tff(fact_7506_ATP_Olambda__286,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_rk(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uu),aa(A,fun(A,A),inf_inf(A),Uu)),Uua)) ) ).

% ATP.lambda_286
tff(fact_7507_ATP_Olambda__287,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_cc(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),aa(option(num),num,aa(fun(num,num),fun(option(num),num),aa(num,fun(fun(num,num),fun(option(num),num)),case_option(num,num),one2),bit1),bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_287
tff(fact_7508_ATP_Olambda__288,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_vm(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_vl(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_288
tff(fact_7509_ATP_Olambda__289,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_tk(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_tj(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_289
tff(fact_7510_ATP_Olambda__290,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_ki(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_kh(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_290
tff(fact_7511_ATP_Olambda__291,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_kc(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_kb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_291
tff(fact_7512_ATP_Olambda__292,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_ka(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_jz(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_292
tff(fact_7513_ATP_Olambda__293,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_jy(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_jx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_293
tff(fact_7514_ATP_Olambda__294,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_jl(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_jk(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_294
tff(fact_7515_ATP_Olambda__295,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_bs(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_br(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_295
tff(fact_7516_ATP_Olambda__296,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_bq(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_bp(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_296
tff(fact_7517_ATP_Olambda__297,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,bool)] : aa(fun(A,bool),set(A),aTP_Lamp_aqj(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_aqi(fun(A,fun(B,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_297
tff(fact_7518_ATP_Olambda__298,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] : aa(B,set(A),aTP_Lamp_uo(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_un(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_298
tff(fact_7519_ATP_Olambda__299,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_py(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_pv(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_299
tff(fact_7520_ATP_Olambda__300,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_pu(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_pt(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_300
tff(fact_7521_ATP_Olambda__301,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_pw(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_pv(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).

% ATP.lambda_301
tff(fact_7522_ATP_Olambda__302,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_px(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_pt(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).

% ATP.lambda_302
tff(fact_7523_ATP_Olambda__303,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_ai(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_303
tff(fact_7524_ATP_Olambda__304,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_abj(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_304
tff(fact_7525_ATP_Olambda__305,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_yc(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_yb(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).

% ATP.lambda_305
tff(fact_7526_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_aps(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_apr(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua)),Uu)) ).

% ATP.lambda_306
tff(fact_7527_ATP_Olambda__307,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_aan(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_aam(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_307
tff(fact_7528_ATP_Olambda__308,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_bc(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),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),bit0(one2))),Uua)))) ) ).

% ATP.lambda_308
tff(fact_7529_ATP_Olambda__309,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_pq(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_309
tff(fact_7530_ATP_Olambda__310,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_pr(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_310
tff(fact_7531_ATP_Olambda__311,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_qa(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_311
tff(fact_7532_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ba(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_312
tff(fact_7533_ATP_Olambda__313,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_az(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_313
tff(fact_7534_ATP_Olambda__314,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_amf(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_314
tff(fact_7535_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_abe(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_315
tff(fact_7536_ATP_Olambda__316,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adj(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua != last(A,Uu) ) ) ).

% ATP.lambda_316
tff(fact_7537_ATP_Olambda__317,axiom,
    ! [P3: $tType,O: $tType,Uu: fun(O,set(P3)),Uua: set(O)] : aa(set(O),set(P3),aTP_Lamp_ob(fun(O,set(P3)),fun(set(O),set(P3)),Uu),Uua) = aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(O),set(set(P3)),image2(O,set(P3),Uu),Uua)) ).

% ATP.lambda_317
tff(fact_7538_ATP_Olambda__318,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_aed(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_318
tff(fact_7539_ATP_Olambda__319,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_aql(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_319
tff(fact_7540_ATP_Olambda__320,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_aqs(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_320
tff(fact_7541_ATP_Olambda__321,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_aem(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_321
tff(fact_7542_ATP_Olambda__322,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_aqk(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_322
tff(fact_7543_ATP_Olambda__323,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_aqr(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_323
tff(fact_7544_ATP_Olambda__324,axiom,
    ! [P3: $tType,O: $tType,Uu: fun(O,set(P3)),Uua: set(O)] : aa(set(O),set(P3),aTP_Lamp_od(fun(O,set(P3)),fun(set(O),set(P3)),Uu),Uua) = aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(O),set(set(P3)),image2(O,set(P3),Uu),Uua)) ).

% ATP.lambda_324
tff(fact_7545_ATP_Olambda__325,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_aee(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_325
tff(fact_7546_ATP_Olambda__326,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_aqu(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_326
tff(fact_7547_ATP_Olambda__327,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_adl(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_327
tff(fact_7548_ATP_Olambda__328,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_aqt(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_328
tff(fact_7549_ATP_Olambda__329,axiom,
    ! [A: $tType,Uu: pred(A),Uua: pred(A)] : aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_atn(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uu),Uua)) ).

% ATP.lambda_329
tff(fact_7550_ATP_Olambda__330,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_ajj(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_330
tff(fact_7551_ATP_Olambda__331,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_amg(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_331
tff(fact_7552_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_rn(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert2(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).

% ATP.lambda_332
tff(fact_7553_ATP_Olambda__333,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_yd(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert2(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).

% ATP.lambda_333
tff(fact_7554_ATP_Olambda__334,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_lm(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_334
tff(fact_7555_ATP_Olambda__335,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ln(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_335
tff(fact_7556_ATP_Olambda__336,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ix(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_336
tff(fact_7557_ATP_Olambda__337,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_iw(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_337
tff(fact_7558_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_lw(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).

% ATP.lambda_338
tff(fact_7559_ATP_Olambda__339,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_adp(A,fun(A,bool)),Uu),Uua))
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_339
tff(fact_7560_ATP_Olambda__340,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahb(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_340
tff(fact_7561_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aga(A,fun(A,bool),Uu),Uua))
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_341
tff(fact_7562_ATP_Olambda__342,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_abd(A,fun(A,bool)),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_342
tff(fact_7563_ATP_Olambda__343,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_al(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_343
tff(fact_7564_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_wi(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_7565_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aha(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_7566_ATP_Olambda__346,axiom,
    ! [C: $tType,B: $tType,Uu: fun(B,set(C)),Uua: B] : aa(B,set(product_prod(C,C)),aTP_Lamp_aiv(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_346
tff(fact_7567_ATP_Olambda__347,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_aiw(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_347
tff(fact_7568_ATP_Olambda__348,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,bool),Uua: B] : aa(B,A,aTP_Lamp_la(fun(B,bool),fun(B,A),Uu),Uua) = aa(bool,A,zero_neq_one_of_bool(A),aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_348
tff(fact_7569_ATP_Olambda__349,axiom,
    ! [A: $tType,B: $tType] :
      ( field_char_0(A)
     => ! [Uu: fun(B,rat),Uua: B] : aa(B,A,aTP_Lamp_te(fun(B,rat),fun(B,A),Uu),Uua) = aa(rat,A,field_char_0_of_rat(A),aa(B,rat,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_7570_ATP_Olambda__350,axiom,
    ! [B: $tType,Uu: fun(B,nat),Uua: B] : aa(B,int,aTP_Lamp_cz(fun(B,nat),fun(B,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(B,nat,Uu,Uua)) ).

% ATP.lambda_350
tff(fact_7571_ATP_Olambda__351,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_fe(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_7572_ATP_Olambda__352,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_cs(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_352
tff(fact_7573_ATP_Olambda__353,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_nm(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_353
tff(fact_7574_ATP_Olambda__354,axiom,
    ! [A: $tType,B: $tType] :
      ( comple489889107523837845lgebra(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_nx(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_7575_ATP_Olambda__355,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_dj(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_7576_ATP_Olambda__356,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fv(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_7577_ATP_Olambda__357,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ff(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_7578_ATP_Olambda__358,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_cr(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_7579_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ea(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_359
tff(fact_7580_ATP_Olambda__360,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_fm(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_7581_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_cw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_7582_ATP_Olambda__362,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_alv(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_362
tff(fact_7583_ATP_Olambda__363,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_ada(fun(A,B),fun(A,option(B)),Uu),Uua) = aa(B,option(B),some(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_363
tff(fact_7584_ATP_Olambda__364,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_akb(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_364
tff(fact_7585_ATP_Olambda__365,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_zs(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_365
tff(fact_7586_ATP_Olambda__366,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_acg(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_366
tff(fact_7587_ATP_Olambda__367,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_ang(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = old_In1(A,B,aa(C,set(old_node(A,B)),Uu,Uua)) ).

% ATP.lambda_367
tff(fact_7588_ATP_Olambda__368,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_ani(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = old_In0(A,B,aa(C,set(old_node(A,B)),Uu,Uua)) ).

% ATP.lambda_368
tff(fact_7589_ATP_Olambda__369,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_tu(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_369
tff(fact_7590_ATP_Olambda__370,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_ur(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_370
tff(fact_7591_ATP_Olambda__371,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_rx(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_371
tff(fact_7592_ATP_Olambda__372,axiom,
    ! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_uq(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = principal(E,aa(A,set(E),Uu,Uua)) ).

% ATP.lambda_372
tff(fact_7593_ATP_Olambda__373,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_rw(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_373
tff(fact_7594_ATP_Olambda__374,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_ox(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_374
tff(fact_7595_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(B),aTP_Lamp_aix(fun(A,set(product_prod(B,B))),fun(A,set(B)),Uu),Uua) = field2(B,aa(A,set(product_prod(B,B)),Uu,Uua)) ).

% ATP.lambda_375
tff(fact_7596_ATP_Olambda__376,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_qc(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_376
tff(fact_7597_ATP_Olambda__377,axiom,
    ! [A: $tType,Uu: fun(set(A),fun(A,bool)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_ahd(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_377
tff(fact_7598_ATP_Olambda__378,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_xh(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_378
tff(fact_7599_ATP_Olambda__379,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_rp(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert2(B,aa(A,B,Uu,Uua)) ).

% ATP.lambda_379
tff(fact_7600_ATP_Olambda__380,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_sn(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_380
tff(fact_7601_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_ja(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_381
tff(fact_7602_ATP_Olambda__382,axiom,
    ! [B: $tType,Uu: fun(B,bool),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_qf(fun(B,bool),fun(B,bool),Uu),Uua))
    <=> ~ pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_382
tff(fact_7603_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_lv(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_383
tff(fact_7604_ATP_Olambda__384,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_arg(fun(B,fun(A,bool)),fun(B,bool),Uu),Uua))
    <=> ! [X_1: A] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uua),X_1)) ) ).

% ATP.lambda_384
tff(fact_7605_ATP_Olambda__385,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(B)
     => ! [Uu: fun(A,fun(B,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aft(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
        <=> ! [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ) ).

% ATP.lambda_385
tff(fact_7606_ATP_Olambda__386,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aez(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ! [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).

% ATP.lambda_386
tff(fact_7607_ATP_Olambda__387,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,fun(A,bool),aTP_Lamp_ate(fun(B,A),fun(B,fun(A,bool)),Uu),Uua) = aa(A,fun(A,bool),fequal(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_387
tff(fact_7608_ATP_Olambda__388,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_um(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).

% ATP.lambda_388
tff(fact_7609_ATP_Olambda__389,axiom,
    ! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_amz(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_amy(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu)) ).

% ATP.lambda_389
tff(fact_7610_ATP_Olambda__390,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ajb(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aa(nat,fun(nat,bool),aTP_Lamp_cq(nat,fun(nat,bool)),Uu)) ).

% ATP.lambda_390
tff(fact_7611_ATP_Olambda__391,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_ags(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_agr(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_391
tff(fact_7612_ATP_Olambda__392,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool)] : aa(fun(B,bool),filter(product_prod(A,B)),aa(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_ajm(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),Uu),Uua) = principal(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),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_wy(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua)))) ).

% ATP.lambda_392
tff(fact_7613_ATP_Olambda__393,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_amp(fun(B,fun(A,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X3: B] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,X3),Uua)) ) ).

% ATP.lambda_393
tff(fact_7614_ATP_Olambda__394,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aqw(fun(A,fun(A,bool)),fun(set(A),bool),Uu),Uua))
    <=> ? [X3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),Uu,X3),X3))
          & ( Uua = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),Uu,X3)) ) ) ) ).

% ATP.lambda_394
tff(fact_7615_ATP_Olambda__395,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ut(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_395
tff(fact_7616_ATP_Olambda__396,axiom,
    ! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
      ( pp(aa(set(B),bool,aTP_Lamp_aef(set(set(B)),fun(set(B),bool),Uu),Uua))
    <=> ? [F6: fun(set(B),B)] :
          ( ( Uua = aa(set(set(B)),set(B),image2(set(B),B,F6),Uu) )
          & ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,member(set(B),X3),Uu))
             => pp(aa(set(B),bool,member(B,aa(set(B),B,F6,X3)),X3)) ) ) ) ).

% ATP.lambda_396
tff(fact_7617_ATP_Olambda__397,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_adz(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X3),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F6,X3)),X3)) ) ) ) ) ).

% ATP.lambda_397
tff(fact_7618_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_aec(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X3),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F6,X3)),X3)) ) ) ) ) ).

% ATP.lambda_398
tff(fact_7619_ATP_Olambda__399,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_aeb(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F6),Uu) )
              & ! [X3: set(A)] :
                  ( pp(aa(set(set(A)),bool,member(set(A),X3),Uu))
                 => pp(aa(set(A),bool,member(A,aa(set(A),A,F6,X3)),X3)) ) ) ) ) ).

% ATP.lambda_399
tff(fact_7620_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ve(set(A),fun(set(A),bool),Uu),Uua))
    <=> ? [B8: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B8) )
          & pp(aa(set(set(A)),bool,member(set(A),Uu),pow2(A,B8))) ) ) ).

% ATP.lambda_400
tff(fact_7621_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( pp(aa(filter(A),bool,aTP_Lamp_aea(set(filter(A)),fun(filter(A),bool),Uu),Uua))
    <=> ! [X3: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X3),Uu))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),Uua),X3)) ) ) ).

% ATP.lambda_401
tff(fact_7622_ATP_Olambda__402,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aia(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X3)) ) ) ) ).

% ATP.lambda_402
tff(fact_7623_ATP_Olambda__403,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_adr(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X3)) ) ) ) ).

% ATP.lambda_403
tff(fact_7624_ATP_Olambda__404,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahu(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Uua)) ) ) ) ).

% ATP.lambda_404
tff(fact_7625_ATP_Olambda__405,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ads(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Uua)) ) ) ) ).

% ATP.lambda_405
tff(fact_7626_ATP_Olambda__406,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_alu(set(nat),fun(nat,bool),Uu),Uua))
    <=> ! [X3: nat] :
          ( pp(aa(set(nat),bool,member(nat,X3),Uu))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),dvd_dvd(nat),Uua),X3)) ) ) ).

% ATP.lambda_406
tff(fact_7627_ATP_Olambda__407,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_alx(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),Uua),X3)) ) ) ) ).

% ATP.lambda_407
tff(fact_7628_ATP_Olambda__408,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_aly(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),dvd_dvd(A),X3),Uua)) ) ) ) ).

% ATP.lambda_408
tff(fact_7629_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_agk(set(filter(A)),fun(fun(A,bool),bool),Uu),Uua))
    <=> ! [X3: filter(A)] :
          ( pp(aa(set(filter(A)),bool,member(filter(A),X3),Uu))
         => eventually(A,Uua,X3) ) ) ).

% ATP.lambda_409
tff(fact_7630_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adx(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),Uu))
         => pp(aa(set(A),bool,member(A,Uua),X3)) ) ) ).

% ATP.lambda_410
tff(fact_7631_ATP_Olambda__411,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aeo(set(set(A)),fun(A,bool),Uu),Uua))
    <=> ? [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,member(set(A),X3),Uu))
          & pp(aa(set(A),bool,member(A,Uua),X3)) ) ) ).

% ATP.lambda_411
tff(fact_7632_ATP_Olambda__412,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ahe(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y))
             => pp(aa(A,bool,Uu,Y)) ) ) ) ).

% ATP.lambda_412
tff(fact_7633_ATP_Olambda__413,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aqm(fun(A,bool),fun(set(A),bool),Uu),Uua))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uua))
         => pp(aa(A,bool,Uu,X3)) ) ) ).

% ATP.lambda_413
tff(fact_7634_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_afi(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uua))
         => ! [Xa2: A] :
              ( pp(aa(set(A),bool,member(A,Xa2),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),X3),Xa2)),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),Xa2),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_414
tff(fact_7635_ATP_Olambda__415,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_amq(set(product_prod(A,B)),fun(A,bool),Uu),Uua))
    <=> ? [Y: 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),Uua),Y)),Uu)) ) ).

% ATP.lambda_415
tff(fact_7636_ATP_Olambda__416,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_acv(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A7: A] : aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_416
tff(fact_7637_ATP_Olambda__417,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(old_node(A,B))),Uua: set(old_node(A,B))] :
      ( pp(aa(set(old_node(A,B)),bool,aTP_Lamp_anb(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),bool),Uu),Uua))
    <=> ? [X3: B] : Uua = aa(set(old_node(A,B)),set(old_node(A,B)),image2(old_node(A,B),old_node(A,B),old_Push_Node(B,A,aa(B,sum_sum(B,nat),sum_Inl(B,nat),X3))),aa(B,set(old_node(A,B)),Uu,X3)) ) ).

% ATP.lambda_417
tff(fact_7638_ATP_Olambda__418,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_uh(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X3: B] : Uua = aa(B,A,Uu,X3) ) ).

% ATP.lambda_418
tff(fact_7639_ATP_Olambda__419,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,bool),Uua: fun(A,B)] :
      ( pp(aa(fun(A,B),bool,aTP_Lamp_apw(fun(B,bool),fun(fun(A,B),bool),Uu),Uua))
    <=> ! [X3: A] : pp(aa(B,bool,Uu,aa(A,B,Uua,X3))) ) ).

% ATP.lambda_419
tff(fact_7640_ATP_Olambda__420,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_acw(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A7: A,B7: B] :
          ( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
          & ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B7) ) ) ) ).

% ATP.lambda_420
tff(fact_7641_ATP_Olambda__421,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_aeg(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
    <=> ? [X10: set(A),Y8: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X10),Y8) )
          & ( X10 != bot_bot(set(A)) )
          & ! [X3: A] :
              ( pp(aa(set(A),bool,member(A,X3),Y8))
             => ? [Xa2: A] :
                  ( pp(aa(set(A),bool,member(A,Xa2),X10))
                  & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_421
tff(fact_7642_ATP_Olambda__422,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_xd(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).

% ATP.lambda_422
tff(fact_7643_ATP_Olambda__423,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_ace(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_423
tff(fact_7644_ATP_Olambda__424,axiom,
    ! [C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: C] : aa(C,set(B),aTP_Lamp_ais(set(product_prod(B,B)),fun(C,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_424
tff(fact_7645_ATP_Olambda__425,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_aiu(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = field2(B,Uu) ).

% ATP.lambda_425
tff(fact_7646_ATP_Olambda__426,axiom,
    ! [C: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: C] : aa(C,set(A),aTP_Lamp_air(set(product_prod(A,A)),fun(C,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_426
tff(fact_7647_ATP_Olambda__427,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_ait(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_427
tff(fact_7648_ATP_Olambda__428,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_xt(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = field2(A,Uu) ).

