%------------------------------------------------------------------------------
% File     : ITP292_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Intf_Imperative 00161_004904
% 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    : 0096_VEBT_Intf_Imperative_00161_004904 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11806 (4641 unt;1583 typ;   0 def)
%            Number of atoms       : 17934 (8207 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 21385 (2387   ~; 343   |;2050   &)
%                                         (2279 <=>;14326  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 1755 ( 789 fml; 966 var)
%            Number of X terms     : 1203 (   0  []; 913 ite; 290 let)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  : 1388 (1099   >; 289   *;   0   +;   0  <<)
%            Number of predicates  :  236 ( 233 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1359 (1359 usr; 109 con; 0-8 aty)
%            Number of variables   : 34819 (31519   !;1014   ?;34819   :)
%                                         (2286  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 23:43:10.766
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
tff(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $tType ).

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

tff(ty_t_Heap__Time__Monad_OHeap,type,
    heap_Time_Heap: $tType > $tType ).

tff(ty_t_Code__Numeral_Ointeger,type,
    code_integer: $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_Numeral__Type_Onum1,type,
    numeral_num1: $tType ).

tff(ty_t_Numeral__Type_Onum0,type,
    numeral_num0: $tType ).

tff(ty_t_Numeral__Type_Obit1,type,
    numeral_bit1: $tType > $tType ).

tff(ty_t_Numeral__Type_Obit0,type,
    numeral_bit0: $tType > $tType ).

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

tff(ty_t_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

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

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

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

tff(ty_t_Uint32_Ouint32,type,
    uint32: $tType ).

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

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

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

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

tff(ty_t_Heap_Oarray,type,
    array: $tType > $tType ).

tff(ty_t_Word_Oword,type,
    word: $tType > $tType ).

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

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

tff(ty_t_Heap_Oref,type,
    ref: $tType > $tType ).

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

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

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

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

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

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

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

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

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

% Explicit typings (1548)
tff(sy_cl_Type__Length_Olen0,type,
    type_len0: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Fields_Oinverse,type,
    inverse: 
      !>[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_Type__Length_Olen,type,
    type_len: 
      !>[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_Cardinality_Ocard2,type,
    card2: 
      !>[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_Transcendental_Oln,type,
    ln: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot2__space,type,
    topological_t2_space: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Limits_Otopological__monoid__add,type,
    topolo6943815403480290642id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_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_Real__Vector__Spaces_Oreal__vector,type,
    real_V4867850818363320053vector: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_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_Limits_Otopological__ab__group__add,type,
    topolo1287966508704411220up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Ometric__space,type,
    real_V7819770556892013058_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Bit__Comprehension_Obit__comprehension,type,
    bit_bi6583157726757044596ension: 
      !>[A: $tType] : $o ).

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

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

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

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

tff(sy_cl_Quickcheck__Narrowing_Opartial__term__of,type,
    quickc6926020345158392990erm_of: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
    topolo3112930676232923870pology: 
      !>[A: $tType] : $o ).

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

tff(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
    topolo8458572112393995274pology: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__lc____,type,
    aTP_Lamp_lc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__le____,type,
    aTP_Lamp_le: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lf____,type,
    aTP_Lamp_lf: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( 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_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).

tff(sy_c_Array__Time_Omake,type,
    array_make: 
      !>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Omap__entry,type,
    array_map_entry: 
      !>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( ( array(A) * nat ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oswap,type,
    array_swap: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: assn > fun(product_prod(heap_ext(product_unit),set(nat)),$o) ).

tff(sy_c_Assertions_Oentails,type,
    entails: ( assn * assn ) > $o ).

tff(sy_c_Assertions_Oex__assn,type,
    ex_assn: 
      !>[A: $tType] : ( fun(A,assn) > assn ) ).

tff(sy_c_Assertions_Oprecise,type,
    precise: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,assn)) > $o ) ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( array(A) > fun(list(A),assn) ) ).

tff(sy_c_Assertions_Osngr__assn,type,
    sngr_assn: 
      !>[A: $tType] : ( ( ref(A) * A ) > assn ) ).

tff(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( fun(product_prod(heap_ext(product_unit),set(nat)),$o) * fun(product_prod(heap_ext(product_unit),set(nat)),$o) * product_prod(heap_ext(product_unit),set(nat)) ) > $o ).

tff(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o)) ).

tff(sy_c_BNF__Def_Orel__fun,type,
    bNF_rel_fun: 
      !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,$o)) * fun(B,fun(D,$o)) ) > fun(fun(A,B),fun(fun(C,D),$o)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
    bNF_Wellorder_Func: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).

tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
    bNF_We4925052301507509544nc_map: 
      !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).

tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
    basic_BNF_size_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) * product_prod(A,B) ) > nat ) ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: ( nat * nat ) > nat ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
    bit_bi4170147762399595738t_bits: 
      !>[A: $tType] : ( fun(nat,$o) > A ) ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun(nat,$o) > $o ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : fun(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] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : fun(nat,fun(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: $tType] : fun(A,fun(nat,$o)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
    bit_Sh4282982442137083160shiftl: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
    bit_Sh4282982442137083166shiftr: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
    bit_Sh8784991116023147202shiftr: 
      !>[A: $tType] : fun(word(A),fun(nat,word(A))) ).

tff(sy_c_Bits__Integer_Ointeger__set__bit,type,
    bits_integer_set_bit: ( code_integer * code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__shiftl,type,
    bits_integer_shiftl: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__shiftr,type,
    bits_integer_shiftr: ( code_integer * code_integer ) > code_integer ).

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,$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_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: fun(nat,code_integer) ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
    code_integer_of_num: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

tff(sy_c_Code__Numeral_Onegative,type,
    code_negative: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun(num,code_integer) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
    code_T4080844693773952564amples: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > list(A) ) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples__rel,type,
    code_T1710151556404007877es_rel: 
      !>[A: $tType] : ( fun(code_integer,product_prod(A,A)) > fun(code_integer,fun(code_integer,$o)) ) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample,type,
    code_T4081349890594273596sample: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > A ) ).

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( fun(nat,$o) * nat * word(A) ) > word(A) ) ).

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_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

tff(sy_c_Deriv_Ohas__derivative,type,
    has_derivative: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofiltercomap,type,
    filtercomap: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

tff(sy_c_Filter_Oprincipal,type,
    principal: 
      !>[A: $tType] : ( set(A) > filter(A) ) ).

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Finite__Set_Ocard,type,
    finite_card: 
      !>[B: $tType] : fun(set(B),nat) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
    finite6289374366891150609ommute: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
    finite4664212375090638736ute_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
    finite4980608107308702382axioms: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : fun(set(A),$o) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Ostrict__mono__on,type,
    strict_mono_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit,type,
    generi7602027413899671122et_bit: 
      !>[A: $tType] : fun(A,fun(nat,fun($o,A))) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
    groups3894954378712506084id_sum: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A * fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > A ) ).

tff(sy_c_Groups__List_Omonoid__add_Osum__list,type,
    groups4543113879258116180m_list: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * list(A) ) > A ) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Hash__Instances_Ohash__code__prod,type,
    hash_hash_code_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,uint32) * fun(B,uint32) * product_prod(A,B) ) > uint32 ) ).

tff(sy_c_Heap_Oarray_Osize__array,type,
    size_array: 
      !>[A: $tType] : ( ( fun(A,nat) * array(A) ) > nat ) ).

tff(sy_c_Heap__Time__Monad_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : ( A > heap_Time_Heap(A) ) ).

tff(sy_c_Hoare__Triple_Ohoare__triple,type,
    hoare_hoare_triple: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( set(A) > fun(nat,A) ) ).