% ATP.lambda_428
tff(fact_7649_ATP_Olambda__429,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_xe(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_429
tff(fact_7650_ATP_Olambda__430,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_wz(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).

% ATP.lambda_430
tff(fact_7651_ATP_Olambda__431,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_aat(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = insert2(A,Uu) ).

% ATP.lambda_431
tff(fact_7652_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_aoc(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_432
tff(fact_7653_ATP_Olambda__433,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: fun(product_unit,seq(A)),Uub: product_unit] : aa(product_unit,seq(A),aa(fun(product_unit,seq(A)),fun(product_unit,seq(A)),aTP_Lamp_atm(fun(product_unit,seq(A)),fun(fun(product_unit,seq(A)),fun(product_unit,seq(A))),Uu),Uua),Uub) = case_seq(seq(A),A,aa(product_unit,seq(A),Uua,product_Unity),aTP_Lamp_atk(fun(product_unit,seq(A)),fun(A,fun(pred(A),seq(A))),Uua),aTP_Lamp_atl(fun(product_unit,seq(A)),fun(pred(A),fun(seq(A),seq(A))),Uua),aa(product_unit,seq(A),Uu,product_Unity)) ).

% ATP.lambda_433
tff(fact_7654_ATP_Olambda__434,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(A),Uub: option(B)] :
      ( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aTP_Lamp_aob(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),Uu),Uua),Uub))
    <=> pp(aa(option(A),bool,aa(fun(A,bool),fun(option(A),bool),aa(bool,fun(fun(A,bool),fun(option(A),bool)),case_option(bool,A),aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fTrue),aTP_Lamp_anz(B,bool)),Uub)),aa(option(B),fun(A,bool),aTP_Lamp_aoa(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_434
tff(fact_7655_ATP_Olambda__435,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_bi(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),if(product_prod(code_integer,code_integer)),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu)))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_435
tff(fact_7656_ATP_Olambda__436,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_bj(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(bool,fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),if(product_prod(nat,nat)),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu)))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_436
tff(fact_7657_ATP_Olambda__437,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_bk(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int)),aa(bool,fun(product_prod(int,int),fun(product_prod(int,int),product_prod(int,int))),if(product_prod(int,int)),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu)))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_437
tff(fact_7658_ATP_Olambda__438,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_bd(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = aa(product_prod(A,A),product_prod(A,A),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(A,A)),aa(bool,fun(product_prod(A,A),fun(product_prod(A,A),product_prod(A,A))),if(product_prod(A,A)),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu)))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_438
tff(fact_7659_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zd(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(bool,fun(list(A),fun(list(A),list(A))),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_439
tff(fact_7660_ATP_Olambda__440,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_li(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(set(B),bool,member(B,Uub),Uua)),aa(B,A,Uu,Uub)),zero_zero(A)) ) ).

% ATP.lambda_440
tff(fact_7661_ATP_Olambda__441,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_ll(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(set(B),bool,member(B,Uub),Uua)),aa(B,A,Uu,Uub)),one_one(A)) ) ).

% ATP.lambda_441
tff(fact_7662_ATP_Olambda__442,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_hf(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub)),aa(B,A,Uua,Uub)),zero_zero(A)) ) ).

% ATP.lambda_442
tff(fact_7663_ATP_Olambda__443,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_hi(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub)),aa(B,A,Uua,Uub)),one_one(A)) ) ).

% ATP.lambda_443
tff(fact_7664_ATP_Olambda__444,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_hg(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu)),aa(B,A,Uua,Uub)),zero_zero(A)) ) ).

% ATP.lambda_444
tff(fact_7665_ATP_Olambda__445,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_hh(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu)),aa(B,A,Uua,Uub)),one_one(A)) ) ).

% ATP.lambda_445
tff(fact_7666_ATP_Olambda__446,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_ck(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),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_446
tff(fact_7667_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_cl(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),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_447
tff(fact_7668_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_cj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(bool,fun(product_prod(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),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_448
tff(fact_7669_ATP_Olambda__449,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_rq(fun(A,bool),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(bool,fun(set(A),fun(set(A),set(A))),if(set(A)),aa(A,bool,Uu,Uua)),aa(set(A),set(A),insert2(A,Uua),Uub)),Uub) ).

% ATP.lambda_449
tff(fact_7670_ATP_Olambda__450,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_hx(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uua,Uub)),aa(B,A,Uu,Uub)),zero_zero(A)) ) ).

% ATP.lambda_450
tff(fact_7671_ATP_Olambda__451,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_abb(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uua,Uub)),aa(B,A,Uu,Uub)),zero_zero(A)) ) ).

% ATP.lambda_451
tff(fact_7672_ATP_Olambda__452,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_ia(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uua,Uub)),aa(B,A,Uu,Uub)),one_one(A)) ) ).

% ATP.lambda_452
tff(fact_7673_ATP_Olambda__453,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_aay(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),Uu),Uua),Uub) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),aa(bool,fun(fun(A,A),fun(fun(A,A),fun(A,A))),if(fun(A,A)),aa(B,bool,Uua,Uub)),aa(B,fun(A,A),Uu,Uub)),id(A)) ).

% ATP.lambda_453
tff(fact_7674_ATP_Olambda__454,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_ad(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) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(bool,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),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_454
tff(fact_7675_ATP_Olambda__455,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_abt(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),aa(bool,fun(option(A),fun(option(A),option(A))),if(option(A)),aa(B,bool,Uua,Uub)),aa(A,option(A),some(A),aa(B,A,Uu,Uub))),none(A)) ).

% ATP.lambda_455
tff(fact_7676_ATP_Olambda__456,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_ye(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_yd(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_456
tff(fact_7677_ATP_Olambda__457,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_rv(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_rn(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_457
tff(fact_7678_ATP_Olambda__458,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_anp(fun(A,fun(A,A)),fun(option(A),fun(A,option(A))),Uu),Uua),Uub) = aa(option(A),option(A),aa(fun(A,option(A)),fun(option(A),option(A)),aa(option(A),fun(fun(A,option(A)),fun(option(A),option(A))),case_option(option(A),A),aa(A,option(A),some(A),Uub)),aa(A,fun(A,option(A)),aTP_Lamp_ano(fun(A,fun(A,A)),fun(A,fun(A,option(A))),Uu),Uub)),Uua) ).

% ATP.lambda_458
tff(fact_7679_ATP_Olambda__459,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_acy(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_459
tff(fact_7680_ATP_Olambda__460,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_ams(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_460
tff(fact_7681_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_wo(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_461
tff(fact_7682_ATP_Olambda__462,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: bool] : aa(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_qi(set(A),fun(set(A),fun(bool,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(bool,fun(set(A),fun(set(A),set(A))),if(set(A)),Uub),Uu),Uua) ).

% ATP.lambda_462
tff(fact_7683_ATP_Olambda__463,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: option(B),Uub: A] :
      ( pp(aa(A,bool,aa(option(B),fun(A,bool),aTP_Lamp_aoa(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(option(B),bool,aa(fun(B,bool),fun(option(B),bool),aa(bool,fun(fun(B,bool),fun(option(B),bool)),case_option(bool,B),fFalse),aa(A,fun(B,bool),Uu,Uub)),Uua)) ) ).

% ATP.lambda_463
tff(fact_7684_ATP_Olambda__464,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_bt(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) = aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(option(product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),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_bs(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_464
tff(fact_7685_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_asn(fun(A,fun(A,A)),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),Uu,aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub))),aa(nat,A,Uua,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)))) ).

% ATP.lambda_465
tff(fact_7686_ATP_Olambda__466,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_fr(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_466
tff(fact_7687_ATP_Olambda__467,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_pv(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_467
tff(fact_7688_ATP_Olambda__468,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_gl(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_468
tff(fact_7689_ATP_Olambda__469,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_ed(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_469
tff(fact_7690_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aso(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_470
tff(fact_7691_ATP_Olambda__471,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_afr(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_471
tff(fact_7692_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_gs(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_472
tff(fact_7693_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_gp(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_473
tff(fact_7694_ATP_Olambda__474,axiom,
    ! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ars(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_474
tff(fact_7695_ATP_Olambda__475,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_pt(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_475
tff(fact_7696_ATP_Olambda__476,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_ne(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_476
tff(fact_7697_ATP_Olambda__477,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_ng(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_7698_ATP_Olambda__478,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_zp(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_7699_ATP_Olambda__479,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_of(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_479
tff(fact_7700_ATP_Olambda__480,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_fh(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_480
tff(fact_7701_ATP_Olambda__481,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_db(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_7702_ATP_Olambda__482,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_arz(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_7703_ATP_Olambda__483,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_tw(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_483
tff(fact_7704_ATP_Olambda__484,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_adf(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_484
tff(fact_7705_ATP_Olambda__485,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_afs(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_485
tff(fact_7706_ATP_Olambda__486,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_un(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_7707_ATP_Olambda__487,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_add(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_487
tff(fact_7708_ATP_Olambda__488,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aap(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_488
tff(fact_7709_ATP_Olambda__489,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_bz(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),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),aa(bool,fun(code_integer,fun(code_integer,code_integer)),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_489
tff(fact_7710_ATP_Olambda__490,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),aTP_Lamp_aku(set(set(A)),fun(set(set(A)),fun(set(set(A)),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(A)),bool,pred_chain(set(A),Uu,ord_less(set(A))),Uub))
        & pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less(set(set(A))),Uua),Uub)) ) ) ).

% ATP.lambda_490
tff(fact_7711_ATP_Olambda__491,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_au(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_491
tff(fact_7712_ATP_Olambda__492,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_gt(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_gs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_492
tff(fact_7713_ATP_Olambda__493,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_gq(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_gp(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_7714_ATP_Olambda__494,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_att(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_ats(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua)),split_seed(Uub)) ).

% ATP.lambda_494
tff(fact_7715_ATP_Olambda__495,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_kk(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_kj(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_495
tff(fact_7716_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_kg(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_kf(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_496
tff(fact_7717_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_ke(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_kd(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_497
tff(fact_7718_ATP_Olambda__498,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ju(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_jt(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_498
tff(fact_7719_ATP_Olambda__499,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_js(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_jr(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).

% ATP.lambda_499
tff(fact_7720_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_jn(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_jm(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_500
tff(fact_7721_ATP_Olambda__501,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_fi(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_fh(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_501
tff(fact_7722_ATP_Olambda__502,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_dc(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_db(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_502
tff(fact_7723_ATP_Olambda__503,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_pj(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_pi(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_503
tff(fact_7724_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(A),Uub: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(filter(A),fun(fun(B,bool),bool),aTP_Lamp_ajg(fun(A,B),fun(filter(A),fun(fun(B,bool),bool)),Uu),Uua),Uub))
    <=> eventually(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_yp(fun(A,B),fun(fun(B,bool),fun(A,bool)),Uu),Uub),Uua) ) ).

% ATP.lambda_504
tff(fact_7725_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_he(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_7726_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_agw(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_7727_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_ahi(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_7728_ATP_Olambda__508,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_zb(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_508
tff(fact_7729_ATP_Olambda__509,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: sum_sum(product_prod(A,C),product_prod(B,D))] :
      ( pp(aa(sum_sum(product_prod(A,C),product_prod(B,D)),bool,aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_are(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),Uu),Uua),Uub))
    <=> ( 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))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(A,C)),basic_setl(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(A,C),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),Uu))))
        & pp(aa(set(product_prod(B,D)),bool,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),bool),ord_less_eq(set(product_prod(B,D))),aa(sum_sum(product_prod(A,C),product_prod(B,D)),set(product_prod(B,D)),basic_setr(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(B,D),bool),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,bool)),fun(product_prod(B,D),bool),product_case_prod(B,D,bool),Uua)))) ) ) ).

% ATP.lambda_509
tff(fact_7730_ATP_Olambda__510,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: product_prod(product_prod(A,C),product_prod(B,D))] :
      ( pp(aa(product_prod(product_prod(A,C),product_prod(B,D)),bool,aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ale(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),Uu),Uua),Uub))
    <=> ( 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))),aa(product_prod(product_prod(A,C),product_prod(B,D)),set(product_prod(A,C)),basic_fsts(product_prod(A,C),product_prod(B,D)),Uub)),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),Uu))))
        & pp(aa(set(product_prod(B,D)),bool,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),bool),ord_less_eq(set(product_prod(B,D))),basic_snds(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),bool),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,bool)),fun(product_prod(B,D),bool),product_case_prod(B,D,bool),Uua)))) ) ) ).

% ATP.lambda_510
tff(fact_7731_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_tq(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_511
tff(fact_7732_ATP_Olambda__512,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_zf(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_512
tff(fact_7733_ATP_Olambda__513,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_vc(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),X3: A,Y: A,Xs5: list(A),Ys5: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys5)) )
            & pp(aa(set(product_prod(A,A)),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)),Uu)) ) ) ) ).

% ATP.lambda_513
tff(fact_7734_ATP_Olambda__514,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_zc(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_514
tff(fact_7735_ATP_Olambda__515,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_jj(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_515
tff(fact_7736_ATP_Olambda__516,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ji(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_516
tff(fact_7737_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_wv(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_517
tff(fact_7738_ATP_Olambda__518,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_kx(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_518
tff(fact_7739_ATP_Olambda__519,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: sum_sum(A,B)] :
      ( pp(aa(sum_sum(A,B),bool,aa(set(B),fun(sum_sum(A,B),bool),aTP_Lamp_ard(set(A),fun(set(B),fun(sum_sum(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(sum_sum(A,B),set(A),basic_setl(A,B),Uub)),Uu))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(sum_sum(A,B),set(B),basic_setr(A,B),Uub)),Uua)) ) ) ).

% ATP.lambda_519
tff(fact_7740_ATP_Olambda__520,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(set(B),fun(product_prod(A,B),bool),aTP_Lamp_alf(set(A),fun(set(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(product_prod(A,B),set(A),basic_fsts(A,B),Uub)),Uu))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),basic_snds(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_520
tff(fact_7741_ATP_Olambda__521,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_jc(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_521
tff(fact_7742_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_jf(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_522
tff(fact_7743_ATP_Olambda__523,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_za(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_523
tff(fact_7744_ATP_Olambda__524,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_ade(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_524
tff(fact_7745_ATP_Olambda__525,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_iy(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_525
tff(fact_7746_ATP_Olambda__526,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_abs(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_526
tff(fact_7747_ATP_Olambda__527,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_vs(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_527
tff(fact_7748_ATP_Olambda__528,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),aTP_Lamp_so(nat,fun(nat,fun(set(nat),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(nat)),bool,member(set(nat),Uub),pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu))))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_528
tff(fact_7749_ATP_Olambda__529,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_vg(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_529
tff(fact_7750_ATP_Olambda__530,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_abg(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_530
tff(fact_7751_ATP_Olambda__531,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_aak(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_531
tff(fact_7752_ATP_Olambda__532,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aaw(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_532
tff(fact_7753_ATP_Olambda__533,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_acx(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_533
tff(fact_7754_ATP_Olambda__534,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hb(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_534
tff(fact_7755_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_adu(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),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),X3)),Uu)) ) ) ) ).

% ATP.lambda_535
tff(fact_7756_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_adw(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),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),X3),Uub)),Uu)) ) ) ) ).

% ATP.lambda_536
tff(fact_7757_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_adv(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),Uua))
           => ( ( Uub != 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),Uub),X3)),Uu)) ) ) ) ) ).

% ATP.lambda_537
tff(fact_7758_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_ahp(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,member(A,Uub),field2(A,Uu)))
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),Uua))
           => ( ( Uub != 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),X3),Uub)),Uu)) ) ) ) ) ).

% ATP.lambda_538
tff(fact_7759_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_vd(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_7760_ATP_Olambda__540,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_sp(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_7761_ATP_Olambda__541,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_akj(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) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_541
tff(fact_7762_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_aib(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != field2(A,Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != 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_7763_ATP_Olambda__543,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_aev(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_aeu(set(nat),fun(nat,fun(nat,bool)),Uu),Uub)))),Uua)) ) ) ).

% ATP.lambda_543
tff(fact_7764_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_is(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_7765_ATP_Olambda__545,axiom,
    ! [A: $tType,B: $tType,Uu: nat,Uua: set(old_node(A,B)),Uub: old_node(A,B)] :
      ( pp(aa(old_node(A,B),bool,aa(set(old_node(A,B)),fun(old_node(A,B),bool),aTP_Lamp_anh(nat,fun(set(old_node(A,B)),fun(old_node(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(old_node(A,B)),bool,member(old_node(A,B),Uub),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),old_ndepth(A,B,Uub)),Uu)) ) ) ).

% ATP.lambda_545
tff(fact_7766_ATP_Olambda__546,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_iz(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_546
tff(fact_7767_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_ako(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_7768_ATP_Olambda__548,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ef(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_7769_ATP_Olambda__549,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_qj(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_549
tff(fact_7770_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_jd(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uu )
        | pp(aa(set(A),bool,member(A,Uub),Uua)) ) ) ).

% ATP.lambda_550
tff(fact_7771_ATP_Olambda__551,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_bv(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_551
tff(fact_7772_ATP_Olambda__552,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ih(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_552
tff(fact_7773_ATP_Olambda__553,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_bw(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_553
tff(fact_7774_ATP_Olambda__554,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ij(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_554
tff(fact_7775_ATP_Olambda__555,axiom,
    ! [A: $tType,Uu: pred(A),Uua: pred(A),Uub: seq(A)] :
      ( pp(aa(seq(A),bool,aa(pred(A),fun(seq(A),bool),aTP_Lamp_atg(pred(A),fun(pred(A),fun(seq(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(pred(A),bool,aa(pred(A),fun(pred(A),bool),ord_less_eq(pred(A)),Uua),Uu))
        & contained(A,Uub,Uu) ) ) ).

% ATP.lambda_555
tff(fact_7776_ATP_Olambda__556,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ik(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_7777_ATP_Olambda__557,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ii(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_7778_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_yl(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_7779_ATP_Olambda__559,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_rg(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_559
tff(fact_7780_ATP_Olambda__560,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ri(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_560
tff(fact_7781_ATP_Olambda__561,axiom,
    ! [A: $tType,Uu: pred(A),Uua: A,Uub: pred(A)] :
      ( pp(aa(pred(A),bool,aa(A,fun(pred(A),bool),aTP_Lamp_atf(pred(A),fun(A,fun(pred(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,aa(pred(A),fun(A,bool),eval(A),Uu),Uua))
        & pp(aa(pred(A),bool,aa(pred(A),fun(pred(A),bool),ord_less_eq(pred(A)),Uub),Uu)) ) ) ).

% ATP.lambda_561
tff(fact_7782_ATP_Olambda__562,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_agn(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ( eventually(A,Uub,Uu)
        & eventually(A,Uub,Uua) ) ) ).

% ATP.lambda_562
tff(fact_7783_ATP_Olambda__563,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afl(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_563
tff(fact_7784_ATP_Olambda__564,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aeu(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_564
tff(fact_7785_ATP_Olambda__565,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: A,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),aTP_Lamp_ajd(set(set(A)),fun(A,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(set(A)),bool,member(set(A),Uub),Uu))
        & pp(aa(set(A),bool,member(A,Uua),Uub)) ) ) ).

% ATP.lambda_565
tff(fact_7786_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_le(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_7787_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_zo(list(A),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uu,Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_567
tff(fact_7788_ATP_Olambda__568,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_aaa(fun(A,nat),fun(list(A),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),count_list(A,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_568
tff(fact_7789_ATP_Olambda__569,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_age(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_569
tff(fact_7790_ATP_Olambda__570,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_hw(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_570
tff(fact_7791_ATP_Olambda__571,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_alo(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_571
tff(fact_7792_ATP_Olambda__572,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_aw(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_572
tff(fact_7793_ATP_Olambda__573,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_rt(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_573
tff(fact_7794_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ru(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_574
tff(fact_7795_ATP_Olambda__575,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_lh(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uua = Uub )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_575
tff(fact_7796_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_lg(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub = Uua )
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_576
tff(fact_7797_ATP_Olambda__577,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_xs(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_577
tff(fact_7798_ATP_Olambda__578,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_hs(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_578
tff(fact_7799_ATP_Olambda__579,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_lu(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_579
tff(fact_7800_ATP_Olambda__580,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_hu(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_580
tff(fact_7801_ATP_Olambda__581,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_lx(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_581
tff(fact_7802_ATP_Olambda__582,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_ky(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_582
tff(fact_7803_ATP_Olambda__583,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_io(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),bit0(one2))),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_583
tff(fact_7804_ATP_Olambda__584,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_il(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_584
tff(fact_7805_ATP_Olambda__585,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_aqh(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_585
tff(fact_7806_ATP_Olambda__586,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_acf(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_586
tff(fact_7807_ATP_Olambda__587,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_rf(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_587
tff(fact_7808_ATP_Olambda__588,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ec(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_588
tff(fact_7809_ATP_Olambda__589,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ge(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_589
tff(fact_7810_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mu(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_590
tff(fact_7811_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ms(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_591
tff(fact_7812_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mt(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_592
tff(fact_7813_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_mr(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_593
tff(fact_7814_ATP_Olambda__594,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_ala(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_594
tff(fact_7815_ATP_Olambda__595,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_ul(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_595
tff(fact_7816_ATP_Olambda__596,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_as(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_596
tff(fact_7817_ATP_Olambda__597,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_sm(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_597
tff(fact_7818_ATP_Olambda__598,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_aka(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_598
tff(fact_7819_ATP_Olambda__599,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aic(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_599
tff(fact_7820_ATP_Olambda__600,axiom,
    ! [A: $tType,Uu: set(list(A)),Uua: A,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_anq(set(list(A)),fun(A,fun(list(A),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uu)) ) ).

% ATP.lambda_600
tff(fact_7821_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aab(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_601
tff(fact_7822_ATP_Olambda__602,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ib(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_602
tff(fact_7823_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,A),fun(A,bool),aTP_Lamp_aje(fun(A,B),fun(fun(B,A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_603
tff(fact_7824_ATP_Olambda__604,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_ajo(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub)))
        & pp(aa(B,bool,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub))) ) ) ).

% ATP.lambda_604
tff(fact_7825_ATP_Olambda__605,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hk(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_605
tff(fact_7826_ATP_Olambda__606,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_aqi(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_606
tff(fact_7827_ATP_Olambda__607,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_ya(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_607
tff(fact_7828_ATP_Olambda__608,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_xz(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_608
tff(fact_7829_ATP_Olambda__609,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_aag(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_609
tff(fact_7830_ATP_Olambda__610,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_ahg(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_610
tff(fact_7831_ATP_Olambda__611,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_fk(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_611
tff(fact_7832_ATP_Olambda__612,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_als(fun(A,fun(C,B)),fun(fun(A,option(C)),fun(A,option(B))),Uu),Uua),Uub) = aa(option(C),option(B),aa(fun(C,B),fun(option(C),option(B)),map_option(C,B),aa(A,fun(C,B),Uu,Uub)),aa(A,option(C),Uua,Uub)) ).

% ATP.lambda_612
tff(fact_7833_ATP_Olambda__613,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_fj(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_613
tff(fact_7834_ATP_Olambda__614,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_cx(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_614
tff(fact_7835_ATP_Olambda__615,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_di(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_615
tff(fact_7836_ATP_Olambda__616,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_xi(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_616
tff(fact_7837_ATP_Olambda__617,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_gn(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_617
tff(fact_7838_ATP_Olambda__618,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_dl(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_618
tff(fact_7839_ATP_Olambda__619,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_qt(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_619
tff(fact_7840_ATP_Olambda__620,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_aht(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_620
tff(fact_7841_ATP_Olambda__621,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_qn(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_621
tff(fact_7842_ATP_Olambda__622,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ahs(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_622
tff(fact_7843_ATP_Olambda__623,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_xl(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_623
tff(fact_7844_ATP_Olambda__624,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_oo(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_624
tff(fact_7845_ATP_Olambda__625,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ahz(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_625
tff(fact_7846_ATP_Olambda__626,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_os(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_626
tff(fact_7847_ATP_Olambda__627,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ahy(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_627
tff(fact_7848_ATP_Olambda__628,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_xk(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_628
tff(fact_7849_ATP_Olambda__629,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_dh(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_629
tff(fact_7850_ATP_Olambda__630,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_ym(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_630
tff(fact_7851_ATP_Olambda__631,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ajl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_631
tff(fact_7852_ATP_Olambda__632,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_qm(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_632
tff(fact_7853_ATP_Olambda__633,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_afn(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_633
tff(fact_7854_ATP_Olambda__634,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_th(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_634
tff(fact_7855_ATP_Olambda__635,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_arf(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_7856_ATP_Olambda__636,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_ig(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_636
tff(fact_7857_ATP_Olambda__637,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_jo(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_637
tff(fact_7858_ATP_Olambda__638,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_av(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_7859_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_aav(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_639
tff(fact_7860_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aaz(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_640
tff(fact_7861_ATP_Olambda__641,axiom,
    ! [D: $tType,B: $tType,Uu: fun(B,D),Uua: fun(B,D),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,D),fun(B,bool),aTP_Lamp_aoz(fun(B,D),fun(fun(B,D),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,D,Uu,Uub) = aa(B,D,Uua,Uub) ) ) ).

% ATP.lambda_641
tff(fact_7862_ATP_Olambda__642,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_afv(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_642
tff(fact_7863_ATP_Olambda__643,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_afz(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_643
tff(fact_7864_ATP_Olambda__644,axiom,
    ! [C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(A,C),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,C),fun(A,bool),aTP_Lamp_aoy(fun(A,C),fun(fun(A,C),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,C,Uu,Uub) = aa(A,C,Uua,Uub) ) ) ).

% ATP.lambda_644
tff(fact_7865_ATP_Olambda__645,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_agd(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_645
tff(fact_7866_ATP_Olambda__646,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_acr(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_646
tff(fact_7867_ATP_Olambda__647,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_lc(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_647
tff(fact_7868_ATP_Olambda__648,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_afc(fun(A,fun(A,bool)),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & ! [Y: A] :
            ( pp(aa(A,bool,Uua,Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Y)) ) ) ) ).

% ATP.lambda_648
tff(fact_7869_ATP_Olambda__649,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_fg(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_649
tff(fact_7870_ATP_Olambda__650,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_da(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_650
tff(fact_7871_ATP_Olambda__651,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hl(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_651
tff(fact_7872_ATP_Olambda__652,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_ny(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_652
tff(fact_7873_ATP_Olambda__653,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_agh(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_653
tff(fact_7874_ATP_Olambda__654,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_agf(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_654
tff(fact_7875_ATP_Olambda__655,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_ct(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_655
tff(fact_7876_ATP_Olambda__656,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_lo(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_656
tff(fact_7877_ATP_Olambda__657,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_dk(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_657
tff(fact_7878_ATP_Olambda__658,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_agg(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_658
tff(fact_7879_ATP_Olambda__659,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_de(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_659
tff(fact_7880_ATP_Olambda__660,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_nv(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_660
tff(fact_7881_ATP_Olambda__661,axiom,
    ! [E: $tType,F2: $tType,Uu: fun(E,set(F2)),Uua: set(F2),Uub: E] : aa(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_pn(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),Uu),Uua),Uub) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(E,set(F2),Uu,Uub)),Uua) ).

% ATP.lambda_661
tff(fact_7882_ATP_Olambda__662,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_fl(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_662
tff(fact_7883_ATP_Olambda__663,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_qv(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_663
tff(fact_7884_ATP_Olambda__664,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_qh(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_664
tff(fact_7885_ATP_Olambda__665,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_qs(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_665
tff(fact_7886_ATP_Olambda__666,axiom,
    ! [G2: $tType,H6: $tType,Uu: fun(G2,set(H6)),Uua: set(H6),Uub: G2] : aa(G2,set(H6),aa(set(H6),fun(G2,set(H6)),aTP_Lamp_oq(fun(G2,set(H6)),fun(set(H6),fun(G2,set(H6))),Uu),Uua),Uub) = aa(set(H6),set(H6),aa(set(H6),fun(set(H6),set(H6)),inf_inf(set(H6)),aa(G2,set(H6),Uu,Uub)),Uua) ).

% ATP.lambda_666
tff(fact_7887_ATP_Olambda__667,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_pf(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_667
tff(fact_7888_ATP_Olambda__668,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_oy(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_668
tff(fact_7889_ATP_Olambda__669,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_nq(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_669
tff(fact_7890_ATP_Olambda__670,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_re(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_670
tff(fact_7891_ATP_Olambda__671,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_a(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_671
tff(fact_7892_ATP_Olambda__672,axiom,
    ! [E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F2,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F2,B),aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_ama(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),Uu),Uua),Uub) = product_scomp(F2,C,D,B,aa(E,fun(F2,product_prod(C,D)),Uu,Uub),Uua) ).

% ATP.lambda_672
tff(fact_7893_ATP_Olambda__673,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(C,set(old_node(A,B))),Uua: set(old_node(A,B)),Uub: C] : aa(C,set(old_node(A,B)),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_and(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,aa(C,set(old_node(A,B)),Uu,Uub),Uua) ).

% ATP.lambda_673
tff(fact_7894_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_ajt(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).

% ATP.lambda_674
tff(fact_7895_ATP_Olambda__675,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_alc(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_675
tff(fact_7896_ATP_Olambda__676,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_sh(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_676
tff(fact_7897_ATP_Olambda__677,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_uy(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_677
tff(fact_7898_ATP_Olambda__678,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,pred(B)),Uua: fun(B,pred(A)),Uub: C] : aa(C,pred(A),aa(fun(B,pred(A)),fun(C,pred(A)),aTP_Lamp_ath(fun(C,pred(B)),fun(fun(B,pred(A)),fun(C,pred(A))),Uu),Uua),Uub) = bind3(B,A,aa(C,pred(B),Uu,Uub),Uua) ).

% ATP.lambda_678
tff(fact_7899_ATP_Olambda__679,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,option(B)),Uua: fun(B,option(A)),Uub: C] : aa(C,option(A),aa(fun(B,option(A)),fun(C,option(A)),aTP_Lamp_anr(fun(C,option(B)),fun(fun(B,option(A)),fun(C,option(A))),Uu),Uua),Uub) = aa(fun(B,option(A)),option(A),aa(option(B),fun(fun(B,option(A)),option(A)),bind2(B,A),aa(C,option(B),Uu,Uub)),Uua) ).

% ATP.lambda_679
tff(fact_7900_ATP_Olambda__680,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_aqz(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
       => pp(Uua) ) ) ).

% ATP.lambda_680
tff(fact_7901_ATP_Olambda__681,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_yr(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_681
tff(fact_7902_ATP_Olambda__682,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_vz(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_682
tff(fact_7903_ATP_Olambda__683,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,set(C)),Uua: fun(C,set(B)),Uub: A] : aa(A,set(B),aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_amj(fun(A,set(C)),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua),Uub) = bind4(C,B,aa(A,set(C),Uu,Uub),Uua) ).

% ATP.lambda_683
tff(fact_7904_ATP_Olambda__684,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_afy(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        | pp(Uua) ) ) ).

% ATP.lambda_684
tff(fact_7905_ATP_Olambda__685,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: bool,Uub: A] :
      ( pp(aa(A,bool,aa(bool,fun(A,bool),aTP_Lamp_arb(fun(A,bool),fun(bool,fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(Uua) ) ) ).

% ATP.lambda_685
tff(fact_7906_ATP_Olambda__686,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_ack(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_686
tff(fact_7907_ATP_Olambda__687,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_amt(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uu,Uub) = Uua ) ) ).

% ATP.lambda_687
tff(fact_7908_ATP_Olambda__688,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_aky(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_688
tff(fact_7909_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_yu(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_689
tff(fact_7910_ATP_Olambda__690,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(B,A),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_asb(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
    <=> ( aa(B,A,Uua,Uub) = Uu ) ) ).

% ATP.lambda_690
tff(fact_7911_ATP_Olambda__691,axiom,
    ! [A: $tType,Uu: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_ato(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_atn(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),Uu,Uub)) ).

% ATP.lambda_691
tff(fact_7912_ATP_Olambda__692,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_alz(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),Uua),Uub)),Uu)) ) ) ).

% ATP.lambda_692
tff(fact_7913_ATP_Olambda__693,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aif(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_693
tff(fact_7914_ATP_Olambda__694,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_je(A,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( ( Uub != Uu )
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_694
tff(fact_7915_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hc(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_695
tff(fact_7916_ATP_Olambda__696,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aqf(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( ~ pp(aa(A,bool,Uu,Uub))
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_696
tff(fact_7917_ATP_Olambda__697,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_ld(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_697
tff(fact_7918_ATP_Olambda__698,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_arc(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_698
tff(fact_7919_ATP_Olambda__699,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_aeh(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X3: A] :
                ( ( Uub = aa(A,A,Uu,X3) )
                & pp(aa(A,bool,Uua,X3)) )
            | ? [M10: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M10) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M10)
                & ! [X3: A] :
                    ( pp(aa(set(A),bool,member(A,X3),M10))
                   => pp(aa(A,bool,Uua,X3)) ) ) ) ) ) ).

% ATP.lambda_699
tff(fact_7920_ATP_Olambda__700,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_afa(set(A),fun(set(B),fun(fun(A,B),bool)),Uu),Uua),Uub))
    <=> ( ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uub,X3)),Uua)) )
        & ! [A7: A] :
            ( ~ pp(aa(set(A),bool,member(A,A7),Uu))
           => ( aa(A,B,Uub,A7) = undefined(B) ) ) ) ) ).

% ATP.lambda_700
tff(fact_7921_ATP_Olambda__701,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_vq(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) ) )
            | ? [X3: A,Y: A,Xs4: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y)) )
            | ? [X3: A,Y: A,Xs4: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) )
                & ~ 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(A),Y),X3))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs4),Ys4)) ) ) ) ) ).

% ATP.lambda_701
tff(fact_7922_ATP_Olambda__702,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_agr(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_702
tff(fact_7923_ATP_Olambda__703,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_aex(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)) )
        & ! [X3: A] :
            ( pp(aa(set(A),bool,member(A,X3),Uua))
           => ? [Xa2: A] :
                ( pp(aa(set(A),bool,member(A,Xa2),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),X3),Xa2)),Uu)) ) ) ) ) ).

% ATP.lambda_703
tff(fact_7924_ATP_Olambda__704,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: pred(A),Uub: seq(A)] : aa(seq(A),seq(A),aa(pred(A),fun(seq(A),seq(A)),aTP_Lamp_atl(fun(product_unit,seq(A)),fun(pred(A),fun(seq(A),seq(A))),Uu),Uua),Uub) = adjunct(A,seq2(A,Uu),join(A,Uua,Uub)) ).

% ATP.lambda_704
tff(fact_7925_ATP_Olambda__705,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_akd(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),insert2(A,Uua),bot_bot(set(A)))) ).

% ATP.lambda_705
tff(fact_7926_ATP_Olambda__706,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_gh(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_706
tff(fact_7927_ATP_Olambda__707,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_gg(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_707
tff(fact_7928_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fb(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_7929_ATP_Olambda__709,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ex(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_709
tff(fact_7930_ATP_Olambda__710,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_afh(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),insert2(A,Uua),bot_bot(set(A)))))) ) ).

% ATP.lambda_710
tff(fact_7931_ATP_Olambda__711,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_ha(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_711
tff(fact_7932_ATP_Olambda__712,axiom,
    ! [A: $tType,Uu: fun(product_unit,seq(A)),Uua: A,Uub: pred(A)] : aa(pred(A),seq(A),aa(A,fun(pred(A),seq(A)),aTP_Lamp_atk(fun(product_unit,seq(A)),fun(A,fun(pred(A),seq(A))),Uu),Uua),Uub) = insert(A,Uua,aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uub),seq2(A,Uu))) ).

% ATP.lambda_712
tff(fact_7933_ATP_Olambda__713,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_adh(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_713
tff(fact_7934_ATP_Olambda__714,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_adi(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_714
tff(fact_7935_ATP_Olambda__715,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_cv(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_715
tff(fact_7936_ATP_Olambda__716,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ez(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_716
tff(fact_7937_ATP_Olambda__717,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_aaj(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_717
tff(fact_7938_ATP_Olambda__718,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_aah(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_718
tff(fact_7939_ATP_Olambda__719,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_vo(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_719
tff(fact_7940_ATP_Olambda__720,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_abo(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_720
tff(fact_7941_ATP_Olambda__721,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_alp(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_721
tff(fact_7942_ATP_Olambda__722,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_ow(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_722
tff(fact_7943_ATP_Olambda__723,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_ov(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_723
tff(fact_7944_ATP_Olambda__724,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_agv(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_724
tff(fact_7945_ATP_Olambda__725,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_ahf(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_725
tff(fact_7946_ATP_Olambda__726,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_aca(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_726
tff(fact_7947_ATP_Olambda__727,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_ahh(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_727
tff(fact_7948_ATP_Olambda__728,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_alt(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),Uu),Uua),Uub) = aa(option(C),option(A),aa(fun(C,A),fun(option(C),option(A)),map_option(C,A),Uu),aa(B,option(C),Uua,Uub)) ).

% ATP.lambda_728
tff(fact_7949_ATP_Olambda__729,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_dd(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_729
tff(fact_7950_ATP_Olambda__730,axiom,
    ! [M8: $tType,N8: $tType,Uu: set(M8),Uua: fun(N8,set(M8)),Uub: N8] : aa(N8,set(M8),aa(fun(N8,set(M8)),fun(N8,set(M8)),aTP_Lamp_nw(set(M8),fun(fun(N8,set(M8)),fun(N8,set(M8))),Uu),Uua),Uub) = aa(set(M8),set(M8),aa(set(M8),fun(set(M8),set(M8)),minus_minus(set(M8)),Uu),aa(N8,set(M8),Uua,Uub)) ).

% ATP.lambda_730
tff(fact_7951_ATP_Olambda__731,axiom,
    ! [G2: $tType,H6: $tType,Uu: set(G2),Uua: fun(H6,set(G2)),Uub: H6] : aa(H6,set(G2),aa(fun(H6,set(G2)),fun(H6,set(G2)),aTP_Lamp_ns(set(G2),fun(fun(H6,set(G2)),fun(H6,set(G2))),Uu),Uua),Uub) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),Uu),aa(H6,set(G2),Uua,Uub)) ).