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_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Limits_OBfun,type,
    bfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : fun(list(A),fun(list(A),list(A))) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).

tff(sy_c_List_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_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldl,type,
    foldl: 
      !>[B: $tType,A: $tType] : ( fun(B,fun(A,B)) > fun(B,fun(list(A),B)) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(list(B),list(B)) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( set(A) > list(A) ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(A,fun(list(A),list(A))) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : ( list(A) > A ) ).

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_Oset,type,
    set2: 
      !>[A: $tType] : fun(list(A),set(A)) ).

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).

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_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_OremoveAll,type,
    removeAll: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_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__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Misc_Obijective,type,
    bijective: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Misc_Obrk__rel,type,
    brk_rel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(product_prod($o,A),product_prod($o,B))) ) ).

tff(sy_c_Misc_Odflt__None__set,type,
    dflt_None_set: 
      !>[A: $tType] : ( set(A) > option(set(A)) ) ).

tff(sy_c_Misc_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).

tff(sy_c_Misc_Olist__collect__set,type,
    list_collect_set: 
      !>[B: $tType,A: $tType] : ( ( fun(B,set(A)) * list(B) ) > set(A) ) ).

tff(sy_c_Misc_Omap__to__set,type,
    map_to_set: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Omerge__list,type,
    merge_list: 
      !>[A: $tType] : ( ( list(list(A)) * list(list(A)) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__split__rel,type,
    merges7066485432131860899it_rel: 
      !>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),$o)) ).

tff(sy_c_Misc_Omergesort__remdups,type,
    mergesort_remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_Misc_Opairself,type,
    pairself: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).

tff(sy_c_Misc_Opairself__rel,type,
    pairself_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),$o)) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Oremove__rev,type,
    remove_rev: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Oset__to__map,type,
    set_to_map: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,option(A)) ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Othe__default,type,
    the_default: 
      !>[A: $tType] : ( ( A * option(A) ) > A ) ).

tff(sy_c_Misc_Ouncurry,type,
    uncurry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Multiset_Oadd__mset,type,
    add_mset: 
      !>[A: $tType] : ( ( A * multiset(A) ) > multiset(A) ) ).

tff(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_Multiset_Osubseteq__mset,type,
    subseteq_mset: 
      !>[A: $tType] : fun(multiset(A),fun(multiset(A),$o)) ).

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_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

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_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Numeral__Type_Omod__ring,type,
    numeral_mod_ring: 
      !>[A: $tType] : ( ( int * fun(A,int) * fun(int,A) ) > $o ) ).

tff(sy_c_Numeral__Type_Omod__type,type,
    numeral_mod_type: 
      !>[A: $tType] : ( ( int * fun(A,int) * fun(int,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] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : ( option(A) > A ) ).

tff(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
    order_532582986084564980_cclfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Power_Opower_Opower,type,
    power2: 
      !>[A: $tType] : ( ( A * fun(A,fun(A,A)) * A * nat ) > A ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,A)) ).

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Ointernal__case__prod,type,
    produc5280177257484947105e_prod: 
      !>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).

tff(sy_c_Product__Type_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_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
    real_V4916620083959148203axioms: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

tff(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

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_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord__class_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] : ( 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] : ( A > set(A) ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
    signed7115095781618012415divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o * $o ) > char ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Ofails,type,
    time_fails: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Time__Reasoning_Ohtt,type,
    time_htt: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).

tff(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(A) ) > filter(A) ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transitive__Closure_Otrancl,type,
    transitive_trancl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).

tff(sy_c_Type__Length_Olen0__class_Olen__of,type,
    type_len0_len_of: 
      !>[A: $tType] : ( itself(A) > nat ) ).

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_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun(nat,$o) * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun(nat,fun(uint32,uint32)) ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint32: uint32 > word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > product_prod(uint32,uint32) ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer * $o ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: ( uint32 * code_integer ) > $o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( ( nat * heap_Time_Heap(A) ) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( $o * $o ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option(product_prod(nat,nat)) * nat * array(vEBT_VEBTi) * vEBT_VEBTi ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
    vEBT_case_VEBTi: 
      !>[A: $tType] : fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A))) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun(vEBT_VEBTi,nat) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun(vEBT_VEBT,fun(vEBT_VEBTi,assn)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun(product_prod(vEBT_VEBT,vEBT_VEBTi),fun(product_prod(vEBT_VEBT,vEBT_VEBTi),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun(vEBT_VEBTi,fun(vEBT_VEBTi,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: ( vEBT_VEBTi * nat ) > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun(vEBT_VEBTi,fun(vEBT_VEBTi,$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__DelImperative_OVEBT__internal_Ovebt__deletei_H,type,
    vEBT_V1365221501068881998eletei: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__DelImperative_Ovebt__deletei,type,
    vEBT_vebt_deletei: ( vEBT_VEBTi * nat ) > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
    vEBT_VEBT_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
    vEBT_VEBT_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Intf__Imperative_Ovebt__assn,type,
    vEBT_Intf_vebt_assn: ( nat * set(nat) ) > fun(vEBT_VEBTi,assn) ).

tff(sy_c_VEBT__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set(nat) * fun(A,fun(B,assn)) * list(A) * list(B) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) ) > fun(list(C),assn) ) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: ( vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: ( vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Uniqueness_OVEBT__internal_OperInsTrans,type,
    vEBT_V6289311342943941716sTrans: ( vEBT_VEBT * vEBT_VEBT ) > $o ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,$o) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Olex__prod,type,
    lex_prod: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),fun(set(A),$o)) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wfrec_Osame__fst,type,
    same_fst: 
      !>[A: $tType,B: $tType] : ( ( fun(A,$o) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).

tff(sy_c_Word_OWord,type,
    word2: 
      !>[A: $tType] : ( int > word(A) ) ).

tff(sy_c_Word_Ois__down,type,
    is_down: 
      !>[A: $tType,B: $tType] : fun(fun(word(A),word(B)),$o) ).

tff(sy_c_Word_Ois__up,type,
    is_up: 
      !>[A: $tType,B: $tType] : fun(fun(word(A),word(B)),$o) ).

tff(sy_c_Word_Opcr__word,type,
    pcr_word: 
      !>[B: $tType] : fun(int,fun(word(B),$o)) ).

tff(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : ( word(A) > word(B) ) ).

tff(sy_c_Word_Oring__1__class_Osigned,type,
    ring_1_signed: 
      !>[B: $tType,A: $tType] : fun(word(B),A) ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned,type,
    semiring_1_unsigned: 
      !>[B: $tType,A: $tType] : fun(word(B),A) ).

tff(sy_c_Word_Osigned__cast,type,
    signed_cast: 
      !>[A: $tType,B: $tType] : fun(word(A),word(B)) ).

tff(sy_c_Word_Osigned__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : fun(nat,fun(word(A),word(A))) ).

tff(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Othe__signed__int,type,
    the_signed_int: 
      !>[A: $tType] : fun(word(A),int) ).

tff(sy_c_Word_Oudvd,type,
    udvd: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

tff(sy_c_Word_Oword__cat,type,
    word_cat: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( word(A) * word(B) ) > word(C) ) ).

tff(sy_c_Word_Oword__int__case,type,
    word_int_case: 
      !>[B: $tType,A: $tType] : ( ( fun(int,B) * word(A) ) > B ) ).

tff(sy_c_Word_Oword__rcat,type,
    word_rcat: 
      !>[A: $tType,B: $tType] : ( list(word(A)) > word(B) ) ).

tff(sy_c_Word_Oword__reverse,type,
    word_reverse: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_Word_Oword__roti,type,
    word_roti: 
      !>[A: $tType] : ( ( int * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__rotl,type,
    word_rotl: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__sle,type,
    word_sle: 
      !>[A: $tType] : fun(word(A),fun(word(A),$o)) ).

tff(sy_c_Word_Oword__sless,type,
    word_sless: 
      !>[A: $tType] : fun(word(A),fun(word(A),$o)) ).

tff(sy_c_Word_Oword__split,type,
    word_split: 
      !>[A: $tType,B: $tType,C: $tType] : ( word(A) > product_prod(word(B),word(C)) ) ).

tff(sy_c_Word_Oword__succ,type,
    word_succ: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fconj,type,
    fconj: fun($o,fun($o,$o)) ).

tff(sy_c_fdisj,type,
    fdisj: fun($o,fun($o,$o)) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_a,type,
    a2: heap_ext(product_unit) ).

tff(sy_v_b,type,
    b2: set(nat) ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_r,type,
    r: vEBT_VEBTi ).

tff(sy_v_s,type,
    s: set(nat) ).

tff(sy_v_t,type,
    t: vEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

tff(sy_v_xa,type,
    xa: nat ).