% ATP.lambda_731
tff(fact_7952_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_fq(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_7953_ATP_Olambda__733,axiom,
    ! [M8: $tType,N8: $tType,Uu: set(M8),Uua: fun(N8,set(M8)),Uub: N8] : aa(N8,set(M8),aa(fun(N8,set(M8)),fun(N8,set(M8)),aTP_Lamp_qu(set(M8),fun(fun(N8,set(M8)),fun(N8,set(M8))),Uu),Uua),Uub) = aa(set(M8),set(M8),aa(set(M8),fun(set(M8),set(M8)),sup_sup(set(M8)),Uu),aa(N8,set(M8),Uua,Uub)) ).

% ATP.lambda_733
tff(fact_7954_ATP_Olambda__734,axiom,
    ! [E: $tType,F2: $tType,Uu: set(E),Uua: fun(F2,set(E)),Uub: F2] : aa(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_qg(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),Uu),Uua),Uub) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),Uu),aa(F2,set(E),Uua,Uub)) ).

% ATP.lambda_734
tff(fact_7955_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_qw(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_7956_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_qq(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_7957_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_or(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_7958_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_np(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_7959_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_op(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_7960_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_pd(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_7961_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_oz(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_7962_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_lp(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_7963_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: nat,Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_anf(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,Uu),aa(C,set(old_node(A,B)),Uua,Uub)) ).

% ATP.lambda_743
tff(fact_7964_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(old_node(A,B)),Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_anc(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,Uu,aa(C,set(old_node(A,B)),Uua,Uub)) ).

% ATP.lambda_744
tff(fact_7965_ATP_Olambda__745,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_aju(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).

% ATP.lambda_745
tff(fact_7966_ATP_Olambda__746,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_aho(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_746
tff(fact_7967_ATP_Olambda__747,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_ahn(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_747
tff(fact_7968_ATP_Olambda__748,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_alb(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_748
tff(fact_7969_ATP_Olambda__749,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_zr(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_749
tff(fact_7970_ATP_Olambda__750,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_si(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_750
tff(fact_7971_ATP_Olambda__751,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ajf(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtermap(B,A,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_751
tff(fact_7972_ATP_Olambda__752,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_uj(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_752
tff(fact_7973_ATP_Olambda__753,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afw(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
       => pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_753
tff(fact_7974_ATP_Olambda__754,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_ys(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_754
tff(fact_7975_ATP_Olambda__755,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_pp(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_755
tff(fact_7976_ATP_Olambda__756,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_nu(I6,fun(fun(J4,set(I6)),fun(J4,set(I6))),Uu),Uua),Uub) = aa(set(I6),set(I6),insert2(I6,Uu),aa(J4,set(I6),Uua,Uub)) ).

% ATP.lambda_756
tff(fact_7977_ATP_Olambda__757,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_nt(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert2(B,Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_757
tff(fact_7978_ATP_Olambda__758,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_nk(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),insert2(A,Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_758
tff(fact_7979_ATP_Olambda__759,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_oj(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_759
tff(fact_7980_ATP_Olambda__760,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_wd(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_760
tff(fact_7981_ATP_Olambda__761,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_ahq(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub)) ) ).

% ATP.lambda_761
tff(fact_7982_ATP_Olambda__762,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afx(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
        | pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_762
tff(fact_7983_ATP_Olambda__763,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_aet(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        | pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_763
tff(fact_7984_ATP_Olambda__764,axiom,
    ! [A: $tType,Uu: bool,Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ara(bool,fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(Uu)
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_764
tff(fact_7985_ATP_Olambda__765,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: bool] :
      ( pp(aa(bool,bool,aa(A,fun(bool,bool),aTP_Lamp_adm(fun(A,bool),fun(A,fun(bool,bool)),Uu),Uua),Uub))
    <=> ( pp(Uub)
        & pp(aa(A,bool,Uu,Uua)) ) ) ).

% ATP.lambda_765
tff(fact_7986_ATP_Olambda__766,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_aax(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_766
tff(fact_7987_ATP_Olambda__767,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_wm(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_767
tff(fact_7988_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_unit,seq(B)),Uua: fun(B,pred(A)),Uub: product_unit] : aa(product_unit,seq(A),aa(fun(B,pred(A)),fun(product_unit,seq(A)),aTP_Lamp_atj(fun(product_unit,seq(B)),fun(fun(B,pred(A)),fun(product_unit,seq(A))),Uu),Uua),Uub) = apply(B,A,Uua,aa(product_unit,seq(B),Uu,product_Unity)) ).

% ATP.lambda_768
tff(fact_7989_ATP_Olambda__769,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fz(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_769
tff(fact_7990_ATP_Olambda__770,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_abc(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_770
tff(fact_7991_ATP_Olambda__771,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aqn(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,member(A,Uub),aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),Uu,Uua)))) ) ).

% ATP.lambda_771
tff(fact_7992_ATP_Olambda__772,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_aig(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uua),field2(B,Uu)) ).

% ATP.lambda_772
tff(fact_7993_ATP_Olambda__773,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_aio(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_773
tff(fact_7994_ATP_Olambda__774,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_xj(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_774
tff(fact_7995_ATP_Olambda__775,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_aie(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_775
tff(fact_7996_ATP_Olambda__776,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_yg(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_above(A,Uu,Uua) ).

% ATP.lambda_776
tff(fact_7997_ATP_Olambda__777,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(set(old_node(A,B))),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apq(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),Uu),Uua),Uub) = old_uprod(A,B,Uu,Uua) ).

% ATP.lambda_777
tff(fact_7998_ATP_Olambda__778,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(set(old_node(A,B))),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_apo(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B))))),Uu),Uua),Uub) = old_usum(A,B,Uu,Uua) ).

% ATP.lambda_778
tff(fact_7999_ATP_Olambda__779,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_aiy(set(B),fun(set(C),fun(A,set(sum_sum(B,C)))),Uu),Uua),Uub) = sum_Plus(B,C,Uu,Uua) ).

% ATP.lambda_779
tff(fact_8000_ATP_Olambda__780,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_abn(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_780
tff(fact_8001_ATP_Olambda__781,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_xf(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert2(B,Uu),Uua) ).

% ATP.lambda_781
tff(fact_8002_ATP_Olambda__782,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_xr(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_782
tff(fact_8003_ATP_Olambda__783,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_ajy(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_783
tff(fact_8004_ATP_Olambda__784,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aTP_Lamp_amu(fun(A,B),fun(fun(A,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub))) ) ).

% ATP.lambda_784
tff(fact_8005_ATP_Olambda__785,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),Uua: fun(A,B),Uub: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_aop(fun(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),aa(fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))),comp(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit)),aa(fun(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),map_option(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_map_prod(A,B,product_prod(heap_ext(product_unit),nat),product_prod(heap_ext(product_unit),nat),Uua,id(product_prod(heap_ext(product_unit),nat))))),Uub)) ).

% ATP.lambda_785
tff(fact_8006_ATP_Olambda__786,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_gk(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_786
tff(fact_8007_ATP_Olambda__787,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_gj(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_787
tff(fact_8008_ATP_Olambda__788,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_asg(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_788
tff(fact_8009_ATP_Olambda__789,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_agz(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_789
tff(fact_8010_ATP_Olambda__790,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_ou(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_790
tff(fact_8011_ATP_Olambda__791,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_fp(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_791
tff(fact_8012_ATP_Olambda__792,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_ds(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_792
tff(fact_8013_ATP_Olambda__793,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_arp(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_793
tff(fact_8014_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_akm(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).

% ATP.lambda_794
tff(fact_8015_ATP_Olambda__795,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_bh(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_795
tff(fact_8016_ATP_Olambda__796,axiom,
    ! [V6: $tType,U3: $tType,T: $tType,Uu: fun(U3,set(V6)),Uua: fun(T,U3),Uub: T] : aa(T,set(V6),aa(fun(T,U3),fun(T,set(V6)),aTP_Lamp_nz(fun(U3,set(V6)),fun(fun(T,U3),fun(T,set(V6))),Uu),Uua),Uub) = aa(U3,set(V6),Uu,aa(T,U3,Uua,Uub)) ).

% ATP.lambda_796
tff(fact_8017_ATP_Olambda__797,axiom,
    ! [D: $tType,F2: $tType,B: $tType,Uu: fun(F2,fun(D,bool)),Uua: fun(B,F2),Uub: B] : aa(B,fun(D,bool),aa(fun(B,F2),fun(B,fun(D,bool)),aTP_Lamp_alj(fun(F2,fun(D,bool)),fun(fun(B,F2),fun(B,fun(D,bool))),Uu),Uua),Uub) = aa(F2,fun(D,bool),Uu,aa(B,F2,Uua,Uub)) ).

% ATP.lambda_797
tff(fact_8018_ATP_Olambda__798,axiom,
    ! [C: $tType,E: $tType,A: $tType,Uu: fun(E,fun(C,bool)),Uua: fun(A,E),Uub: A] : aa(A,fun(C,bool),aa(fun(A,E),fun(A,fun(C,bool)),aTP_Lamp_ali(fun(E,fun(C,bool)),fun(fun(A,E),fun(A,fun(C,bool))),Uu),Uua),Uub) = aa(E,fun(C,bool),Uu,aa(A,E,Uua,Uub)) ).

% ATP.lambda_798
tff(fact_8019_ATP_Olambda__799,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_su(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_799
tff(fact_8020_ATP_Olambda__800,axiom,
    ! [D: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(B,C),Uub: B] : aa(B,D,aa(fun(B,C),fun(B,D),aTP_Lamp_aom(fun(C,D),fun(fun(B,C),fun(B,D)),Uu),Uua),Uub) = aa(C,D,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_800
tff(fact_8021_ATP_Olambda__801,axiom,
    ! [D: $tType,C: $tType,A: $tType,Uu: fun(C,D),Uua: fun(A,C),Uub: A] : aa(A,D,aa(fun(A,C),fun(A,D),aTP_Lamp_aol(fun(C,D),fun(fun(A,C),fun(A,D)),Uu),Uua),Uub) = aa(C,D,Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_801
tff(fact_8022_ATP_Olambda__802,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_ps(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_802
tff(fact_8023_ATP_Olambda__803,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),Uub: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C),aTP_Lamp_aov(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),B,Uua,Uub)) ).

% ATP.lambda_803
tff(fact_8024_ATP_Olambda__804,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(fun(A,bool),B),Uub: fun(A,bool)] : aa(fun(A,bool),C,aa(fun(fun(A,bool),B),fun(fun(A,bool),C),aTP_Lamp_atq(fun(B,C),fun(fun(fun(A,bool),B),fun(fun(A,bool),C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(fun(A,bool),B,Uua,Uub)) ).

% ATP.lambda_804
tff(fact_8025_ATP_Olambda__805,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_an(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).

% ATP.lambda_805
tff(fact_8026_ATP_Olambda__806,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_mf(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_806
tff(fact_8027_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ahv(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_807
tff(fact_8028_ATP_Olambda__808,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_sv(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_808
tff(fact_8029_ATP_Olambda__809,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_wh(fun(A,B),fun(fun(B,A),fun(B,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_809
tff(fact_8030_ATP_Olambda__810,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_afu(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_810
tff(fact_8031_ATP_Olambda__811,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_acn(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_811
tff(fact_8032_ATP_Olambda__812,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_fs(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_812
tff(fact_8033_ATP_Olambda__813,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_ch(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_813
tff(fact_8034_ATP_Olambda__814,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_cg(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_814
tff(fact_8035_ATP_Olambda__815,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_ud(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_815
tff(fact_8036_ATP_Olambda__816,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,C),Uua: fun(C,bool),Uub: D] :
      ( pp(aa(D,bool,aa(fun(C,bool),fun(D,bool),aTP_Lamp_aks(fun(D,C),fun(fun(C,bool),fun(D,bool)),Uu),Uua),Uub))
    <=> pp(aa(C,bool,Uua,aa(D,C,Uu,Uub))) ) ).

% ATP.lambda_816
tff(fact_8037_ATP_Olambda__817,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,bool),Uub: C] :
      ( pp(aa(C,bool,aa(fun(D,bool),fun(C,bool),aTP_Lamp_akt(fun(C,D),fun(fun(D,bool),fun(C,bool)),Uu),Uua),Uub))
    <=> pp(aa(D,bool,Uua,aa(C,D,Uu,Uub))) ) ).

% ATP.lambda_817
tff(fact_8038_ATP_Olambda__818,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_afg(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_818
tff(fact_8039_ATP_Olambda__819,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_aff(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_819
tff(fact_8040_ATP_Olambda__820,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_asc(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ).

% ATP.lambda_820
tff(fact_8041_ATP_Olambda__821,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_pz(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_821
tff(fact_8042_ATP_Olambda__822,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_wg(fun(A,B),fun(fun(B,A),fun(A,A)),Uu),Uua),Uub) = aa(B,A,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_822
tff(fact_8043_ATP_Olambda__823,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_yp(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_823
tff(fact_8044_ATP_Olambda__824,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_mv(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_824
tff(fact_8045_ATP_Olambda__825,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_ahm(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_825
tff(fact_8046_ATP_Olambda__826,axiom,
    ! [T: $tType,Uu: fun(code_natural,T),Uua: code_natural,Uub: T] : aa(T,T,aa(code_natural,fun(T,T),aTP_Lamp_app(fun(code_natural,T),fun(code_natural,fun(T,T)),Uu),Uua),Uub) = aa(code_natural,T,Uu,Uua) ).

% ATP.lambda_826
tff(fact_8047_ATP_Olambda__827,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_xu(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).

% ATP.lambda_827
tff(fact_8048_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ajq(fun(B,bool),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,Uu,Uua)) ) ).

% ATP.lambda_828
tff(fact_8049_ATP_Olambda__829,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_vx(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_829
tff(fact_8050_ATP_Olambda__830,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_ac(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_830
tff(fact_8051_ATP_Olambda__831,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_yi(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_yh(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).

% ATP.lambda_831
tff(fact_8052_ATP_Olambda__832,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_apl(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),aa(option(A),A,aa(fun(A,A),fun(option(A),A),aa(A,fun(fun(A,A),fun(option(A),A)),case_option(A,A),Uua),aa(A,fun(A,A),Uu,Uua)),Uub)) ).

% ATP.lambda_832
tff(fact_8053_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: product_prod(code_natural,code_natural),Uub: A] : aa(A,pred(B),aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_atr(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uua),Uub) = aa(product_prod(pred(B),product_prod(code_natural,code_natural)),pred(B),product_fst(pred(B),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),Uu,Uub),Uua)) ).

% ATP.lambda_833
tff(fact_8054_ATP_Olambda__834,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_ano(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_834
tff(fact_8055_ATP_Olambda__835,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_pk(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_pj(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_835
tff(fact_8056_ATP_Olambda__836,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_oi(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_of(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_836
tff(fact_8057_ATP_Olambda__837,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_og(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_of(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).

% ATP.lambda_837
tff(fact_8058_ATP_Olambda__838,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_oh(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_838
tff(fact_8059_ATP_Olambda__839,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_afk(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_839
tff(fact_8060_ATP_Olambda__840,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_oe(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_840
tff(fact_8061_ATP_Olambda__841,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_acs(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_841
tff(fact_8062_ATP_Olambda__842,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_act(B,fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_842
tff(fact_8063_ATP_Olambda__843,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),Uua: set(old_node(A,B)),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_ann(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua),Uub) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_anm(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uua),Uub))),Uu)) ).

% ATP.lambda_843
tff(fact_8064_ATP_Olambda__844,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_ats(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua),Uub) = aa(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),product_Pair(pred(B),product_prod(code_natural,code_natural)),bind3(A,B,Uua,aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_atr(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uub))) ).

% ATP.lambda_844
tff(fact_8065_ATP_Olambda__845,axiom,
    ! [S6: $tType,R6: $tType,Q6: $tType,Uu: fun(R6,set(S6)),Uua: fun(Q6,set(R6)),Uub: Q6] : aa(Q6,set(S6),aa(fun(Q6,set(R6)),fun(Q6,set(S6)),aTP_Lamp_oa(fun(R6,set(S6)),fun(fun(Q6,set(R6)),fun(Q6,set(S6))),Uu),Uua),Uub) = aa(set(set(S6)),set(S6),complete_Sup_Sup(set(S6)),aa(set(R6),set(set(S6)),image2(R6,set(S6),Uu),aa(Q6,set(R6),Uua,Uub))) ).

% ATP.lambda_845
tff(fact_8066_ATP_Olambda__846,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_ol(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_846
tff(fact_8067_ATP_Olambda__847,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_ok(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_847
tff(fact_8068_ATP_Olambda__848,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_ahr(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_848
tff(fact_8069_ATP_Olambda__849,axiom,
    ! [S6: $tType,R6: $tType,Q6: $tType,Uu: fun(R6,set(S6)),Uua: fun(Q6,set(R6)),Uub: Q6] : aa(Q6,set(S6),aa(fun(Q6,set(R6)),fun(Q6,set(S6)),aTP_Lamp_oc(fun(R6,set(S6)),fun(fun(Q6,set(R6)),fun(Q6,set(S6))),Uu),Uua),Uub) = aa(set(set(S6)),set(S6),complete_Inf_Inf(set(S6)),aa(set(R6),set(set(S6)),image2(R6,set(S6),Uu),aa(Q6,set(R6),Uua,Uub))) ).

% ATP.lambda_849
tff(fact_8070_ATP_Olambda__850,axiom,
    ! [B: $tType,D: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_ahx(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),image2(D,B,Uua),aa(C,set(D),Uu,Uub))) ) ).

% ATP.lambda_850
tff(fact_8071_ATP_Olambda__851,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_lq(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_851
tff(fact_8072_ATP_Olambda__852,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_xw(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_852
tff(fact_8073_ATP_Olambda__853,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_aad(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_853
tff(fact_8074_ATP_Olambda__854,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_iu(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_it(set(B),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub))) ).

% ATP.lambda_854
tff(fact_8075_ATP_Olambda__855,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_tt(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_855
tff(fact_8076_ATP_Olambda__856,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_yk(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A7: A] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
          & pp(aa(set(A),bool,member(A,A7),Uu)) ) ) ).

% ATP.lambda_856
tff(fact_8077_ATP_Olambda__857,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_vb(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_857
tff(fact_8078_ATP_Olambda__858,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_amk(set(A),fun(fun(A,set(B)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: set(B)] :
          ( pp(aa(set(set(B)),bool,member(set(B),X3),aa(set(A),set(set(B)),image2(A,set(B),Uua),Uu)))
          & pp(aa(set(B),bool,member(B,Uub),X3)) ) ) ).

% ATP.lambda_858
tff(fact_8079_ATP_Olambda__859,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_afj(set(set(product_prod(A,A))),fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua),Uub))
    <=> ? [R5: set(product_prod(A,A))] :
          ( ( Uub = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R5),Uua) )
          & pp(aa(set(set(product_prod(A,A))),bool,member(set(product_prod(A,A)),R5),Uu)) ) ) ).

% ATP.lambda_859
tff(fact_8080_ATP_Olambda__860,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_uw(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
              & pp(aa(set(A),bool,member(A,A7),Uu)) ) ) ) ).

% ATP.lambda_860
tff(fact_8081_ATP_Olambda__861,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ui(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B7) )
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_861
tff(fact_8082_ATP_Olambda__862,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_uu(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
              & pp(aa(set(A),bool,member(A,A7),Uu)) ) ) ) ).