% Relevant facts (9019)
tff(fact_0_invar__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => vEBT_invar_vebt(vEBT_vebt_delete(T2,Xa),Na) ) ).

% invar_vebt_delete
tff(fact_1_set__vebt__equal,axiom,
    ! [T_1: vEBT_VEBT,Na: nat,T_2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T_1,Na)
     => ( vEBT_invar_vebt(T_2,Na)
       => ( ( T_1 = T_2 )
        <=> ( vEBT_set_vebt(T_1) = vEBT_set_vebt(T_2) ) ) ) ) ).

% set_vebt_equal
tff(fact_2_unique__tree,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_invar_vebt(Sa,Na)
       => ( ( vEBT_set_vebt(T2) = vEBT_set_vebt(Sa) )
         => ( Sa = T2 ) ) ) ) ).

% unique_tree
tff(fact_3_insert_H__pres__valid,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => vEBT_invar_vebt(vEBT_VEBT_insert(T2,Xa),Na) ) ).

% insert'_pres_valid
tff(fact_4_perIT__concat,axiom,
    ! [Sa: vEBT_VEBT,T2: vEBT_VEBT,U: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(Sa,T2)
     => ( vEBT_V6289311342943941716sTrans(T2,U)
       => vEBT_V6289311342943941716sTrans(Sa,U) ) ) ).

% perIT_concat
tff(fact_5_set__vebt__set__vebt_H__valid,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(T2) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_6_uniquetree,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Sa: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_invar_vebt(Sa,Na)
       => ( ( vEBT_VEBT_set_vebt(T2) = vEBT_VEBT_set_vebt(Sa) )
         => ( Sa = T2 ) ) ) ) ).

% uniquetree
tff(fact_7_Rep__assn__inject,axiom,
    ! [Xa: assn,Y: assn] :
      ( ( rep_assn(Xa) = rep_assn(Y) )
    <=> ( Xa = Y ) ) ).

% Rep_assn_inject
tff(fact_8_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
tff(fact_9_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
    <=> ( ( A2 = A3 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
tff(fact_10_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
    <=> vEBT_invar_vebt(T2,D2) ) ).

% valid_eq
tff(fact_11_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(T2,D2)
     => vEBT_VEBT_valid(T2,D2) ) ).

% valid_eq1
tff(fact_12_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
     => vEBT_invar_vebt(T2,D2) ) ).

% valid_eq2
tff(fact_13_set__vebt__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_vebt_member(T2),Xa)
      <=> member(nat,Xa,vEBT_set_vebt(T2)) ) ) ).

% set_vebt_member
tff(fact_14_dele__member__cont__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(T2,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_vebt_member(T2),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_15_dele__bmo__cont__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(T2,Xa)),Y)
      <=> ( ( Xa != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_16_valid__member__both__member__options,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Xa)
       => aa(nat,$o,vEBT_vebt_member(T2),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_17_both__member__options__equiv__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Xa)
      <=> aa(nat,$o,vEBT_vebt_member(T2),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_18_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: array(A),X: list(A),H: heap_ext(product_unit),As: set(nat)] : Xa != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ) ).

% snga_assn_raw.cases
tff(fact_19_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xa: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R: ref(A),X: A,H: heap_ext(product_unit),As: set(nat)] : Xa != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ) ).

% sngr_assn_raw.cases
tff(fact_20_times__assn__raw_Ocases,axiom,
    ! [Xa: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))] :
      ~ ! [P: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H: heap_ext(product_unit),As: set(nat)] : Xa != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),P),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Q),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))) ).

% times_assn_raw.cases
tff(fact_21_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod($o,product_prod(A,set(B)))] :
      ~ ! [B4: $o,H: A,As: set(B)] : Xa != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(B4)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H),As)) ).

% pure_assn_raw.cases
tff(fact_22_prod__induct7,axiom,
    ! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o),Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ( ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F,G2: G] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D3),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),$o,P2,Xa) ) ).

% prod_induct7
tff(fact_23_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o),Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ( ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),$o,P2,Xa) ) ).

% prod_induct6
tff(fact_24_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A4: A,B4: B,C2: C,D3: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P2,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P2,Xa) ) ).

% prod_induct5
tff(fact_25_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),Xa: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A4: A,B4: B,C2: C,D3: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P2,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D3))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P2,Xa) ) ).

% prod_induct4
tff(fact_26_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,C)),$o),Xa: product_prod(A,product_prod(B,C))] :
      ( ! [A4: A,B4: B,C2: C] : aa(product_prod(A,product_prod(B,C)),$o,P2,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)))
     => aa(product_prod(A,product_prod(B,C)),$o,P2,Xa) ) ).

% prod_induct3
tff(fact_27_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))] :
      ~ ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F,G2: G] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),B4),aa(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F,G))),product_prod(C,product_prod(D,product_prod(E,product_prod(F,G))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F,G)))),C2),aa(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G))),aa(D,fun(product_prod(E,product_prod(F,G)),product_prod(D,product_prod(E,product_prod(F,G)))),product_Pair(D,product_prod(E,product_prod(F,G))),D3),aa(product_prod(F,G),product_prod(E,product_prod(F,G)),aa(E,fun(product_prod(F,G),product_prod(E,product_prod(F,G))),product_Pair(E,product_prod(F,G)),E2),aa(G,product_prod(F,G),aa(F,fun(G,product_prod(F,G)),product_Pair(F,G),F2),G2)))))) ).

% prod_cases7
tff(fact_28_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))] :
      ~ ! [A4: A,B4: B,C2: C,D3: D,E2: E,F2: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F)))),B4),aa(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F))),aa(C,fun(product_prod(D,product_prod(E,F)),product_prod(C,product_prod(D,product_prod(E,F)))),product_Pair(C,product_prod(D,product_prod(E,F))),C2),aa(product_prod(E,F),product_prod(D,product_prod(E,F)),aa(D,fun(product_prod(E,F),product_prod(D,product_prod(E,F))),product_Pair(D,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F2))))) ).

% prod_cases6
tff(fact_29_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B4: B,C2: C,D3: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),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))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2)))) ).

% prod_cases5
tff(fact_30_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B4: B,C2: C,D3: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),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)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D3))) ).

% prod_cases4
tff(fact_31_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B4: B,C2: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)) ).

% prod_cases3
tff(fact_32_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
     => ~ ( ( A2 = A3 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
tff(fact_33_prod__cases,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),$o),P3: product_prod(A,B)] :
      ( ! [A4: A,B4: B] : aa(product_prod(A,B),$o,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4))
     => aa(product_prod(A,B),$o,P2,P3) ) ).

% prod_cases
tff(fact_34_surj__pair,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B)] :
    ? [X: A,Y3: B] : P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) ).

% surj_pair
tff(fact_35_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) ).

% old.prod.exhaust
tff(fact_36_one__assn__raw_Ocases,axiom,
    ! [Xa: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H: heap_ext(product_unit),As: set(nat)] : Xa != aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) ).

% one_assn_raw.cases
tff(fact_37_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A2: B,B2: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F1,A2),B2) ).

% old.prod.rec
tff(fact_38_set__vebt__finite,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(set(nat),$o,finite_finite2(nat),vEBT_VEBT_set_vebt(T2)) ) ).

% set_vebt_finite
tff(fact_39_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      ( member(A,A2,aa(fun(A,$o),set(A),collect(A),P2))
    <=> aa(A,$o,P2,A2) ) ).

% mem_Collect_eq
tff(fact_40_Collect__mem__eq,axiom,
    ! [A: $tType,A5: set(A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_a(set(A),fun(A,$o),A5)) = A5 ).

% Collect_mem_eq
tff(fact_41_Collect__cong,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P2,X)
        <=> aa(A,$o,Q2,X) )
     => ( aa(fun(A,$o),set(A),collect(A),P2) = aa(fun(A,$o),set(A),collect(A),Q2) ) ) ).

% Collect_cong
tff(fact_42_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G3: fun(A,B)] :
      ( ! [X: A] : aa(A,B,F3,X) = aa(A,B,G3,X)
     => ( F3 = G3 ) ) ).

% HOL.ext
tff(fact_43_VEBT__internal_OperInsTrans_Ointros_I1_J,axiom,
    ! [T2: vEBT_VEBT] : vEBT_V6289311342943941716sTrans(T2,T2) ).

% VEBT_internal.perInsTrans.intros(1)
tff(fact_44_internal__case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),C3,A2),B2) ).

% internal_case_prod_conv
tff(fact_45_valid__tree__deg__neq__0,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_46_valid__0__not,axiom,
    ! [T2: vEBT_VEBT] : ~ vEBT_invar_vebt(T2,zero_zero(nat)) ).

% valid_0_not
tff(fact_47_min__Null__member,axiom,
    ! [T2: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_minNull(T2)
     => ~ aa(nat,$o,vEBT_vebt_member(T2),Xa) ) ).

% min_Null_member
tff(fact_48_not__min__Null__member,axiom,
    ! [T2: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(T2)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X_1) ) ).

% not_min_Null_member
tff(fact_49_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_50_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,Sa: B,R3: set(product_prod(A,B)),S: B] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),Sa),R3)
     => ( ( S = Sa )
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S),R3) ) ) ).

% ssubst_Pair_rhs
tff(fact_51_bex2I,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,S2: set(product_prod(A,B)),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S2)
     => ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S2)
         => aa(B,$o,aa(A,fun(B,$o),P2,A2),B2) )
       => ? [A4: A,B4: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4),S2)
            & aa(B,$o,aa(A,fun(B,$o),P2,A4),B4) ) ) ) ).

% bex2I
tff(fact_52_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F2: fun(A,B),A4: A,B4: A] : Xa != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4)) ).

% pairself.cases
tff(fact_53_set__vebt__minNull,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
      <=> ( vEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% set_vebt_minNull
tff(fact_54_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_55_finite__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A5: set(A)] : aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_code
tff(fact_56_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% deg_not_0
tff(fact_57_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Na)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Na)),TreeList2,S3) ) ).

% deg_SUcn_Node
tff(fact_58_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_59_uncurry__apply,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,uncurry(B,C,A,F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F3,A2),B2) ).

% uncurry_apply
tff(fact_60_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_61_finite__set__choice,axiom,
    ! [B: $tType,A: $tType,A5: set(A),P2: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),P2,X),X_12) )
       => ? [F2: fun(A,B)] :
          ! [X3: A] :
            ( member(A,X3,A5)
           => aa(B,$o,aa(A,fun(B,$o),P2,X3),aa(A,B,F2,X3)) ) ) ) ).

% finite_set_choice
tff(fact_62_finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A5: set(A)] : aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite
tff(fact_63_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_64_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_65_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_66_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A2,X_1)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),X3) ) ) ) ).

% succ_none_empty
tff(fact_67_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A2,X_1)
     => ( aa(set(nat),$o,finite_finite2(nat),Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),A2) ) ) ) ).

% pred_none_empty
tff(fact_68_memb__imp__not__empty,axiom,
    ! [A: $tType,Xa: A,S2: set(A)] :
      ( member(A,Xa,S2)
     => ( S2 != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_69_set__notEmptyE,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( S2 != bot_bot(set(A)) )
     => ~ ! [X: A] : ~ member(A,X,S2) ) ).

% set_notEmptyE
tff(fact_70_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ( Na != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na) ) ) ).

% gr_zeroI
tff(fact_71_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Na),zero_zero(A)) ) ).

% not_less_zero
tff(fact_72_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na)
         => ( Na != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_73_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_74_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_75_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_76_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_77_infinite__imp__nonempty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ( S2 != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_78_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X)
     => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Ux)
       => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)),X) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_79_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_80_mod__h__bot__indep,axiom,
    ! [P2: assn,H2: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H3),bot_bot(set(nat)))) ) ).

% mod_h_bot_indep
tff(fact_81_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: A] :
          ( ( zero_zero(A) = Xa )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_82_zero__less__Suc,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Na)) ).

% zero_less_Suc
tff(fact_83_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_84_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_85_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% neq0_conv
tff(fact_86_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_87_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_88_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) ) ).

% Suc_mono
tff(fact_89_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_eq
tff(fact_90_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf((A2),$false) ).

% vebt_delete.simps(2)
tff(fact_91_invar__vebt__buildup,axiom,
    ! [Na: nat] :
      ( vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% invar_vebt_buildup
tff(fact_92_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na) ) ).

% buildup_gives_valid
tff(fact_93_T__vebt__buildupi__gq__0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),vEBT_V441764108873111860ildupi(Na)) ).

% T_vebt_buildupi_gq_0
tff(fact_94_set__vebt__buildup,axiom,
    ! [I: nat] : vEBT_set_vebt(vEBT_vebt_buildup(I)) = bot_bot(set(nat)) ).

% set_vebt_buildup
tff(fact_95_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_96_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_97_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X2) = aa(nat,nat,suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_98_obtain__set__pred,axiom,
    ! [Z: nat,Xa: nat,A5: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z),Xa)
     => ( vEBT_VEBT_min_in_set(A5,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),A5)
         => ? [X_1: nat] : vEBT_is_pred_in_set(A5,Xa,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_99_obtain__set__succ,axiom,
    ! [Xa: nat,Z: nat,A5: set(nat),B5: set(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( vEBT_VEBT_max_in_set(A5,Z)
       => ( aa(set(nat),$o,finite_finite2(nat),B5)
         => ( ( A5 = B5 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A5,Xa,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_100_finite__psubset__induct,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A6)
           => ( ! [B6: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B6),A6)
                 => aa(set(A),$o,P2,B6) )
             => aa(set(A),$o,P2,A6) ) )
       => aa(set(A),$o,P2,A5) ) ) ).

% finite_psubset_induct
tff(fact_101_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_102_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X))
                 => aa(A,$o,P2,Y4) )
             => aa(A,$o,P2,X) )
         => aa(A,$o,P2,A2) ) ) ).

% measure_induct_rule
tff(fact_103_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P2: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,Y4)),aa(A,B,F3,X))
                 => aa(A,$o,P2,Y4) )
             => aa(A,$o,P2,X) )
         => aa(A,$o,P2,A2) ) ) ).

% measure_induct
tff(fact_104_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_105_Suc__inject,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xa) = aa(nat,nat,suc,Y) )
     => ( Xa = Y ) ) ).