% ATP.lambda_862
tff(fact_8083_ATP_Olambda__863,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_tz(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu,X3) )
          & pp(aa(set(B),bool,member(B,X3),Uua)) ) ) ).

% ATP.lambda_863
tff(fact_8084_ATP_Olambda__864,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_adq(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X3: C] :
          ( pp(aa(set(C),bool,member(C,X3),aa(B,set(C),Uu,Uub)))
         => pp(aa(C,bool,Uua,X3)) ) ) ).

% ATP.lambda_864
tff(fact_8085_ATP_Olambda__865,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_tx(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( ( Uub = aa(B,set(A),Uu,X3) )
          & pp(aa(B,bool,Uua,X3)) ) ) ).

% ATP.lambda_865
tff(fact_8086_ATP_Olambda__866,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_ua(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( ( Uub = aa(B,A,Uu,X3) )
          & pp(aa(B,bool,Uua,X3)) ) ) ).

% ATP.lambda_866
tff(fact_8087_ATP_Olambda__867,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_ub(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( ( Uub = aa(A,B,Uua,X3) )
          & pp(aa(A,bool,Uu,X3)) ) ) ).

% ATP.lambda_867
tff(fact_8088_ATP_Olambda__868,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_aqc(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(C,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),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),X3)) ) ) ).

% ATP.lambda_868
tff(fact_8089_ATP_Olambda__869,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_aqd(fun(A,fun(B,bool)),fun(fun(A,C),fun(fun(A,C),bool)),Uu),Uua),Uub))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),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,X3) = aa(A,C,Uub,X3) ) ) ) ).

% ATP.lambda_869
tff(fact_8090_ATP_Olambda__870,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_ado(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),Uu))
         => pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_870
tff(fact_8091_ATP_Olambda__871,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_agb(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ! [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uu))
         => pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_871
tff(fact_8092_ATP_Olambda__872,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_aeq(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),Uu))
          & pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_872
tff(fact_8093_ATP_Olambda__873,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_aek(set(A),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uu))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),X3)) ) ) ).

% ATP.lambda_873
tff(fact_8094_ATP_Olambda__874,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_ael(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),Uua))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uu,X3),Uub)) ) ) ).

% ATP.lambda_874
tff(fact_8095_ATP_Olambda__875,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_akg(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X3: product_prod(A,A)] :
          ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X3),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_akf(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X3)) ) ) ).

% ATP.lambda_875
tff(fact_8096_ATP_Olambda__876,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_aer(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),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),X3),Uub)),Uu)) ) ) ).

% ATP.lambda_876
tff(fact_8097_ATP_Olambda__877,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_aes(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uua))
          & ( aa(A,B,Uu,X3) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_877
tff(fact_8098_ATP_Olambda__878,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_aew(fun(A,option(B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uua))
          & ( aa(A,option(B),Uu,X3) = aa(B,option(B),some(B),Uub) ) ) ) ).

% ATP.lambda_878
tff(fact_8099_ATP_Olambda__879,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_adn(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ! [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),Uua))
         => pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X3))) ) ) ).

% ATP.lambda_879
tff(fact_8100_ATP_Olambda__880,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_aep(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( pp(aa(set(B),bool,member(B,X3),Uua))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X3))) ) ) ).

% ATP.lambda_880
tff(fact_8101_ATP_Olambda__881,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_aen(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),Uua))
          & ( Uub = aa(A,B,Uu,X3) ) ) ) ).

% ATP.lambda_881
tff(fact_8102_ATP_Olambda__882,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_ahk(fun(A,B),fun(filter(B),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q7: fun(B,bool)] :
          ( eventually(B,Q7,Uua)
          & ! [X3: A] :
              ( pp(aa(B,bool,Q7,aa(A,B,Uu,X3)))
             => pp(aa(A,bool,Uub,X3)) ) ) ) ).

% ATP.lambda_882
tff(fact_8103_ATP_Olambda__883,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_tv(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y: A] :
          ( pp(aa(A,bool,Uu,Y))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y)) ) ) ).

% ATP.lambda_883
tff(fact_8104_ATP_Olambda__884,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_ty(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X3: B] :
          ( pp(aa(B,bool,Uua,X3))
          & pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X3))) ) ) ).

% ATP.lambda_884
tff(fact_8105_ATP_Olambda__885,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_we(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N6: 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)),N6),Uu),Uua) ) ).

% ATP.lambda_885
tff(fact_8106_ATP_Olambda__886,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_ajx(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A18: B,A26: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A18)),aa(B,A,Uua,A26)) )
          & 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),A18),A26)),Uu)) ) ) ).

% ATP.lambda_886
tff(fact_8107_ATP_Olambda__887,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_aip(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),Uu),Uua),Uub))
    <=> ? [A7: A,B7: A] :
          ( ( Uub = aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,Uua,A7)),aa(A,A2,Uua,B7)) )
          & pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B7)),Uu)) ) ) ).

% ATP.lambda_887
tff(fact_8108_ATP_Olambda__888,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_vh(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A7: A,V7: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V7)) )
          | ? [U4: list(A),Aa3: A,B7: A,Va: 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),B7)),Uu))
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W3)) ) ) ) ) ).

% ATP.lambda_888
tff(fact_8109_ATP_Olambda__889,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_ux(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B7) )
              & pp(aa(set(A),bool,member(A,A7),Uu))
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_889
tff(fact_8110_ATP_Olambda__890,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_uv(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B7) )
              & pp(aa(set(A),bool,member(A,A7),Uu))
              & pp(aa(set(A),bool,member(A,B7),Uua)) ) ) ) ).

% ATP.lambda_890
tff(fact_8111_ATP_Olambda__891,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_vj(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X3: A,Xs4: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
          & pp(aa(set(A),bool,member(A,X3),Uu))
          & pp(aa(set(list(A)),bool,member(list(A),Xs4),Uua)) ) ) ).

% ATP.lambda_891
tff(fact_8112_ATP_Olambda__892,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)),Uua: set(old_node(A,B)),Uub: C] :
      ( pp(aa(C,bool,aa(set(old_node(A,B)),fun(C,bool),aTP_Lamp_ane(fun(set(old_node(A,B)),fun(set(old_node(A,B)),C)),fun(set(old_node(A,B)),fun(C,bool)),Uu),Uua),Uub))
    <=> ? [X3: set(old_node(A,B)),Y: set(old_node(A,B))] :
          ( ( Uua = old_Scons(A,B,X3,Y) )
          & ( Uub = aa(set(old_node(A,B)),C,aa(set(old_node(A,B)),fun(set(old_node(A,B)),C),Uu,X3),Y) ) ) ) ).

% ATP.lambda_892
tff(fact_8113_ATP_Olambda__893,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_agu(filter(A),fun(filter(A),fun(fun(A,bool),bool)),Uu),Uua),Uub))
    <=> ? [Q7: fun(A,bool),R11: fun(A,bool)] :
          ( eventually(A,Q7,Uu)
          & eventually(A,R11,Uua)
          & ! [X3: A] :
              ( ( pp(aa(A,bool,Q7,X3))
                & pp(aa(A,bool,R11,X3)) )
             => pp(aa(A,bool,Uub,X3)) ) ) ) ).

% ATP.lambda_893
tff(fact_8114_ATP_Olambda__894,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_vk(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us3: list(A),Z5: A,Z9: 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),Z5),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),Z5),Z9)),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),Z9),Vs3)) ) ) ) ).

% ATP.lambda_894
tff(fact_8115_ATP_Olambda__895,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_asp(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_aso(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc)) ).

% ATP.lambda_895
tff(fact_8116_ATP_Olambda__896,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_arr(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uuc)) ).

% ATP.lambda_896
tff(fact_8117_ATP_Olambda__897,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_ash(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),Uub,Uuc)),aa(nat,set(nat),set_ord_lessThan(nat),Uuc)) ).

% ATP.lambda_897
tff(fact_8118_ATP_Olambda__898,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_bf(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_898
tff(fact_8119_ATP_Olambda__899,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_sf(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = 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))),aa(bool,fun(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B)))),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)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc)),Uuc) ).

% ATP.lambda_899
tff(fact_8120_ATP_Olambda__900,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_wu(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uu),Uua))),aa(B,A,the_inv_into(A,B,Uua,Uu),Uuc)),Uub) ).

% ATP.lambda_900
tff(fact_8121_ATP_Olambda__901,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_wr(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),image2(A,B,Uua),Uu))),aa(B,A,the_inv_into(A,B,Uu,Uua),Uuc)),Uub) ).

% ATP.lambda_901
tff(fact_8122_ATP_Olambda__902,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_sg(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),aa(bool,fun(set(B),fun(set(B),set(B))),if(set(B)),aa(set(A),bool,member(A,Uua),Uu)),aa(set(B),set(B),insert2(B,Uub),Uuc)),Uuc) ).

% ATP.lambda_902
tff(fact_8123_ATP_Olambda__903,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_ep(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu)),aa(nat,A,Uua,Uuc)),aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_903
tff(fact_8124_ATP_Olambda__904,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_gb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu)),aa(nat,A,Uua,Uuc)),aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_904
tff(fact_8125_ATP_Olambda__905,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_gc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_905
tff(fact_8126_ATP_Olambda__906,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_eq(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_906
tff(fact_8127_ATP_Olambda__907,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_arj(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_907
tff(fact_8128_ATP_Olambda__908,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_wc(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),aa(bool,fun(B,fun(B,B)),if(B),aa(set(A),bool,member(A,Uuc),Uua)),aa(A,B,Uu,Uuc)),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_908
tff(fact_8129_ATP_Olambda__909,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_ls(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_909
tff(fact_8130_ATP_Olambda__910,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_lr(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_910
tff(fact_8131_ATP_Olambda__911,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),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_911
tff(fact_8132_ATP_Olambda__912,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(B,A),Uub: set(B),Uuc: B] : aa(B,A,aa(set(B),fun(B,A),aa(fun(B,A),fun(set(B),fun(B,A)),aTP_Lamp_arm(A,fun(fun(B,A),fun(set(B),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(set(B),bool,member(B,Uuc),Uub)),aa(B,A,Uua,Uuc)),Uu) ).

% ATP.lambda_912
tff(fact_8133_ATP_Olambda__913,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arw(A,fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uuc)),aa(B,A,Uub,Uuc)),Uu) ).

% ATP.lambda_913
tff(fact_8134_ATP_Olambda__914,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_jb(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu)),aa(B,A,Uua,Uuc)),Uub) ) ).

% ATP.lambda_914
tff(fact_8135_ATP_Olambda__915,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_arv(A,fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uua)),aa(B,A,Uub,Uuc)),Uu) ).

% ATP.lambda_915
tff(fact_8136_ATP_Olambda__916,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_yq(B,fun(B,fun(set(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),aa(bool,fun(B,fun(B,B)),if(B),aa(set(A),bool,member(A,Uuc),Uub)),Uu),Uua) ).

% ATP.lambda_916
tff(fact_8137_ATP_Olambda__917,axiom,
    ! [A: $tType,Uu: fun(code_natural,A),Uua: code_natural,Uub: code_natural,Uuc: product_unit] : aa(product_unit,seq(A),aa(code_natural,fun(product_unit,seq(A)),aa(code_natural,fun(code_natural,fun(product_unit,seq(A))),aTP_Lamp_asx(fun(code_natural,A),fun(code_natural,fun(code_natural,fun(product_unit,seq(A)))),Uu),Uua),Uub),Uuc) = aa(seq(A),seq(A),aa(seq(A),fun(seq(A),seq(A)),aa(bool,fun(seq(A),fun(seq(A),seq(A))),if(seq(A)),aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),Uub),Uua)),empty(A)),insert(A,aa(code_natural,A,Uu,Uua),iterate_upto(A,Uu,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),plus_plus(code_natural),Uua),one_one(code_natural)),Uub))) ).

% ATP.lambda_917
tff(fact_8138_ATP_Olambda__918,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_yz(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),aa(bool,fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A)))),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_918
tff(fact_8139_ATP_Olambda__919,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_lk(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uu,Uuc)),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_919
tff(fact_8140_ATP_Olambda__920,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_lj(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uu,Uuc)),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_920
tff(fact_8141_ATP_Olambda__921,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_qe(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uu,Uuc)),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_921
tff(fact_8142_ATP_Olambda__922,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_adc(fun(A,bool),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = aa(option(B),option(B),aa(option(B),fun(option(B),option(B)),aa(bool,fun(option(B),fun(option(B),option(B))),if(option(B)),aa(A,bool,Uu,Uuc)),aa(A,option(B),Uua,Uuc)),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_922
tff(fact_8143_ATP_Olambda__923,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_uk(fun(A,B),fun(fun(A,bool),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),aa(bool,fun(B,fun(B,B)),if(B),aa(A,bool,Uua,Uuc)),aa(A,B,Uu,Uuc)),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_923
tff(fact_8144_ATP_Olambda__924,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(B,A),Uub: fun(B,bool),Uuc: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aa(fun(B,A),fun(fun(B,bool),fun(B,A)),aTP_Lamp_arx(A,fun(fun(B,A),fun(fun(B,bool),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(B,bool,Uub,Uuc)),aa(B,A,Uua,Uuc)),Uu) ).

% ATP.lambda_924
tff(fact_8145_ATP_Olambda__925,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_sk(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_sj(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_925
tff(fact_8146_ATP_Olambda__926,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_sb(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_sa(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_926
tff(fact_8147_ATP_Olambda__927,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_tg(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_927
tff(fact_8148_ATP_Olambda__928,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: fun(fun(A,B),fun(A,B)),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aTP_Lamp_ans(set(product_prod(A,A)),fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(A,B,aa(fun(A,B),fun(A,B),Uua,cut(A,B,Uub,Uu,Uuc)),Uuc) ).

% ATP.lambda_928
tff(fact_8149_ATP_Olambda__929,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_aal(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_929
tff(fact_8150_ATP_Olambda__930,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_aae(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_930
tff(fact_8151_ATP_Olambda__931,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_zv(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_931
tff(fact_8152_ATP_Olambda__932,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(A,fun(A,A)),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),Uu,aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_932
tff(fact_8153_ATP_Olambda__933,axiom,
    ! [B: $tType,F2: $tType,D: $tType,Uu: fun(B,fun(F2,bool)),Uua: fun(D,F2),Uub: B,Uuc: D] :
      ( pp(aa(D,bool,aa(B,fun(D,bool),aa(fun(D,F2),fun(B,fun(D,bool)),aTP_Lamp_alh(fun(B,fun(F2,bool)),fun(fun(D,F2),fun(B,fun(D,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(F2,bool,aa(B,fun(F2,bool),Uu,Uub),aa(D,F2,Uua,Uuc))) ) ).

% ATP.lambda_933
tff(fact_8154_ATP_Olambda__934,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_yo(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_934
tff(fact_8155_ATP_Olambda__935,axiom,
    ! [A: $tType,E: $tType,C: $tType,Uu: fun(A,fun(E,bool)),Uua: fun(C,E),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,E),fun(A,fun(C,bool)),aTP_Lamp_alg(fun(A,fun(E,bool)),fun(fun(C,E),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(E,bool,aa(A,fun(E,bool),Uu,Uub),aa(C,E,Uua,Uuc))) ) ).

% ATP.lambda_935
tff(fact_8156_ATP_Olambda__936,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_st(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_936
tff(fact_8157_ATP_Olambda__937,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_aco(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_937
tff(fact_8158_ATP_Olambda__938,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_adb(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_938
tff(fact_8159_ATP_Olambda__939,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_dg(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_df(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_939
tff(fact_8160_ATP_Olambda__940,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_agq(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_agp(set(A),fun(fun(A,B),fun(fun(B,bool),fun(A,bool))),Uu),Uua),Uuc),Uub) ) ).

% ATP.lambda_940
tff(fact_8161_ATP_Olambda__941,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,bool)),Uub: set(A),Uuc: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aa(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool)),aTP_Lamp_akr(set(A),fun(fun(A,fun(A,bool)),fun(set(A),fun(set(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,pred_chain(A,Uu,Uua),Uuc))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uub),Uuc)) ) ) ).

% ATP.lambda_941
tff(fact_8162_ATP_Olambda__942,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_pi(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_942
tff(fact_8163_ATP_Olambda__943,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_hr(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_fh(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_ho(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_943
tff(fact_8164_ATP_Olambda__944,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_hp(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_db(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_ho(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_944
tff(fact_8165_ATP_Olambda__945,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_zg(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_wx(set(A),fun(A,set(A)),Uua))))
        & pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).

% ATP.lambda_945
tff(fact_8166_ATP_Olambda__946,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_jk(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_946
tff(fact_8167_ATP_Olambda__947,axiom,
    ! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B)),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_anm(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uu),Uua),Uub),Uuc) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert2(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,Uu,Uub)),old_Scons(A,B,Uua,Uuc))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).

% ATP.lambda_947
tff(fact_8168_ATP_Olambda__948,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_sq(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_948
tff(fact_8169_ATP_Olambda__949,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_akn(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_949
tff(fact_8170_ATP_Olambda__950,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_wp(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_950
tff(fact_8171_ATP_Olambda__951,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_agx(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_951
tff(fact_8172_ATP_Olambda__952,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_en(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_952
tff(fact_8173_ATP_Olambda__953,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_ej(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_953
tff(fact_8174_ATP_Olambda__954,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_em(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_954
tff(fact_8175_ATP_Olambda__955,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_vi(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),X3: A,Y: A,Xs5: list(A),Ys5: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs5)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys5)) )
            & pp(aa(set(product_prod(A,A)),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)),Uu)) ) ) ) ).

% ATP.lambda_955
tff(fact_8176_ATP_Olambda__956,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_gy(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_956
tff(fact_8177_ATP_Olambda__957,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_kq(bool,fun(A,fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( pp(Uu)
         => ( Uuc = Uua ) )
        & ( ~ pp(Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_957
tff(fact_8178_ATP_Olambda__958,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_md(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_958
tff(fact_8179_ATP_Olambda__959,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_hm(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_959
tff(fact_8180_ATP_Olambda__960,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_it(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_960
tff(fact_8181_ATP_Olambda__961,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_ho(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_961
tff(fact_8182_ATP_Olambda__962,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_in(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_962
tff(fact_8183_ATP_Olambda__963,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_aki(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
        & ( aa(A,B,bNF_Greatest_toCard(A,B,Uu,Uua),Uuc) = Uub ) ) ) ).

% ATP.lambda_963
tff(fact_8184_ATP_Olambda__964,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_mi(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_964
tff(fact_8185_ATP_Olambda__965,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),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_ajc(fun(A,B),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,member(A,Uuc),Uua))
        & ( aa(A,B,Uu,Uub) = aa(A,B,Uu,Uuc) ) ) ) ).

% ATP.lambda_965
tff(fact_8186_ATP_Olambda__966,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_ml(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_966
tff(fact_8187_ATP_Olambda__967,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_mw(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_967
tff(fact_8188_ATP_Olambda__968,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_yv(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_968
tff(fact_8189_ATP_Olambda__969,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_vv(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_969
tff(fact_8190_ATP_Olambda__970,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_gz(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_970
tff(fact_8191_ATP_Olambda__971,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_eg(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_971
tff(fact_8192_ATP_Olambda__972,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_jr(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_972
tff(fact_8193_ATP_Olambda__973,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_jz(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_973
tff(fact_8194_ATP_Olambda__974,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_jt(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_974
tff(fact_8195_ATP_Olambda__975,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_jx(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_975
tff(fact_8196_ATP_Olambda__976,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_kb(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_976
tff(fact_8197_ATP_Olambda__977,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_kh(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_977
tff(fact_8198_ATP_Olambda__978,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_yw(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_978
tff(fact_8199_ATP_Olambda__979,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_yx(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_979
tff(fact_8200_ATP_Olambda__980,axiom,
    ! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,bool),Uuc: fun(B,bool)] :
      ( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_ajn(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),Uu),Uua),Uub),Uuc))
    <=> ( eventually(A,Uub,Uu)
        & eventually(B,Uuc,Uua) ) ) ).

% ATP.lambda_980
tff(fact_8201_ATP_Olambda__981,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_ko(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_981
tff(fact_8202_ATP_Olambda__982,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_tj(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_982
tff(fact_8203_ATP_Olambda__983,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_me(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_983
tff(fact_8204_ATP_Olambda__984,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_po(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_984
tff(fact_8205_ATP_Olambda__985,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_hv(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_985
tff(fact_8206_ATP_Olambda__986,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_ht(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_986
tff(fact_8207_ATP_Olambda__987,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: set(B),Uub: fun(B,A),Uuc: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(set(B),fun(fun(B,A),fun(B,bool)),aTP_Lamp_ark(A,fun(set(B),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uua))
        & ( aa(B,A,Uub,Uuc) != Uu ) ) ) ).

% ATP.lambda_987
tff(fact_8208_ATP_Olambda__988,axiom,
    ! [B: $tType,A: $tType,Uu: 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_asi(A,fun(fun(B,A),fun(set(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,member(B,Uuc),Uub))
        & ( aa(B,A,Uua,Uuc) != Uu ) ) ) ).

% ATP.lambda_988
tff(fact_8209_ATP_Olambda__989,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_td(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_989
tff(fact_8210_ATP_Olambda__990,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_agj(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_990
tff(fact_8211_ATP_Olambda__991,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_agp(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_991
tff(fact_8212_ATP_Olambda__992,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_xg(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_992
tff(fact_8213_ATP_Olambda__993,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_xv(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_xu(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).

% ATP.lambda_993
tff(fact_8214_ATP_Olambda__994,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_yj(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_994
tff(fact_8215_ATP_Olambda__995,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_aby(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_995
tff(fact_8216_ATP_Olambda__996,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_amc(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_996
tff(fact_8217_ATP_Olambda__997,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_df(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_997
tff(fact_8218_ATP_Olambda__998,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_qx(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_998
tff(fact_8219_ATP_Olambda__999,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_qo(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_999
tff(fact_8220_ATP_Olambda__1000,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_om(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_1000
tff(fact_8221_ATP_Olambda__1001,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_pb(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_1001
tff(fact_8222_ATP_Olambda__1002,axiom,
    ! [B: $tType,A: $tType,Uu: fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua: fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B))))),aTP_Lamp_apt(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))))),Uu),Uua),Uub),Uuc) = old_uprod(A,B,aa(set(old_node(A,B)),set(set(old_node(A,B))),Uu,Uub),aa(set(old_node(A,B)),set(set(old_node(A,B))),Uua,Uuc)) ).

% ATP.lambda_1002
tff(fact_8223_ATP_Olambda__1003,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_ajr(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).

% ATP.lambda_1003
tff(fact_8224_ATP_Olambda__1004,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_xp(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_1004
tff(fact_8225_ATP_Olambda__1005,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_zk(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_1005
tff(fact_8226_ATP_Olambda__1006,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_wy(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_1006
tff(fact_8227_ATP_Olambda__1007,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_aba(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_1007
tff(fact_8228_ATP_Olambda__1008,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_hq(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_hm(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1008
tff(fact_8229_ATP_Olambda__1009,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_hn(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_hm(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1009
tff(fact_8230_ATP_Olambda__1010,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_afo(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_1010
tff(fact_8231_ATP_Olambda__1011,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_agl(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_1011
tff(fact_8232_ATP_Olambda__1012,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_mx(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_mw(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_1012
tff(fact_8233_ATP_Olambda__1013,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(set(old_node(A,B)),C),Uua: fun(set(old_node(A,B)),C),Uub: set(old_node(A,B)),Uuc: C] :
      ( pp(aa(C,bool,aa(set(old_node(A,B)),fun(C,bool),aa(fun(set(old_node(A,B)),C),fun(set(old_node(A,B)),fun(C,bool)),aTP_Lamp_anj(fun(set(old_node(A,B)),C),fun(fun(set(old_node(A,B)),C),fun(set(old_node(A,B)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [X3: set(old_node(A,B))] :
            ( ( Uub = old_In0(A,B,X3) )
            & ( Uuc = aa(set(old_node(A,B)),C,Uu,X3) ) )
        | ? [Y: set(old_node(A,B))] :
            ( ( Uub = old_In1(A,B,Y) )
            & ( Uuc = aa(set(old_node(A,B)),C,Uua,Y) ) ) ) ) ).

% ATP.lambda_1013
tff(fact_8234_ATP_Olambda__1014,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_ug(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A7: A] :
            ( ( Uub = A7 )
            & ( Uuc = A7 ) )
        | ? [A7: A,B7: A,C4: A] :
            ( ( Uub = A7 )
            & ( Uuc = C4 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A7),B7))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B7),C4)) ) ) ) ).

% ATP.lambda_1014
tff(fact_8235_ATP_Olambda__1015,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_uf(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A7: A,B7: A] :
            ( ( Uub = A7 )
            & ( Uuc = B7 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,A7),B7)) )
        | ? [A7: A,B7: A,C4: A] :
            ( ( Uub = A7 )
            & ( Uuc = C4 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A7),B7))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B7),C4)) ) ) ) ).

% ATP.lambda_1015
tff(fact_8236_ATP_Olambda__1016,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_vp(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [Y: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) ) )
        | ? [X3: A,Y: A,Xs4: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,X3),Y)) )
        | ? [X3: A,Y: A,Xs4: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X3),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y),X3))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs4),Ys4)) ) ) ) ).

% ATP.lambda_1016
tff(fact_8237_ATP_Olambda__1017,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_xx(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_1017
tff(fact_8238_ATP_Olambda__1018,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_bx(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_1018
tff(fact_8239_ATP_Olambda__1019,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_yn(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),insert2(A,Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_1019
tff(fact_8240_ATP_Olambda__1020,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_bp(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_1020
tff(fact_8241_ATP_Olambda__1021,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_br(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_1021
tff(fact_8242_ATP_Olambda__1022,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_mn(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_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1022
tff(fact_8243_ATP_Olambda__1023,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_mp(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_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1023
tff(fact_8244_ATP_Olambda__1024,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_mm(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_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1024
tff(fact_8245_ATP_Olambda__1025,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_mo(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_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1025
tff(fact_8246_ATP_Olambda__1026,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_ze(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_1026
tff(fact_8247_ATP_Olambda__1027,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_bg(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_1027
tff(fact_8248_ATP_Olambda__1028,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(pred(A),fun(seq(A),B)),Uub: pred(A),Uuc: seq(A)] : aa(seq(A),C,aa(pred(A),fun(seq(A),C),aa(fun(pred(A),fun(seq(A),B)),fun(pred(A),fun(seq(A),C)),aTP_Lamp_atd(fun(B,C),fun(fun(pred(A),fun(seq(A),B)),fun(pred(A),fun(seq(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(seq(A),B,aa(pred(A),fun(seq(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1028
tff(fact_8249_ATP_Olambda__1029,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(pred(A),B)),Uub: A,Uuc: pred(A)] : aa(pred(A),C,aa(A,fun(pred(A),C),aa(fun(A,fun(pred(A),B)),fun(A,fun(pred(A),C)),aTP_Lamp_atc(fun(B,C),fun(fun(A,fun(pred(A),B)),fun(A,fun(pred(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(pred(A),B,aa(A,fun(pred(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1029
tff(fact_8250_ATP_Olambda__1030,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_yy(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_1030
tff(fact_8251_ATP_Olambda__1031,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_gw(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_1031
tff(fact_8252_ATP_Olambda__1032,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_gu(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_1032
tff(fact_8253_ATP_Olambda__1033,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_aro(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_1033
tff(fact_8254_ATP_Olambda__1034,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ft(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1034
tff(fact_8255_ATP_Olambda__1035,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_dt(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1035
tff(fact_8256_ATP_Olambda__1036,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_arq(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_1036
tff(fact_8257_ATP_Olambda__1037,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_zu(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_1037
tff(fact_8258_ATP_Olambda__1038,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_zt(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_1038
tff(fact_8259_ATP_Olambda__1039,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(D,C),Uub: fun(B,D),Uuc: A] : aa(A,C,aa(fun(B,D),fun(A,C),aa(fun(D,C),fun(fun(B,D),fun(A,C)),aTP_Lamp_amn(fun(A,B),fun(fun(D,C),fun(fun(B,D),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(D,C,Uua,aa(B,D,Uub,aa(A,B,Uu,Uuc))) ).

% ATP.lambda_1039
tff(fact_8260_ATP_Olambda__1040,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B] : aa(B,fun(product_prod(C,D),bool),aa(A,fun(B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_all(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_alk(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1040
tff(fact_8261_ATP_Olambda__1041,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_kv(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_ku(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_1041
tff(fact_8262_ATP_Olambda__1042,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_ks(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_kr(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_1042
tff(fact_8263_ATP_Olambda__1043,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_pc(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_pb(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1043
tff(fact_8264_ATP_Olambda__1044,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_on(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_om(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1044
tff(fact_8265_ATP_Olambda__1045,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_ajs(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_ajr(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_1045
tff(fact_8266_ATP_Olambda__1046,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_qy(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_qx(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_1046
tff(fact_8267_ATP_Olambda__1047,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_qp(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_qo(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_1047
tff(fact_8268_ATP_Olambda__1048,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_jm(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_1048
tff(fact_8269_ATP_Olambda__1049,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_kj(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_1049
tff(fact_8270_ATP_Olambda__1050,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_acb(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),aa(fun(bool,bool),fun(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_1050
tff(fact_8271_ATP_Olambda__1051,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_vl(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_1051
tff(fact_8272_ATP_Olambda__1052,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_kd(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_1052
tff(fact_8273_ATP_Olambda__1053,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_kf(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_1053
tff(fact_8274_ATP_Olambda__1054,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_afm(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),insert2(B,Uub),bot_bot(set(B)))))) ).

% ATP.lambda_1054
tff(fact_8275_ATP_Olambda__1055,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_ph(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_1055
tff(fact_8276_ATP_Olambda__1056,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_aey(set(A),fun(set(B),fun(B,fun(fun(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X3: A] :
          ( ( pp(aa(set(A),bool,member(A,X3),Uu))
           => pp(aa(set(B),bool,member(B,aa(A,B,Uuc,X3)),Uua)) )
          & ( ~ pp(aa(set(A),bool,member(A,X3),Uu))
           => ( aa(A,B,Uuc,X3) = Uub ) ) ) ) ).

% ATP.lambda_1056
tff(fact_8277_ATP_Olambda__1057,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_up(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y: 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),Y)),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),Y),Uuc)),Uua)) ) ) ).

% ATP.lambda_1057
tff(fact_8278_ATP_Olambda__1058,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_va(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A7: C] :
          ( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
          & pp(aa(set(C),bool,member(C,A7),Uu)) ) ) ).

% ATP.lambda_1058
tff(fact_8279_ATP_Olambda__1059,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_aej(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ! [X3: product_prod(B,B)] :
          ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),X3),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_aei(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uua),Uub),Uuc)),X3)) ) ) ).

% ATP.lambda_1059
tff(fact_8280_ATP_Olambda__1060,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_uc(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X3: A,Y: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X3),Y) )
          & pp(aa(A,bool,Uu,X3))
          & pp(aa(B,bool,Uua,Y)) ) ) ).

% ATP.lambda_1060
tff(fact_8281_ATP_Olambda__1061,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_akc(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A18: A,A26: A] :
          ( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A18),A26) )
          & pp(aa(set(A),bool,member(A,A18),Uu))
          & pp(aa(set(A),bool,member(A,A26),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,A18)),aa(A,B,Uub,A26))),Uua)) ) ) ).

% ATP.lambda_1061
tff(fact_8282_ATP_Olambda__1062,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A))),aTP_Lamp_art(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_ars(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uuc)) ).

% ATP.lambda_1062
tff(fact_8283_ATP_Olambda__1063,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,fun(C,A)),Uuc: set(B),Uud: C] : aa(C,A,aa(set(B),fun(C,A),aa(fun(B,fun(C,A)),fun(set(B),fun(C,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(B),fun(C,A))),aTP_Lamp_ase(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(B),fun(C,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),aa(C,fun(B,A),aTP_Lamp_arz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uub),Uud)),Uuc) ).

% ATP.lambda_1063
tff(fact_8284_ATP_Olambda__1064,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(C)),Uuc: fun(B,fun(C,A)),Uud: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aa(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A))),aTP_Lamp_asj(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),aa(B,fun(C,A),Uuc,Uud)),aa(B,set(C),Uub,Uud)) ).

% ATP.lambda_1064
tff(fact_8285_ATP_Olambda__1065,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,fun(C,A)),Uuc: set(C),Uud: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(B,fun(C,A)),fun(set(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A))),aTP_Lamp_asd(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),aa(B,fun(C,A),Uub,Uud)),Uuc) ).

% ATP.lambda_1065
tff(fact_8286_ATP_Olambda__1066,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,A),Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aa(A,fun(fun(nat,A),fun(nat,fun(nat,A))),aTP_Lamp_arn(fun(A,fun(A,A)),fun(A,fun(fun(nat,A),fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),Uub),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uud),Uuc)),Uuc))) ).

% ATP.lambda_1066
tff(fact_8287_ATP_Olambda__1067,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(C)),Uuc: fun(C,A),Uud: B] : aa(B,A,aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_arl(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),Uuc),aa(B,set(C),Uub,Uud)) ).

% ATP.lambda_1067
tff(fact_8288_ATP_Olambda__1068,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_sa(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = 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))),aa(bool,fun(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B)))),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)),insert2(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud)),Uud) ).

% ATP.lambda_1068
tff(fact_8289_ATP_Olambda__1069,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_sj(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = 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))),aa(bool,fun(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C)))),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)),insert2(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud)),Uud) ).

% ATP.lambda_1069
tff(fact_8290_ATP_Olambda__1070,axiom,
    ! [A: $tType,Uu: A,Uua: nat,Uub: fun(nat,A),Uuc: fun(nat,A),Uud: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aa(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),aTP_Lamp_ari(A,fun(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),Uua)),aa(nat,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uud),Uua)),Uu),aa(nat,A,Uuc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uud),aa(nat,nat,suc,zero_zero(nat)))))) ).

% ATP.lambda_1070
tff(fact_8291_ATP_Olambda__1071,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_zl(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_1071
tff(fact_8292_ATP_Olambda__1072,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_zh(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_1072
tff(fact_8293_ATP_Olambda__1073,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(A,bool)),Uua: fun(A,fun(C,bool)),Uub: B,Uuc: C,Uud: A] :
      ( pp(aa(A,bool,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),aa(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool))),aTP_Lamp_akh(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uud))
        & pp(aa(C,bool,aa(A,fun(C,bool),Uua,Uud),Uuc)) ) ) ).

% ATP.lambda_1073
tff(fact_8294_ATP_Olambda__1074,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_aei(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_1074
tff(fact_8295_ATP_Olambda__1075,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_eu(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_1075
tff(fact_8296_ATP_Olambda__1076,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_er(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_1076
tff(fact_8297_ATP_Olambda__1077,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_tp(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_1077
tff(fact_8298_ATP_Olambda__1078,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_es(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_1078
tff(fact_8299_ATP_Olambda__1079,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_yh(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),insert2(A,Uua),aa(set(A),set(A),insert2(A,Uub),aa(set(A),set(A),insert2(A,Uuc),aa(set(A),set(A),insert2(A,Uud),bot_bot(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_1079
tff(fact_8300_ATP_Olambda__1080,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_ts(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_1080
tff(fact_8301_ATP_Olambda__1081,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_akf(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_1081
tff(fact_8302_ATP_Olambda__1082,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_apz(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))
    <=> ? [X3: A] :
          ( pp(aa(set(A),bool,member(A,X3),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),X3))
          & pp(aa(E,bool,aa(A,fun(E,bool),Uub,X3),Uud)) ) ) ).

% ATP.lambda_1082
tff(fact_8303_ATP_Olambda__1083,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_yf(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A7: A,B7: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A7)),aa(C,B,Uua,B7)) )
              & pp(aa(set(A),bool,member(A,A7),Uub))
              & pp(aa(set(C),bool,member(C,B7),Uuc)) ) ) ) ).

% ATP.lambda_1083
tff(fact_8304_ATP_Olambda__1084,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,fun(C,A)),Uud: fun(B,fun(C,bool)),Uue: 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)),aa(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))),aa(A,fun(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A)))),aTP_Lamp_asa(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),aa(C,fun(B,A),aTP_Lamp_arz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uuc),Uue)),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_ho(set(B),fun(fun(B,fun(C,bool)),fun(C,fun(B,bool))),Uub),Uud),Uue))) ).

% ATP.lambda_1084
tff(fact_8305_ATP_Olambda__1085,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(C),Uuc: fun(B,fun(C,A)),Uud: fun(B,fun(C,bool)),Uue: 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)),aa(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))),aa(A,fun(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A)))),aTP_Lamp_ary(fun(A,fun(A,A)),fun(A,fun(set(C),fun(fun(B,fun(C,A)),fun(fun(B,fun(C,bool)),fun(B,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),aa(B,fun(C,A),Uuc,Uue)),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_hm(set(C),fun(fun(B,fun(C,bool)),fun(B,fun(C,bool))),Uub),Uud),Uue))) ).

% ATP.lambda_1085
tff(fact_8306_ATP_Olambda__1086,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,A),Uud: fun(B,C),Uue: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aa(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A)))),aTP_Lamp_asm(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),Uuc),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uub),Uud),Uue))) ).

% ATP.lambda_1086
tff(fact_8307_ATP_Olambda__1087,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,C),Uud: fun(B,A),Uue: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aa(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),aa(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A)))),aTP_Lamp_ask(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),Uud),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_ml(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uub),Uuc),Uue))) ).

% ATP.lambda_1087
tff(fact_8308_ATP_Olambda__1088,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B,Uud: C,Uue: D] :
      ( pp(aa(D,bool,aa(C,fun(D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_alk(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),Uud))
        & pp(aa(D,bool,aa(B,fun(D,bool),Uua,Uuc),Uue)) ) ) ).

% ATP.lambda_1088
tff(fact_8309_ATP_Olambda__1089,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_kr(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_1089
tff(fact_8310_ATP_Olambda__1090,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_ku(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_1090
tff(fact_8311_ATP_Olambda__1091,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,A),Uud: fun(B,A),Uue: B] :
      ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aa(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool)))),aTP_Lamp_aru(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
    <=> ( pp(aa(set(B),bool,member(B,Uue),Uub))
        & ( aa(A,A,aa(A,fun(A,A),Uu,aa(B,A,Uuc,Uue)),aa(B,A,Uud,Uue)) != Uua ) ) ) ).