% Suc_inject
tff(fact_106_infinite__descent__measure,axiom,
    ! [A: $tType,P2: fun(A,$o),V2: fun(A,nat),Xa: A] :
      ( ! [X: A] :
          ( ~ aa(A,$o,P2,X)
         => ? [Y4: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y4)),aa(A,nat,V2,X))
              & ~ aa(A,$o,P2,Y4) ) )
     => aa(A,$o,P2,Xa) ) ).

% infinite_descent_measure
tff(fact_107_linorder__neqE__nat,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( Xa != Y )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_108_infinite__descent,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ~ aa(nat,$o,P2,N)
         => ? [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
              & ~ aa(nat,$o,P2,M2) ) )
     => aa(nat,$o,P2,Na) ) ).

% infinite_descent
tff(fact_109_nat__less__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
             => aa(nat,$o,P2,M2) )
         => aa(nat,$o,P2,N) )
     => aa(nat,$o,P2,Na) ) ).

% nat_less_induct
tff(fact_110_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_irrefl_nat
tff(fact_111_less__not__refl3,axiom,
    ! [Sa: nat,T2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sa),T2)
     => ( Sa != T2 ) ) ).

% less_not_refl3
tff(fact_112_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_113_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_not_refl
tff(fact_114_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ).

% nat_neq_iff
tff(fact_115_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_116_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M3: nat] : Na = aa(nat,nat,suc,M3) ) ).

% not0_implies_Suc
tff(fact_117_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_118_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_119_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_120_zero__induct,axiom,
    ! [P2: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P2,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P2,aa(nat,nat,suc,N))
           => aa(nat,$o,P2,N) )
       => aa(nat,$o,P2,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_121_diff__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,X),zero_zero(nat))
     => ( ! [Y3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,zero_zero(nat)),aa(nat,nat,suc,Y3))
       => ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P2,X),Y3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,aa(nat,nat,suc,X)),aa(nat,nat,suc,Y3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,M),Na) ) ) ) ).

% diff_induct
tff(fact_122_nat__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P2,N)
           => aa(nat,$o,P2,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P2,Na) ) ) ).

% nat_induct
tff(fact_123_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_124_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa(nat,nat,suc,X2) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_125_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_126_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_127_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero(nat) != aa(nat,nat,suc,X2) ).

% nat.distinct(1)
tff(fact_128_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_129_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P2,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P2,aa(nat,nat,suc,I2))
               => aa(nat,$o,P2,I2) ) )
         => aa(nat,$o,P2,I) ) ) ) ).

% strict_inc_induct
tff(fact_130_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P2,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_131_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_132_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_SucD
tff(fact_133_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_134_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M)
    <=> ? [M4: nat] :
          ( ( M = aa(nat,nat,suc,M4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M4) ) ) ).

% Suc_less_eq2
tff(fact_135_Nat_OAll__less__Suc,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Na))
         => aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,Na)
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
           => aa(nat,$o,P2,I3) ) ) ) ).

% Nat.All_less_Suc
tff(fact_136_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_137_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_138_Ex__less__Suc,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Na))
          & aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,Na)
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
            & aa(nat,$o,P2,I3) ) ) ) ).

% Ex_less_Suc
tff(fact_139_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% less_SucI
tff(fact_140_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_141_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( ( aa(nat,nat,suc,M) != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_142_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_143_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_lessD
tff(fact_144_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_145_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P2: fun(A,$o),Xa: A] :
      ( ! [X: A] :
          ( ( aa(A,nat,V2,X) = zero_zero(nat) )
         => aa(A,$o,P2,X) )
     => ( ! [X: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X))
           => ( ~ aa(A,$o,P2,X)
             => ? [Y4: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V2,Y4)),aa(A,nat,V2,X))
                  & ~ aa(A,$o,P2,Y4) ) ) )
       => aa(A,$o,P2,Xa) ) ) ).

% infinite_descent0_measure
tff(fact_146_infinite__descent0,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( ~ aa(nat,$o,P2,N)
             => ? [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                  & ~ aa(nat,$o,P2,M2) ) ) )
       => aa(nat,$o,P2,Na) ) ) ).

% infinite_descent0
tff(fact_147_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( Na != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_148_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_149_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% not_less0
tff(fact_150_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_151_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% gr0I
tff(fact_152_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_153_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Na: nat,M: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Na)),aa(nat,A,F3,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_154_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F3,Na)),aa(nat,A,F3,N2)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_155_Ex__less__Suc2,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Na))
          & aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
            & aa(nat,$o,P2,aa(nat,nat,suc,I3)) ) ) ) ).

% Ex_less_Suc2
tff(fact_156_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ? [M5: nat] : Na = aa(nat,nat,suc,M5) ) ).

% gr0_conv_Suc
tff(fact_157_All__less__Suc2,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Na))
         => aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
           => aa(nat,$o,P2,aa(nat,nat,suc,I3)) ) ) ) ).

% All_less_Suc2
tff(fact_158_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ? [M3: nat] : Na = aa(nat,nat,suc,M3) ) ).

% gr0_implies_Suc
tff(fact_159_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_160_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Na: nat] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Na))) = vEBT_Leaf((A2),(B2)) ).

% vebt_delete.simps(3)
tff(fact_161_vebt__delete_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = vEBT_Leaf($false,(B2)) ).

% vebt_delete.simps(1)
tff(fact_162_succ__member,axiom,
    ! [T2: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T2),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T2),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z2) ) ) ) ).

% succ_member
tff(fact_163_pred__member,axiom,
    ! [T2: vEBT_VEBT,Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),Xa,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(T2),Y)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(T2),Z2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Z2),Xa) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z2),Y) ) ) ) ).

% pred_member
tff(fact_164_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_leaf
tff(fact_165_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_V441764108873111860ildupi(aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_166_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_V441764108873111860ildupi(zero_zero(nat)) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_167_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X)
     => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,Deg2,TreeList2,Summary2)),X) ) ).

% VEBT_internal.insert'.cases
tff(fact_168_forall__finite_I3_J,axiom,
    ! [Xa: nat,P2: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,aa(nat,nat,suc,Xa)))
         => aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Xa))
           => aa(nat,$o,P2,aa(nat,nat,suc,I3)) ) ) ) ).

% forall_finite(3)
tff(fact_169_forall__finite_I2_J,axiom,
    ! [P2: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P2,I3) )
    <=> aa(nat,$o,P2,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_170_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xa: nat,P2: fun(nat,$o)] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,suc,Xa))
         => aa(nat,$o,P2,I3) )
    <=> ( aa(nat,$o,P2,zero_zero(nat))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Xa)
           => aa(nat,$o,P2,aa(nat,nat,suc,I3)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_171_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X4: set(A)] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,X4)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa2) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X4) ) ) ) ).

% infinite_growing
tff(fact_172_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,S2)
                & ~ ? [Xa2: A] :
                      ( member(A,Xa2,S2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_173_empty__Collect__eq,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P2) )
    <=> ! [X5: A] : ~ aa(A,$o,P2,X5) ) ).

% empty_Collect_eq
tff(fact_174_max__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xa)
    <=> ( member(nat,Xa,Xs)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Xa) ) ) ) ).

% max_in_set_def
tff(fact_175_min__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xa)
    <=> ( member(nat,Xa,Xs)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X5) ) ) ) ).

% min_in_set_def
tff(fact_176_empty__iff,axiom,
    ! [A: $tType,C3: A] : ~ member(A,C3,bot_bot(set(A))) ).

% empty_iff
tff(fact_177_all__not__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X5: A] : ~ member(A,X5,A5)
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_178_Collect__empty__eq,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P2) = bot_bot(set(A)) )
    <=> ! [X5: A] : ~ aa(A,$o,P2,X5) ) ).

% Collect_empty_eq
tff(fact_179_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),zero_zero(A))
        <=> ( Na = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_180_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% Suc_le_mono
tff(fact_181_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_182_le0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% le0
tff(fact_183_psubsetD,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( member(A,C3,A5)
       => member(A,C3,B5) ) ) ).