% ATP.lambda_1091
tff(fact_8312_ATP_Olambda__1092,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool] : aa(bool,fun(list(char),list(char)),aa(bool,fun(bool,fun(list(char),list(char))),aa(bool,fun(bool,fun(bool,fun(list(char),list(char)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char)))))))),aTP_Lamp_aoe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(list(char),list(char))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(char,fun(list(char),list(char)),cons(char),char2(Uu,Uua,Uub,Uuc,Uud,Uue,Uuf,fFalse)) ).

% ATP.lambda_1092
tff(fact_8313_ATP_Olambda__1093,axiom,
    ! [Uu: list(code_integer),Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool,Uuh: bool] : aa(bool,literal,aa(bool,fun(bool,literal),aa(bool,fun(bool,fun(bool,literal)),aa(bool,fun(bool,fun(bool,fun(bool,literal))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal))))))),aTP_Lamp_apx(list(code_integer),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,literal)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = aa(literal,literal,aa(bool,fun(literal,literal),aa(bool,fun(bool,fun(literal,literal)),aa(bool,fun(bool,fun(bool,fun(literal,literal))),aa(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(literal,literal))))))),literal2,Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),aa(list(code_integer),literal,literal_of_asciis,Uu)) ).

% ATP.lambda_1093
tff(fact_8314_ATP_Olambda__1094,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool,Uuh: bool,Uui: bool] : aa(bool,B,aa(bool,fun(bool,B),aa(bool,fun(bool,fun(bool,B)),aa(bool,fun(bool,fun(bool,fun(bool,B))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,B)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B))))))),aa(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B)))))))),aTP_Lamp_apy(fun(A,B),fun(fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))))),fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,B))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = aa(A,B,Uu,aa(bool,A,aa(bool,fun(bool,A),aa(bool,fun(bool,fun(bool,A)),aa(bool,fun(bool,fun(bool,fun(bool,A))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,A))))))),Uua,Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1094
tff(fact_8315_ATP_Olambda__1095,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_ass(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uu) ) ).

% ATP.lambda_1095
tff(fact_8316_ATP_Olambda__1096,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_ast(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uua) ) ).

% ATP.lambda_1096
tff(fact_8317_ATP_Olambda__1097,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_asw(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uub) ) ).

% ATP.lambda_1097
tff(fact_8318_ATP_Olambda__1098,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_asv(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uuc) ) ).

% ATP.lambda_1098
tff(fact_8319_ATP_Olambda__1099,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_asu(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uud) ) ).

% ATP.lambda_1099
tff(fact_8320_ATP_Olambda__1100,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_asr(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uue) ) ).

% ATP.lambda_1100
tff(fact_8321_ATP_Olambda__1101,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_asq(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uuf) ) ).

% ATP.lambda_1101
tff(fact_8322_ATP_Olambda__1102,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_aqb(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> pp(Uug) ) ).

% ATP.lambda_1102
tff(fact_8323_ATP_Olambda__1103,axiom,
    ! [Uu: bool,Uua: bool,Uub: bool,Uuc: bool,Uud: bool,Uue: bool,Uuf: bool,Uug: bool] :
      ( pp(aa(bool,bool,aa(bool,fun(bool,bool),aa(bool,fun(bool,fun(bool,bool)),aa(bool,fun(bool,fun(bool,fun(bool,bool))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))),aa(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool))))))),aTP_Lamp_aqe(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,fun(bool,bool)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))
    <=> $true ) ).

% ATP.lambda_1103
tff(fact_8324_ATP_Olambda__1104,axiom,
    ! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_qd(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(Uu) ) ).

% ATP.lambda_1104
tff(fact_8325_ATP_Olambda__1105,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_ab(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).

% ATP.lambda_1105
tff(fact_8326_ATP_Olambda__1106,axiom,
    ! [A: $tType,Uu: bool,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_pl(bool,fun(A,bool),Uu),Uua))
    <=> pp(Uu) ) ).

% ATP.lambda_1106
tff(fact_8327_ATP_Olambda__1107,axiom,
    ! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_apn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = Uu ).

% ATP.lambda_1107
tff(fact_8328_ATP_Olambda__1108,axiom,
    ! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_xq(set(D),fun(C,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1108
tff(fact_8329_ATP_Olambda__1109,axiom,
    ! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_zm(set(D),fun(B,set(D)),Uu),Uua) = Uu ).

% ATP.lambda_1109
tff(fact_8330_ATP_Olambda__1110,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_xm(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1110
tff(fact_8331_ATP_Olambda__1111,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_xn(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_1111
tff(fact_8332_ATP_Olambda__1112,axiom,
    ! [C: $tType,B: $tType,Uu: set(B),Uua: C] : aa(C,set(B),aTP_Lamp_ain(set(B),fun(C,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1112
tff(fact_8333_ATP_Olambda__1113,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_xa(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_1113
tff(fact_8334_ATP_Olambda__1114,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ww(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1114
tff(fact_8335_ATP_Olambda__1115,axiom,
    ! [C: $tType,A: $tType,Uu: set(A),Uua: C] : aa(C,set(A),aTP_Lamp_aim(set(A),fun(C,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1115
tff(fact_8336_ATP_Olambda__1116,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_nh(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1116
tff(fact_8337_ATP_Olambda__1117,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_wx(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_1117
tff(fact_8338_ATP_Olambda__1118,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_ajp(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).

% ATP.lambda_1118
tff(fact_8339_ATP_Olambda__1119,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_ry(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_1119
tff(fact_8340_ATP_Olambda__1120,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ao(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_1120
tff(fact_8341_ATP_Olambda__1121,axiom,
    ! [B: $tType,C: $tType,Uu: C,Uua: B] : aa(B,C,aTP_Lamp_aoi(C,fun(B,C),Uu),Uua) = Uu ).

% ATP.lambda_1121
tff(fact_8342_ATP_Olambda__1122,axiom,
    ! [A: $tType,C: $tType,Uu: C,Uua: A] : aa(A,C,aTP_Lamp_aoh(C,fun(A,C),Uu),Uua) = Uu ).

% ATP.lambda_1122
tff(fact_8343_ATP_Olambda__1123,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_akx(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1123
tff(fact_8344_ATP_Olambda__1124,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_zz(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1124
tff(fact_8345_ATP_Olambda__1125,axiom,
    ! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aTP_Lamp_aj(B,fun(C,B),Uu),Uua) = Uu ).

% ATP.lambda_1125
tff(fact_8346_ATP_Olambda__1126,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_mj(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1126
tff(fact_8347_ATP_Olambda__1127,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_pg(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1127
tff(fact_8348_ATP_Olambda__1128,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_nj(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1128
tff(fact_8349_ATP_Olambda__1129,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_wq(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1129
tff(fact_8350_ATP_Olambda__1130,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_iq(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1130
tff(fact_8351_ATP_Olambda__1131,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_akz(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1131
tff(fact_8352_ATP_Olambda__1132,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_rd(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1132
tff(fact_8353_ATP_Olambda__1133,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ir(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1133
tff(fact_8354_ATP_Olambda__1134,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_abz(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1134
tff(fact_8355_ATP_Olambda__1135,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aci(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1135
tff(fact_8356_ATP_Olambda__1136,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cu(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1136
tff(fact_8357_ATP_Olambda__1137,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_abm(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1137
tff(fact_8358_ATP_Olambda__1138,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_fn(A,fun(B,B)),Uu),Uua) = Uua ).

% ATP.lambda_1138
tff(fact_8359_ATP_Olambda__1139,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_zi(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1139
tff(fact_8360_ATP_Olambda__1140,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_apa(A,fun(A,bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_1140
tff(fact_8361_ATP_Olambda__1141,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_zj(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1141
tff(fact_8362_ATP_Olambda__1142,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_by(A,fun(B,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1142
tff(fact_8363_ATP_Olambda__1143,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aaf(A,fun(A,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_1143
tff(fact_8364_ATP_Olambda__1144,axiom,
    ! [Uu: product_prod(nat,nat)] : aa(product_prod(nat,nat),product_prod(nat,nat),aTP_Lamp_alm(product_prod(nat,nat),product_prod(nat,nat)),Uu) = Uu ).

% ATP.lambda_1144
tff(fact_8365_ATP_Olambda__1145,axiom,
    ! [Uu: list(char)] : aa(list(char),list(char),aTP_Lamp_amr(list(char),list(char)),Uu) = Uu ).

% ATP.lambda_1145
tff(fact_8366_ATP_Olambda__1146,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_cp(nat,nat),Uu) = Uu ).

% ATP.lambda_1146
tff(fact_8367_ATP_Olambda__1147,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_gf(int,int),Uu) = Uu ).

% ATP.lambda_1147
tff(fact_8368_ATP_Olambda__1148,axiom,
    ! [A: $tType,Uu: fun(A,bool)] : aa(fun(A,bool),fun(A,bool),aTP_Lamp_atp(fun(A,bool),fun(A,bool)),Uu) = Uu ).

% ATP.lambda_1148
tff(fact_8369_ATP_Olambda__1149,axiom,
    ! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_lf(B,B),Uu) = Uu ).

% ATP.lambda_1149
tff(fact_8370_ATP_Olambda__1150,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wb(A,A),Uu) = Uu ) ).

% ATP.lambda_1150
tff(fact_8371_ATP_Olambda__1151,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wa(A,A),Uu) = Uu ) ).

% ATP.lambda_1151
tff(fact_8372_ATP_Olambda__1152,axiom,
    ! [A: $tType] :
      ( complete_Sup(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_nd(A,A),Uu) = Uu ) ).

% ATP.lambda_1152
tff(fact_8373_ATP_Olambda__1153,axiom,
    ! [A: $tType] :
      ( complete_Inf(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_nf(A,A),Uu) = Uu ) ).

% ATP.lambda_1153
tff(fact_8374_ATP_Olambda__1154,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_vw(A,A),Uu) = Uu ) ).

% ATP.lambda_1154
tff(fact_8375_ATP_Olambda__1155,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ay(A,A),Uu) = Uu ) ).

% ATP.lambda_1155
tff(fact_8376_ATP_Olambda__1156,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_amd(A,A),Uu) = Uu ) ).

% ATP.lambda_1156
tff(fact_8377_ATP_Olambda__1157,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ak(A,A),Uu) = Uu ).

% ATP.lambda_1157
tff(fact_8378_ATP_Olambda__1158,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_pm(B,A),Uu) = top_top(A) ) ).

% ATP.lambda_1158
tff(fact_8379_ATP_Olambda__1159,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(bool),aTP_Lamp_aiz(A,set(bool)),Uu) = top_top(set(bool)) ).

% ATP.lambda_1159
tff(fact_8380_ATP_Olambda__1160,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_xc(A,set(B)),Uu) = top_top(set(B)) ).

% ATP.lambda_1160
tff(fact_8381_ATP_Olambda__1161,axiom,
    ! [C: $tType,D: $tType,Uu: C] : aa(C,set(D),aTP_Lamp_aof(C,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1161
tff(fact_8382_ATP_Olambda__1162,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_mg(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1162
tff(fact_8383_ATP_Olambda__1163,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_nr(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_1163
tff(fact_8384_ATP_Olambda__1164,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ni(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_1164
tff(fact_8385_ATP_Olambda__1165,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_mh(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1165
tff(fact_8386_ATP_Olambda__1166,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_xb(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1166
tff(fact_8387_ATP_Olambda__1167,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_cy(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1167
tff(fact_8388_ATP_Olambda__1168,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_zq(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1168
tff(fact_8389_ATP_Olambda__1169,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ax(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1169
tff(fact_8390_ATP_Olambda__1170,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ae(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1170
tff(fact_8391_ATP_Olambda__1171,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_fd(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1171
tff(fact_8392_ATP_Olambda__1172,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_iv(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_1172
tff(fact_8393_ATP_Olambda__1173,axiom,
    ! [A: $tType,Uu: product_unit] : aa(product_unit,seq(A),aTP_Lamp_asy(product_unit,seq(A)),Uu) = empty(A) ).

% ATP.lambda_1173
tff(fact_8394_ATP_Olambda__1174,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_af(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_1174
tff(fact_8395_ATP_Olambda__1175,axiom,
    ! [B: $tType,D: $tType,Uu: B] : aa(B,option(D),aTP_Lamp_ail(B,option(D)),Uu) = none(D) ).

% ATP.lambda_1175
tff(fact_8396_ATP_Olambda__1176,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_acu(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1176
tff(fact_8397_ATP_Olambda__1177,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_aik(A,option(C)),Uu) = none(C) ).

% ATP.lambda_1177
tff(fact_8398_ATP_Olambda__1178,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_acc(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1178
tff(fact_8399_ATP_Olambda__1179,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,A,aTP_Lamp_aoo(B,A),Uu) = undefined(A) ).

% ATP.lambda_1179
tff(fact_8400_ATP_Olambda__1180,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_zn(A,B),Uu) = undefined(B) ).

% ATP.lambda_1180
tff(fact_8401_ATP_Olambda__1181,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_co(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1181
tff(fact_8402_ATP_Olambda__1182,axiom,
    ! [B: $tType,Uu: B] :
      ( pp(aa(B,bool,aTP_Lamp_anz(B,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1182
tff(fact_8403_ATP_Olambda__1183,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_ap(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_1183
tff(fact_8404_ATP_Olambda__1184,axiom,
    ! [Uu: heap_ext(product_unit)] :
      ( pp(aa(heap_ext(product_unit),bool,aTP_Lamp_aow(heap_ext(product_unit),bool),Uu))
    <=> $true ) ).

% ATP.lambda_1184
tff(fact_8405_ATP_Olambda__1185,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_cn(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1185
tff(fact_8406_ATP_Olambda__1186,axiom,
    ! [A: $tType,Uu: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aTP_Lamp_ago(fun(A,bool),bool),Uu))
    <=> $true ) ).

% ATP.lambda_1186
tff(fact_8407_ATP_Olambda__1187,axiom,
    ! [D: $tType,Uu: D] :
      ( pp(aa(D,bool,aTP_Lamp_aml(D,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1187
tff(fact_8408_ATP_Olambda__1188,axiom,
    ! [C: $tType,Uu: C] :
      ( pp(aa(C,bool,aTP_Lamp_amm(C,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1188
tff(fact_8409_ATP_Olambda__1189,axiom,
    ! [B: $tType,Uu: B] :
      ( pp(aa(B,bool,aTP_Lamp_aox(B,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1189
tff(fact_8410_ATP_Olambda__1190,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_aq(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_1190
tff(fact_8411_ATP_Olambda__1191,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_ajz(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1191
tff(fact_8412_ATP_Olambda__1192,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_ro(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1192

% Type constructors (689)
tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_1,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( enum(A19)
        & enum(A20) )
     => enum(product_prod(A19,A20)) ) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__lattice_2,axiom,
    ! [A19: $tType] : bounded_lattice(pred(A19)) ).

tff(tcon_Option_Ooption___Enum_Oenum_3,axiom,
    ! [A19: $tType] :
      ( enum(A19)
     => enum(option(A19)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_4,axiom,
    ! [A19: $tType] : bounded_lattice(filter(A19)) ).

tff(tcon_Sum__Type_Osum___Enum_Oenum_5,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( enum(A19)
        & enum(A20) )
     => enum(sum_sum(A19,A20)) ) ).

tff(tcon_String_Ochar___Enum_Oenum_6,axiom,
    enum(char) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_7,axiom,
    bounded_lattice(bool) ).

tff(tcon_HOL_Obool___Enum_Oenum_8,axiom,
    enum(bool) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_9,axiom,
    ! [A19: $tType] : bounded_lattice(set(A19)) ).

tff(tcon_Set_Oset___Enum_Oenum_10,axiom,
    ! [A19: $tType] :
      ( enum(A19)
     => enum(set(A19)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice_11,axiom,
    ! [A19: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounded_lattice(fun(A19,A20)) ) ).

tff(tcon_fun___Enum_Oenum_12,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( enum(A19)
        & enum(A20) )
     => enum(fun(A19,A20)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A19: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A19: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
    ! [A19: $tType,A20: $tType] :
      ( comple489889107523837845lgebra(A20)
     => comple489889107523837845lgebra(fun(A19,A20)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( cl_HOL_Oequal(A19)
        & quickc3360725361186068524ustive(A19)
        & quickc3360725361186068524ustive(A20) )
     => quickc3360725361186068524ustive(fun(A19,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A19: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4967611905675639751up_bot(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A19: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(fun(A19,A20)) ) ).

tff(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( cl_HOL_Oequal(A19)
        & quickc658316121487927005ustive(A19)
        & quickc658316121487927005ustive(A20) )
     => quickc658316121487927005ustive(fun(A19,A20)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A19: $tType,A20: $tType] :
      ( boolea8198339166811842893lgebra(A20)
     => boolea8198339166811842893lgebra(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A19: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple9053668089753744459l_ccpo(fun(A19,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A19: $tType,A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(fun(A19,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A19: $tType,A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(fun(A19,A20)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A19: $tType,A20: $tType] :
      ( distrib_lattice(A20)
     => distrib_lattice(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Lattices_OSup,axiom,
    ! [A19: $tType,A20: $tType] :
      ( complete_Sup(A20)
     => complete_Sup(fun(A19,A20)) ) ).

tff(tcon_fun___Complete__Lattices_OInf,axiom,
    ! [A19: $tType,A20: $tType] :
      ( complete_Inf(A20)
     => complete_Inf(fun(A19,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A19: $tType,A20: $tType] :
      ( order_top(A20)
     => order_top(fun(A19,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A19: $tType,A20: $tType] :
      ( order_bot(A20)
     => order_bot(fun(A19,A20)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( finite_finite(A19)
        & countable(A20) )
     => countable(fun(A19,A20)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A19: $tType,A20: $tType] :
      ( preorder(A20)
     => preorder(fun(A19,A20)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( finite_finite(A19)
        & finite_finite(A20) )
     => finite_finite(fun(A19,A20)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A19: $tType,A20: $tType] :
      ( lattice(A20)
     => lattice(fun(A19,A20)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A19: $tType,A20: $tType] :
      ( order(A20)
     => order(fun(A19,A20)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ord(A20)
     => ord(fun(A19,A20)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A19: $tType,A20: $tType] :
      ( uminus(A20)
     => uminus(fun(A19,A20)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( enum(A19)
        & cl_HOL_Oequal(A20) )
     => cl_HOL_Oequal(fun(A19,A20)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
    normal6328177297339901930cative(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_14,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_Osemidom__divide__unit__factor,axiom,
    semido2269285787275462019factor(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_15,axiom,
    quickc658316121487927005ustive(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
    normal8620421768224518004emidom(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_16,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_17,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_18,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OSup_19,axiom,
    complete_Sup(int) ).

tff(tcon_Int_Oint___Complete__Lattices_OInf_20,axiom,
    complete_Inf(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Countable_Ocountable_21,axiom,
    countable(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_22,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_23,axiom,
    lattice(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
    semiring_Gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_24,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_25,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_26,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___HOL_Oequal_27,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_28,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_29,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_30,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_31,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_32,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_33,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_34,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_35,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_36,axiom,
    normal6328177297339901930cative(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_37,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_38,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_39,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_40,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_41,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_42,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_43,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_44,axiom,
    quickc3360725361186068524ustive(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_45,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_46,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_47,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_48,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_49,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_50,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_51,axiom,
    semido2269285787275462019factor(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_52,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_53,axiom,
    quickc658316121487927005ustive(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_54,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_55,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_56,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_57,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_58,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_59,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_60,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_61,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_62,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Rings_Onormalization__semidom_63,axiom,
    normal8620421768224518004emidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_64,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_65,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_66,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_67,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_68,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_69,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_70,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_71,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_72,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_73,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_74,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_75,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_76,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_77,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_78,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_79,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_80,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_81,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_82,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OSup_83,axiom,
    complete_Sup(nat) ).