% psubsetD
tff(fact_184_psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C4) ) ) ).

% psubset_trans
tff(fact_185_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P2,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P2,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => ? [X: nat] :
            ( aa(nat,$o,P2,X)
            & ! [Y4: nat] :
                ( aa(nat,$o,P2,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),X) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_186_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% nat_le_linear
tff(fact_187_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_188_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% eq_imp_le
tff(fact_189_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_190_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),Na) ).

% le_refl
tff(fact_191_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => ( ( C3 = D2 )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),D2) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_192_ex__has__least__nat,axiom,
    ! [A: $tType,P2: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P2,K)
     => ? [X: A] :
          ( aa(A,$o,P2,X)
          & ! [Y4: A] :
              ( aa(A,$o,P2,Y4)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X)),aa(A,nat,M,Y4)) ) ) ) ).

% ex_has_least_nat
tff(fact_193_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,Na)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_194_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,Na)),aa(nat,A,F3,N2)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_195_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P2: fun(A,$o),K: A,F3: fun(A,nat),B2: nat] :
      ( aa(A,$o,P2,K)
     => ( ! [Y3: A] :
            ( aa(A,$o,P2,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y3)),B2) )
       => ? [X: A] :
            ( aa(A,$o,P2,X)
            & ! [Y4: A] :
                ( aa(A,$o,P2,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_196_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ).

% zero_le
tff(fact_197_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ? [X: A] :
                ( member(A,X,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_198_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ? [X: A] :
                ( member(A,X,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_199_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_leD
tff(fact_200_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_SucE
tff(fact_201_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_SucI
tff(fact_202_Suc__le__D,axiom,
    ! [Na: nat,M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M6)
     => ? [M3: nat] : M6 = aa(nat,nat,suc,M3) ) ).

% Suc_le_D
tff(fact_203_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        | ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_204_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_205_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_206_full__nat__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M2)),N)
             => aa(nat,$o,P2,M2) )
         => aa(nat,$o,P2,N) )
     => aa(nat,$o,P2,Na) ) ).

% full_nat_induct
tff(fact_207_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,P2,M)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N)
             => ( aa(nat,$o,P2,N)
               => aa(nat,$o,P2,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P2,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_208_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R3: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ! [X: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R3,X),X)
       => ( ! [X: nat,Y3: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R3,X),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R3,Y3),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R3,X),Z3) ) )
         => ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R3,N),aa(nat,nat,suc,N))
           => aa(nat,$o,aa(nat,fun(nat,$o),R3,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_209_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% less_eq_nat.simps(1)
tff(fact_210_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_211_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_212_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),zero_zero(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_213_exists__leI,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ( ! [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),Na)
           => ~ aa(nat,$o,P2,N3) )
       => aa(nat,$o,P2,Na) )
     => ? [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),Na)
          & aa(nat,$o,P2,N4) ) ) ).

% exists_leI
tff(fact_214_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_215_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_imp_le_nat
tff(fact_216_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_217_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_or_eq_imp_le
tff(fact_218_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ( M != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% le_neq_implies_less
tff(fact_219_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F3,I)),aa(nat,nat,F3,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_220_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,A5)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_221_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ? [X: A] :
                ( member(A,X,A5)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2)
                     => ( X = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_222_nat__compl__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N)
               => aa(nat,$o,P2,Nn) )
           => aa(nat,$o,P2,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P2,Na) ) ) ).

% nat_compl_induct
tff(fact_223_nat__compl__induct_H,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N)
               => aa(nat,$o,P2,Nn) )
           => aa(nat,$o,P2,aa(nat,nat,suc,N)) )
       => aa(nat,$o,P2,Na) ) ) ).

% nat_compl_induct'
tff(fact_224_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_225_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_226_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na) ) ).

% Suc_leI
tff(fact_227_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_eq
tff(fact_228_dec__induct,axiom,
    ! [I: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P2,I)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P2,N)
                 => aa(nat,$o,P2,aa(nat,nat,suc,N)) ) ) )
         => aa(nat,$o,P2,J) ) ) ) ).

% dec_induct
tff(fact_229_inc__induct,axiom,
    ! [I: nat,J: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P2,J)
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),J)
               => ( aa(nat,$o,P2,aa(nat,nat,suc,N))
                 => aa(nat,$o,P2,N) ) ) )
         => aa(nat,$o,P2,I) ) ) ) ).

% inc_induct
tff(fact_230_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_lessD
tff(fact_231_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_232_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_Suc_eq_le
tff(fact_233_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_234_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_235_ex__least__nat__le,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,Na)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K2)
               => ~ aa(nat,$o,P2,I4) )
            & aa(nat,$o,P2,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_236_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2),Ux2) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_237_ex__least__nat__less,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,Na)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Na)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K2)
               => ~ aa(nat,$o,P2,I4) )
            & aa(nat,$o,P2,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_238_emptyE,axiom,
    ! [A: $tType,A2: A] : ~ member(A,A2,bot_bot(set(A))) ).

% emptyE
tff(fact_239_equals0D,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( ( A5 = bot_bot(set(A)) )
     => ~ member(A,A2,A5) ) ).

% equals0D
tff(fact_240_equals0I,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [Y3: A] : ~ member(A,Y3,A5)
     => ( A5 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_241_ex__in__conv,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ? [X5: A] : member(A,X5,A5)
    <=> ( A5 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_242_not__psubset__empty,axiom,
    ! [A: $tType,A5: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_243_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X: heap_Time_Heap(A)] : Xa != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),X)
     => ~ ! [N: nat,X: heap_Time_Heap(A)] : Xa != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),aa(nat,nat,suc,N)),X) ) ).

% VEBT_internal.replicatei.cases
tff(fact_244_forall__finite_I1_J,axiom,
    ! [P2: fun(nat,$o),I4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),zero_zero(nat))
     => aa(nat,$o,P2,I4) ) ).

% forall_finite(1)
tff(fact_245_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Tree,Na)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
       => ( vEBT_V5719532721284313246member(Tree,Xa)
          | vEBT_VEBT_membermima(Tree,Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_246_both__member__options__def,axiom,
    ! [T2: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Xa)
    <=> ( vEBT_V5719532721284313246member(T2,Xa)
        | vEBT_VEBT_membermima(T2,Xa) ) ) ).

% both_member_options_def
tff(fact_247_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xa,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),X5)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X5) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_248_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xa: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xa,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
        & ! [X5: nat] :
            ( member(nat,X5,Xs)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),Xa)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_249_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xa: B] : aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ).

% bot_apply
tff(fact_250_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xa) ) ).

% order_refl
tff(fact_251_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_252_mod__h__bot__iff_I3_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),Xa: A,H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(sngr_assn(A,P3,Xa)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ).

% mod_h_bot_iff(3)
tff(fact_253_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_min_max
tff(fact_254_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu2),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_255_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv2))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_256_empty__subsetI,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A5) ).

% empty_subsetI
tff(fact_257_subset__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),bot_bot(set(A)))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_258_psubsetI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( ( A5 != B5 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5) ) ) ).

% psubsetI
tff(fact_259_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_260_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_261_subset__emptyI,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X: A] : ~ member(A,X,A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_262_rev__finite__subset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% rev_finite_subset
tff(fact_263_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_super
tff(fact_264_finite__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_subset
tff(fact_265_mod__false,axiom,
    ! [H2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(bot_bot(assn)),H2) ).

% mod_false
tff(fact_266_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
        | ( A5 = B5 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_267_subset__psubset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B5),C4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C4) ) ) ).

% subset_psubset_trans
tff(fact_268_subset__not__subset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% subset_not_subset_eq
tff(fact_269_psubset__subset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),C4) ) ) ).

% psubset_subset_trans
tff(fact_270_psubset__imp__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% psubset_imp_subset
tff(fact_271_psubset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
        & ( A5 != B5 ) ) ) ).

% psubset_eq
tff(fact_272_psubsetE,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% psubsetE
tff(fact_273_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu2),(Uv2)),Uw2) ).

% VEBT_internal.membermima.simps(1)
tff(fact_274_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_275_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_le_cases
tff(fact_276_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_277_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_278_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_linear
tff(fact_279_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_eq_refl
tff(fact_280_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_subst2
tff(fact_281_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_subst1
tff(fact_282_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_283_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
        <=> ! [X5: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X5)),aa(A,B,G3,X5)) ) ) ).

% le_fun_def
tff(fact_284_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3) ) ) ).

% le_funI
tff(fact_285_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,G3,Xa)) ) ) ).

% le_funE
tff(fact_286_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,G3,Xa)) ) ) ).

% le_funD
tff(fact_287_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_288_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% dual_order.trans
tff(fact_289_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_290_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_291_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
         => ( ! [A4: A,B4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P2,B4),A4)
               => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
           => aa(A,$o,aa(A,fun(A,$o),P2,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_292_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% order_trans
tff(fact_293_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% order.trans
tff(fact_294_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
           => ( Xa = Y ) ) ) ) ).

% order_antisym
tff(fact_295_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_le_eq_trans
tff(fact_296_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% ord_eq_le_trans
tff(fact_297_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% order_class.order_eq_iff
tff(fact_298_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_299_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_300_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_imp_not_less
tff(fact_301_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Y != Xa ) ) ) ).

% order_less_imp_not_eq2
tff(fact_302_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_303_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          | ( Xa = Y )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_less_linear
tff(fact_304_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
           => (P2) ) ) ) ).

% order_less_imp_triv
tff(fact_305_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_not_sym
tff(fact_306_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_subst2
tff(fact_307_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_subst1
tff(fact_308_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Xa) ) ).

% order_less_irrefl
tff(fact_309_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( aa(A,B,F3,B2) = C3 )
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_310_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_311_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_trans
tff(fact_312_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order_less_asym'
tff(fact_313_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neq_iff
tff(fact_314_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% order_less_asym
tff(fact_315_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE
tff(fact_316_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_317_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_318_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_319_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
            | ( Xa = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_320_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans
tff(fact_321_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),B4)
             => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P2,A4),A4)
           => ( ! [A4: A,B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P2,B4),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P2,A4),B4) )
             => aa(A,$o,aa(A,fun(A,$o),P2,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_322_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P2,X_13)
        <=> ? [N5: A] :
              ( aa(A,$o,P2,N5)
              & ! [M5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M5),N5)
                 => ~ aa(A,$o,P2,M5) ) ) ) ) ).

% exists_least_iff
tff(fact_323_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% dual_order.irrefl
tff(fact_324_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% dual_order.asym
tff(fact_325_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ( Xa != Y )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_cases
tff(fact_326_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv3
tff(fact_327_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( ! [X: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X)
                 => aa(A,$o,P2,Y4) )
             => aa(A,$o,P2,X) )
         => aa(A,$o,P2,A2) ) ) ).

% less_induct
tff(fact_328_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( ( B2 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_less_eq_trans
tff(fact_329_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% ord_eq_less_trans
tff(fact_330_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% order.asym
tff(fact_331_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( Xa != Y ) ) ) ).

% less_imp_neq
tff(fact_332_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),Y) ) ) ) ).

% dense
tff(fact_333_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X_1) ) ).

% gt_ex
tff(fact_334_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),Xa) ) ).

% lt_ex
tff(fact_335_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X3: A] : aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_336_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_337_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_le_less_linear
tff(fact_338_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,B2)),C3)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_less_le_subst2
tff(fact_339_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_340_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,B2)),C3)
           => ( ! [X: A,Y3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,A2)),C3) ) ) ) ) ).

% order_le_less_subst2
tff(fact_341_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F3,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B2),C3)
           => ( ! [X: B,Y3: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X)),aa(B,A,F3,Y3)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,C3)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_342_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_le_trans
tff(fact_343_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_le_less_trans
tff(fact_344_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_345_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_346_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% order_less_imp_le
tff(fact_347_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% linorder_not_less
tff(fact_348_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% linorder_not_le
tff(fact_349_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ( Xa != Y ) ) ) ) ).

% order_less_le
tff(fact_350_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
            | ( Xa = Y ) ) ) ) ).

% order_le_less
tff(fact_351_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_352_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_353_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_354_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_355_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_356_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_357_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_358_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_359_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Xa)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_360_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_361_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans2
tff(fact_362_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% order.strict_trans1
tff(fact_363_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_364_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_365_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% not_le_imp_less
tff(fact_366_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ) ).

% less_le_not_le
tff(fact_367_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_368_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_369_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv2
tff(fact_370_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
          <=> ( Xa = Y ) ) ) ) ).

% antisym_conv1
tff(fact_371_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_372_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% leI
tff(fact_373_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% leD
tff(fact_374_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_375_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_376_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_377_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_378_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_379_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_380_deg1Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
    <=> ? [A7: $o,B7: $o] : T2 = vEBT_Leaf((A7),(B7)) ) ).

% deg1Leaf
tff(fact_381_deg__1__Leaf,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_invar_vebt(T2,one_one(nat))
     => ? [A4: $o,B4: $o] : T2 = vEBT_Leaf((A4),(B4)) ) ).

% deg_1_Leaf
tff(fact_382_deg__1__Leafy,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( Na = one_one(nat) )
       => ? [A4: $o,B4: $o] : T2 = vEBT_Leaf((A4),(B4)) ) ) ).

% deg_1_Leafy
tff(fact_383_finite__nat__set__iff__bounded__le,axiom,
    ! [N6: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N6)
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( member(nat,X5,N6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),M5) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_384_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M5: nat] :
        ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N5)
          & member(nat,N5,S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_385_finite__nat__set__iff__bounded,axiom,
    ! [N6: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N6)
    <=> ? [M5: nat] :
        ! [X5: nat] :
          ( member(nat,X5,N6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),M5) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_386_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ! [M5: nat] :
        ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N5)
          & member(nat,N5,S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_387_bounded__nat__set__is__finite,axiom,
    ! [N6: set(nat),Na: nat] :
      ( ! [X: nat] :
          ( member(nat,X,N6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),Na) )
     => aa(set(nat),$o,finite_finite2(nat),N6) ) ).

% bounded_nat_set_is_finite
tff(fact_388_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M3)
         => ? [N7: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N7)
              & member(nat,N7,S2) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S2) ) ).

% unbounded_k_infinite
tff(fact_389_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A4: $o,B4: $o] : Xa != vEBT_Leaf((A4),(B4))
     => ~ ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa != vEBT_Node(Uu,Deg2,TreeList2,Summary2) ) ).

% VEBT_internal.height.cases
tff(fact_390_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ~ ? [X3: A] :
                  ( member(A,X3,S2)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_391_subsetI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => member(A,X,B5) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% subsetI
tff(fact_392_subset__antisym,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( A5 = B5 ) ) ) ).

% subset_antisym
tff(fact_393_less__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),one_one(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_one
tff(fact_394_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G3: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F3),G3)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F3),G3)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G3),F3) ) ) ) ).

% less_fun_def
tff(fact_395_subset__Collect__conv,axiom,
    ! [A: $tType,S2: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(fun(A,$o),set(A),collect(A),P2))
    <=> ! [X5: A] :
          ( member(A,X5,S2)
         => aa(A,$o,P2,X5) ) ) ).

% subset_Collect_conv
tff(fact_396_in__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( member(A,Xa,A5)
       => member(A,Xa,B5) ) ) ).