tff(tcon_Nat_Onat___Complete__Lattices_OInf_84,axiom,
    complete_Inf(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_85,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_86,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_87,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_88,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_89,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_90,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_91,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_92,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_93,axiom,
    countable(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_94,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_95,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_96,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_97,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_98,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_99,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_100,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_101,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_102,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_103,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_104,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_105,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_106,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_107,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_108,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_109,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Power_Opower_110,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_111,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_112,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_113,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_114,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_115,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_116,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_117,axiom,
    quickc3360725361186068524ustive(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_118,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_119,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_120,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_121,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_122,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_123,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_124,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_125,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_126,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_127,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_128,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_129,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_130,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_131,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_132,axiom,
    quickc3360725361186068524ustive(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_133,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_134,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_135,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_136,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_137,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_138,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_139,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_140,axiom,
    quickc658316121487927005ustive(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_141,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_142,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_143,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_144,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_145,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_146,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_147,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_148,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_149,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_150,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_151,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_152,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_153,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_154,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_155,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_156,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_157,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_158,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_159,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_160,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_161,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_162,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_163,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_164,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_165,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_166,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_167,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_168,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_169,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_170,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_171,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_172,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_173,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_174,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_175,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_176,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_177,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_178,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_179,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Countable_Ocountable_180,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_181,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_182,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_183,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_184,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_185,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_186,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_187,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_188,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_189,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_190,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_191,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_192,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_193,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_194,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_195,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_196,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_197,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_198,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_199,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_200,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_201,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_202,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_203,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_204,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_205,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_206,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_207,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_208,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_209,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_210,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_211,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_212,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_213,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_214,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_215,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216,axiom,
    ! [A19: $tType] : condit1219197933456340205attice(set(A19)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_217,axiom,
    ! [A19: $tType] : comple592849572758109894attice(set(A19)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_218,axiom,
    ! [A19: $tType] : comple489889107523837845lgebra(set(A19)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_219,axiom,
    ! [A19: $tType] :
      ( quickc3360725361186068524ustive(A19)
     => quickc3360725361186068524ustive(set(A19)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_220,axiom,
    ! [A19: $tType] : bounde4967611905675639751up_bot(set(A19)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_221,axiom,
    ! [A19: $tType] : comple6319245703460814977attice(set(A19)) ).

tff(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_222,axiom,
    ! [A19: $tType] :
      ( quickc658316121487927005ustive(A19)
     => quickc658316121487927005ustive(set(A19)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_223,axiom,
    ! [A19: $tType] : boolea8198339166811842893lgebra(set(A19)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_224,axiom,
    ! [A19: $tType] : comple9053668089753744459l_ccpo(set(A19)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_225,axiom,
    ! [A19: $tType] : semilattice_sup(set(A19)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_226,axiom,
    ! [A19: $tType] : semilattice_inf(set(A19)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_227,axiom,
    ! [A19: $tType] : distrib_lattice(set(A19)) ).

tff(tcon_Set_Oset___Complete__Lattices_OSup_228,axiom,
    ! [A19: $tType] : complete_Sup(set(A19)) ).

tff(tcon_Set_Oset___Complete__Lattices_OInf_229,axiom,
    ! [A19: $tType] : complete_Inf(set(A19)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_230,axiom,
    ! [A19: $tType] : order_top(set(A19)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_231,axiom,
    ! [A19: $tType] : order_bot(set(A19)) ).

tff(tcon_Set_Oset___Countable_Ocountable_232,axiom,
    ! [A19: $tType] :
      ( finite_finite(A19)
     => countable(set(A19)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_233,axiom,
    ! [A19: $tType] : preorder(set(A19)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_234,axiom,
    ! [A19: $tType] :
      ( finite_finite(A19)
     => finite_finite(set(A19)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_235,axiom,
    ! [A19: $tType] : lattice(set(A19)) ).

tff(tcon_Set_Oset___Orderings_Oorder_236,axiom,
    ! [A19: $tType] : order(set(A19)) ).

tff(tcon_Set_Oset___Orderings_Oord_237,axiom,
    ! [A19: $tType] : ord(set(A19)) ).

tff(tcon_Set_Oset___Groups_Ouminus_238,axiom,
    ! [A19: $tType] : uminus(set(A19)) ).

tff(tcon_Set_Oset___HOL_Oequal_239,axiom,
    ! [A19: $tType] :
      ( cl_HOL_Oequal(A19)
     => cl_HOL_Oequal(set(A19)) ) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_240,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_241,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_242,axiom,
    comple489889107523837845lgebra(bool) ).

tff(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_243,axiom,
    quickc3360725361186068524ustive(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_244,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_245,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_246,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_247,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_248,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_249,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_250,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OSup_251,axiom,
    complete_Sup(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_OInf_252,axiom,
    complete_Inf(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_253,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_254,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Countable_Ocountable_255,axiom,
    countable(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_256,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_257,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_258,axiom,
    finite_finite(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_259,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_260,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_261,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_262,axiom,
    uminus(bool) ).

tff(tcon_HOL_Obool___HOL_Oequal_263,axiom,
    cl_HOL_Oequal(bool) ).

tff(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
    ! [A19: $tType] :
      ( quickc3360725361186068524ustive(A19)
     => quickc3360725361186068524ustive(list(A19)) ) ).

tff(tcon_List_Olist___Countable_Ocountable_265,axiom,
    ! [A19: $tType] :
      ( countable(A19)
     => countable(list(A19)) ) ).

tff(tcon_List_Olist___HOL_Oequal_266,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(list(A19)) ).

tff(tcon_List_Olist___Nat_Osize_267,axiom,
    ! [A19: $tType] : size(list(A19)) ).

tff(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_268,axiom,
    quickc3360725361186068524ustive(char) ).

tff(tcon_String_Ochar___Countable_Ocountable_269,axiom,
    countable(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_270,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___HOL_Oequal_271,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_String_Ochar___Nat_Osize_272,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_273,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( quickc3360725361186068524ustive(A19)
        & quickc3360725361186068524ustive(A20) )
     => quickc3360725361186068524ustive(sum_sum(A19,A20)) ) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_274,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( countable(A19)
        & countable(A20) )
     => countable(sum_sum(A19,A20)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_275,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( finite_finite(A19)
        & finite_finite(A20) )
     => finite_finite(sum_sum(A19,A20)) ) ).

tff(tcon_Sum__Type_Osum___HOL_Oequal_276,axiom,
    ! [A19: $tType,A20: $tType] : cl_HOL_Oequal(sum_sum(A19,A20)) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_277,axiom,
    ! [A19: $tType,A20: $tType] : size(sum_sum(A19,A20)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_278,axiom,
    ! [A19: $tType] : condit1219197933456340205attice(filter(A19)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_279,axiom,
    ! [A19: $tType] : bounde4967611905675639751up_bot(filter(A19)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_280,axiom,
    ! [A19: $tType] : comple6319245703460814977attice(filter(A19)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_281,axiom,
    ! [A19: $tType] : comple9053668089753744459l_ccpo(filter(A19)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_282,axiom,
    ! [A19: $tType] : semilattice_sup(filter(A19)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_283,axiom,
    ! [A19: $tType] : semilattice_inf(filter(A19)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_284,axiom,
    ! [A19: $tType] : distrib_lattice(filter(A19)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_285,axiom,
    ! [A19: $tType] : complete_Sup(filter(A19)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_286,axiom,
    ! [A19: $tType] : complete_Inf(filter(A19)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_287,axiom,
    ! [A19: $tType] : order_top(filter(A19)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_288,axiom,
    ! [A19: $tType] : order_bot(filter(A19)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_289,axiom,
    ! [A19: $tType] : preorder(filter(A19)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_290,axiom,
    ! [A19: $tType] : lattice(filter(A19)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_291,axiom,
    ! [A19: $tType] : order(filter(A19)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_292,axiom,
    ! [A19: $tType] : ord(filter(A19)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_293,axiom,
    ! [A19: $tType] :
      ( cl_HOL_Oequal(A19)
     => cl_HOL_Oequal(filter(A19)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_294,axiom,
    ! [A19: $tType] :
      ( quickc3360725361186068524ustive(A19)
     => quickc3360725361186068524ustive(option(A19)) ) ).

tff(tcon_Option_Ooption___Countable_Ocountable_295,axiom,
    ! [A19: $tType] :
      ( countable(A19)
     => countable(option(A19)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_296,axiom,
    ! [A19: $tType] :
      ( finite_finite(A19)
     => finite_finite(option(A19)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_297,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(option(A19)) ).

tff(tcon_Option_Ooption___Nat_Osize_298,axiom,
    ! [A19: $tType] : size(option(A19)) ).

tff(tcon_Predicate_Oseq___HOL_Oequal_299,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(seq(A19)) ).

tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_300,axiom,
    ! [A19: $tType] : condit1219197933456340205attice(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_301,axiom,
    ! [A19: $tType] : comple592849572758109894attice(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_302,axiom,
    ! [A19: $tType] : comple489889107523837845lgebra(pred(A19)) ).

tff(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_303,axiom,
    ! [A19: $tType] : bounde4967611905675639751up_bot(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_304,axiom,
    ! [A19: $tType] : comple6319245703460814977attice(pred(A19)) ).

tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_305,axiom,
    ! [A19: $tType] : boolea8198339166811842893lgebra(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_306,axiom,
    ! [A19: $tType] : comple9053668089753744459l_ccpo(pred(A19)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_307,axiom,
    ! [A19: $tType] : semilattice_sup(pred(A19)) ).

tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_308,axiom,
    ! [A19: $tType] : semilattice_inf(pred(A19)) ).

tff(tcon_Predicate_Opred___Lattices_Odistrib__lattice_309,axiom,
    ! [A19: $tType] : distrib_lattice(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_OSup_310,axiom,
    ! [A19: $tType] : complete_Sup(pred(A19)) ).

tff(tcon_Predicate_Opred___Complete__Lattices_OInf_311,axiom,
    ! [A19: $tType] : complete_Inf(pred(A19)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__top_312,axiom,
    ! [A19: $tType] : order_top(pred(A19)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder__bot_313,axiom,
    ! [A19: $tType] : order_bot(pred(A19)) ).

tff(tcon_Predicate_Opred___Orderings_Opreorder_314,axiom,
    ! [A19: $tType] : preorder(pred(A19)) ).

tff(tcon_Predicate_Opred___Lattices_Olattice_315,axiom,
    ! [A19: $tType] : lattice(pred(A19)) ).

tff(tcon_Predicate_Opred___Orderings_Oorder_316,axiom,
    ! [A19: $tType] : order(pred(A19)) ).

tff(tcon_Predicate_Opred___Orderings_Oord_317,axiom,
    ! [A19: $tType] : ord(pred(A19)) ).

tff(tcon_Predicate_Opred___Groups_Ouminus_318,axiom,
    ! [A19: $tType] : uminus(pred(A19)) ).

tff(tcon_Predicate_Opred___HOL_Oequal_319,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(pred(A19)) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_320,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Countable_Ocountable_321,axiom,
    countable(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_322,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_323,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_324,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_325,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_326,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_327,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_328,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___HOL_Oequal_329,axiom,
    cl_HOL_Oequal(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_330,axiom,
    size(literal) ).

tff(tcon_Typerep_Otyperep___Countable_Ocountable_331,axiom,
    countable(typerep) ).

tff(tcon_Typerep_Otyperep___HOL_Oequal_332,axiom,
    cl_HOL_Oequal(typerep) ).

tff(tcon_Typerep_Otyperep___Nat_Osize_333,axiom,
    size(typerep) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( quickc3360725361186068524ustive(A19)
        & quickc3360725361186068524ustive(A20) )
     => quickc3360725361186068524ustive(product_prod(A19,A20)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_335,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( quickc658316121487927005ustive(A19)
        & quickc658316121487927005ustive(A20) )
     => quickc658316121487927005ustive(product_prod(A19,A20)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_336,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( countable(A19)
        & countable(A20) )
     => countable(product_prod(A19,A20)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_337,axiom,
    ! [A19: $tType,A20: $tType] :
      ( ( finite_finite(A19)
        & finite_finite(A20) )
     => finite_finite(product_prod(A19,A20)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_338,axiom,
    ! [A19: $tType,A20: $tType] : cl_HOL_Oequal(product_prod(A19,A20)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_339,axiom,
    ! [A19: $tType,A20: $tType] : size(product_prod(A19,A20)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_340,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_341,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_342,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_343,axiom,
    comple489889107523837845lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_344,axiom,
    quickc3360725361186068524ustive(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_345,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_346,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_347,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_348,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_349,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_350,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_351,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_352,axiom,
    complete_Sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_353,axiom,
    complete_Inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_354,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_355,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_356,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_357,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_358,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_359,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_360,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_361,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_362,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_363,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_364,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_365,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_366,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(heap_ext(A19)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_367,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_368,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_369,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_370,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_371,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_372,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_373,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_374,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_375,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_376,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_377,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_378,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_379,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_380,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_381,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_382,axiom,
    quickc3360725361186068524ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_383,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_384,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_385,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_386,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_387,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_388,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_389,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_390,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_391,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_392,axiom,
    quickc658316121487927005ustive(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_393,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_394,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_395,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_396,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_397,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_398,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_399,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_400,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_401,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_402,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_403,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_404,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_405,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_406,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_407,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_408,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_409,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_410,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_411,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_412,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_413,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_414,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_415,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_416,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_417,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_418,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_419,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_420,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_421,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_422,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_423,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_424,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_425,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_426,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_427,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_428,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_429,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_430,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_431,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_432,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_433,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_434,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_435,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_436,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_437,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_438,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_439,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_440,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_441,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_442,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_443,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_444,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_445,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_446,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_447,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_448,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_449,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_450,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_451,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_452,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_453,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_454,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_455,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_456,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_457,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_458,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_459,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_460,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_461,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_462,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_463,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_464,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_465,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_466,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_467,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_468,axiom,
    bit_un5681908812861735899ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_469,axiom,
    euclid5411537665997757685th_nat(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_470,axiom,
    ordere1937475149494474687imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_471,axiom,
    euclid3128863361964157862miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_472,axiom,
    euclid4440199948858584721cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_473,axiom,
    semiri6575147826004484403cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_474,axiom,
    strict9044650504122735259up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_475,axiom,
    ordere580206878836729694up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_476,axiom,
    ordere2412721322843649153imp_le(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_477,axiom,
    bit_se359711467146920520ations(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_478,axiom,
    linord2810124833399127020strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_479,axiom,
    quickc3360725361186068524ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_480,axiom,
    strict7427464778891057005id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_481,axiom,
    ordere8940638589300402666id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_482,axiom,
    euclid3725896446679973847miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_483,axiom,
    linord4140545234300271783up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_484,axiom,
    semiri2026040879449505780visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_485,axiom,
    linord181362715937106298miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_486,axiom,
    linord8928482502909563296strict(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_487,axiom,
    quickc658316121487927005ustive(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_488,axiom,
    semiri3467727345109120633visors(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_489,axiom,
    ordere6658533253407199908up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_490,axiom,
    ordere6911136660526730532id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_491,axiom,
    cancel2418104881723323429up_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_492,axiom,
    cancel1802427076303600483id_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_493,axiom,
    comm_s4317794764714335236cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_494,axiom,
    bit_semiring_bits(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_495,axiom,
    ordere2520102378445227354miring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_496,axiom,
    cancel_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_497,axiom,
    linordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_498,axiom,
    ordered_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_499,axiom,
    linordered_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_500,axiom,
    ab_semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_501,axiom,
    semiring_1_cancel(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_502,axiom,
    algebraic_semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_503,axiom,
    comm_monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_504,axiom,
    comm_monoid_diff(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_505,axiom,
    ab_semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_506,axiom,
    ordered_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_507,axiom,
    semiring_parity(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_508,axiom,
    comm_monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_509,axiom,
    semiring_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_510,axiom,
    comm_semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_511,axiom,
    comm_semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_512,axiom,
    semigroup_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_513,axiom,
    semidom_modulo(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_514,axiom,
    semidom_divide(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_515,axiom,
    semiring_numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_516,axiom,
    semigroup_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_517,axiom,
    zero_less_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_518,axiom,
    comm_semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_519,axiom,
    semiring_char_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_520,axiom,
    zero_neq_one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_521,axiom,
    preorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_522,axiom,
    linorder(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_523,axiom,
    monoid_mult(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_524,axiom,
    monoid_add(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_525,axiom,
    semiring_1(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_526,axiom,
    semiring_0(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Omult__zero_527,axiom,
    mult_zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_528,axiom,
    order(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemiring_529,axiom,
    semiring(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Osemidom_530,axiom,
    semidom(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Orderings_Oord_531,axiom,
    ord(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Power_Opower_532,axiom,
    power(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Num_Onumeral_533,axiom,
    numeral(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Ozero_534,axiom,
    zero(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oplus_535,axiom,
    plus(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Groups_Oone_536,axiom,
    one(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Rings_Odvd_537,axiom,
    dvd(code_natural) ).

tff(tcon_Code__Numeral_Onatural___HOL_Oequal_538,axiom,
    cl_HOL_Oequal(code_natural) ).

tff(tcon_Code__Numeral_Onatural___Nat_Osize_539,axiom,
    size(code_natural) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_540,axiom,
    ! [A19: $tType] : cl_HOL_Oequal(heap_Time_Heap(A19)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_541,axiom,
    ! [A19: $tType] : size(heap_Time_Heap(A19)) ).

tff(tcon_Record_Otuple__isomorphism___Quickcheck__Exhaustive_Ofull__exhaustive_542,axiom,
    ! [A19: $tType,A20: $tType,A21: $tType] :
      ( ( cl_HOL_Oequal(A19)
        & quickc3360725361186068524ustive(A19)
        & quickc3360725361186068524ustive(A20)
        & quickc3360725361186068524ustive(A21) )
     => quickc3360725361186068524ustive(tuple_isomorphism(A19,A20,A21)) ) ).

tff(tcon_Record_Otuple__isomorphism___HOL_Oequal_543,axiom,
    ! [A19: $tType,A20: $tType,A21: $tType] : cl_HOL_Oequal(tuple_isomorphism(A19,A20,A21)) ).

tff(tcon_Record_Otuple__isomorphism___Nat_Osize_544,axiom,
    ! [A19: $tType,A20: $tType,A21: $tType] : size(tuple_isomorphism(A19,A20,A21)) ).

% Helper facts (17)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fFalse),X),Y3) = Y3 ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fTrue),X),Y3) = 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,Y3: A] :
      ( ( X != Y3 )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y3)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y3: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y3))
      | ( X = Y3 ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    pp(aa(a,bool,p,x)) ).

tff(conj_1,conjecture,
    heap_Time_success(a,heap_Time_assert(a,p,x),h) ).

%------------------------------------------------------------------------------