% in_mono
tff(fact_397_subsetD,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( member(A,C3,A5)
       => member(A,C3,B5) ) ) ).

% subsetD
tff(fact_398_equalityE,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( A5 = B5 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% equalityE
tff(fact_399_subset__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> ! [X5: A] :
          ( member(A,X5,A5)
         => member(A,X5,B5) ) ) ).

% subset_eq
tff(fact_400_equalityD1,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( A5 = B5 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% equalityD1
tff(fact_401_Set_OequalityD2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( A5 = B5 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ).

% Set.equalityD2
tff(fact_402_subset__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> ! [T4: A] :
          ( member(A,T4,A5)
         => member(A,T4,B5) ) ) ).

% subset_iff
tff(fact_403_subset__refl,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),A5) ).

% subset_refl
tff(fact_404_Collect__mono,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P2,X)
         => aa(A,$o,Q2,X) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P2)),aa(fun(A,$o),set(A),collect(A),Q2)) ) ).

% Collect_mono
tff(fact_405_subset__trans,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4) ) ) ).

% subset_trans
tff(fact_406_set__eq__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( A5 = B5 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ) ).

% set_eq_subset
tff(fact_407_Collect__mono__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P2)),aa(fun(A,$o),set(A),collect(A),Q2))
    <=> ! [X5: A] :
          ( aa(A,$o,P2,X5)
         => aa(A,$o,Q2,X5) ) ) ).

% Collect_mono_iff
tff(fact_408_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xa: A] :
          ( ( one_one(A) = Xa )
        <=> ( Xa = one_one(A) ) ) ) ).

% one_reorient
tff(fact_409_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_410_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_411_nat__geq__1__eq__neqz,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Xa)
    <=> ( Xa != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_412_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_413_nat__induct__non__zero,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,P2,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,P2,N)
               => aa(nat,$o,P2,aa(nat,nat,suc,N)) ) )
         => aa(nat,$o,P2,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_414_bounded__Max__nat,axiom,
    ! [P2: fun(nat,$o),Xa: nat,M7: nat] :
      ( aa(nat,$o,P2,Xa)
     => ( ! [X: nat] :
            ( aa(nat,$o,P2,X)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M7) )
       => ~ ! [M3: nat] :
              ( aa(nat,$o,P2,M3)
             => ~ ! [X3: nat] :
                    ( aa(nat,$o,P2,X3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_415_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A2),
          $ite(Xa = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_416_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F3,S2),S2) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_417_finite__transitivity__chain,axiom,
    ! [A: $tType,A5: set(A),R3: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X: A] : ~ aa(A,$o,aa(A,fun(A,$o),R3,X),X)
       => ( ! [X: A,Y3: A,Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y3)
             => ( aa(A,$o,aa(A,fun(A,$o),R3,Y3),Z3)
               => aa(A,$o,aa(A,fun(A,$o),R3,X),Z3) ) )
         => ( ! [X: A] :
                ( member(A,X,A5)
               => ? [Y4: A] :
                    ( member(A,Y4,A5)
                    & aa(A,$o,aa(A,fun(A,$o),R3,X),Y4) ) )
           => ( A5 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_418_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),Y: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( member(A,Y,S2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2))),aa(A,B,F3,Y)) ) ) ) ) ).

% arg_min_least
tff(fact_419_minNull__delete__time__bound_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T2,Xa)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_420_vebt__member__code_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A2),
          $ite(Xa = one_one(nat),(B2),$false) ) ) ).

% vebt_member_code(1)
tff(fact_421_vebt__delete__code_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),Xa) = $ite(
        Xa = zero_zero(nat),
        vEBT_Leaf($false,(B2)),
        $ite(Xa = one_one(nat),vEBT_Leaf((A2),$false),vEBT_Leaf((A2),(B2))) ) ).

% vebt_delete_code(1)
tff(fact_422_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_423_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_424_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_425_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_426_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_427_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_428_bot__empty__eq,axiom,
    ! [A: $tType,X3: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X3)
    <=> member(A,X3,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_429_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P2) = bot_bot(set(A)) )
    <=> ( P2 = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_430_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_431_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_432_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),Sa: set(product_prod(A,B))] :
      ( ! [X: A,Y3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3),Sa) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),Sa) ) ).

% subrelI
tff(fact_433_mod__emp__simp,axiom,
    ! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(one_one(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).

% mod_emp_simp
tff(fact_434_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_435_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_436_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_437_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_438_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_439_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_440_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_441_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Na: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Na))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_442_nat__descend__induct,axiom,
    ! [Na: nat,P2: fun(nat,$o),M: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K2)
         => aa(nat,$o,P2,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
           => ( ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I4)
                 => aa(nat,$o,P2,I4) )
             => aa(nat,$o,P2,K2) ) )
       => aa(nat,$o,P2,M) ) ) ).

% nat_descend_induct
tff(fact_443_exists__least__lemma,axiom,
    ! [P2: fun(nat,$o)] :
      ( ~ aa(nat,$o,P2,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P2,X_12)
       => ? [N: nat] :
            ( ~ aa(nat,$o,P2,N)
            & aa(nat,$o,P2,aa(nat,nat,suc,N)) ) ) ) ).

% exists_least_lemma
tff(fact_444_dependent__nat__choice,axiom,
    ! [A: $tType,P2: fun(nat,fun(A,$o)),Q2: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P2,zero_zero(nat)),X_12)
     => ( ! [X: A,N: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P2,N),X)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P2,aa(nat,nat,suc,N)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q2,N),X),Y4) ) )
       => ? [F2: fun(nat,A)] :
          ! [N7: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P2,N7),aa(nat,A,F2,N7))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q2,N7),aa(nat,A,F2,N7)),aa(nat,A,F2,aa(nat,nat,suc,N7))) ) ) ) ).

% dependent_nat_choice
tff(fact_445_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ( ( Xa != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : Xa != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_446_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ~ ! [N: nat] : Xa != aa(nat,nat,suc,N) ) ).

% list_decode.cases
tff(fact_447_mod__h__bot__iff_I4_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Q3: array(A),Y: list(A),H2: heap_ext(product_unit)] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(list(A),assn,snga_assn(A,Q3),Y)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ).

% mod_h_bot_iff(4)
tff(fact_448_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,P2,A2)
           => ( ~ aa(A,$o,P2,B2)
             => ? [C2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
                  & ! [X3: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),C2) )
                     => aa(A,$o,P2,X3) )
                  & ! [D4: A] :
                      ( ! [X: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D4) )
                         => aa(A,$o,P2,X) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D4),C2) ) ) ) ) ) ) ).

% complete_interval
tff(fact_449_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% verit_comp_simplify1(3)
tff(fact_450_less__by__empty,axiom,
    ! [A: $tType,A5: set(product_prod(A,A)),B5: set(product_prod(A,A))] :
      ( ( A5 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A5),B5) ) ).

% less_by_empty
tff(fact_451_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_452_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_453_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_454_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B4)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) ) ) ).

% ex_gt_or_lt
tff(fact_455_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X3) ) ) ).

% minf(8)
tff(fact_456_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),T2) ) ) ).

% minf(6)
tff(fact_457_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),X3) ) ) ).

% pinf(8)
tff(fact_458_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),T2) ) ) ).

% pinf(6)
tff(fact_459_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E2),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_460_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_461_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_462_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_463_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xa) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_464_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = Xa )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_465_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_466_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_467_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_468_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_469_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_470_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_471_Suc__diff__diff,axiom,
    ! [M: nat,Na: 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)),Na)),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),Na)),K) ).

% Suc_diff_diff
tff(fact_472_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% diff_Suc_Suc
tff(fact_473_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_474_diff__0__eq__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_475_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_476_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_477_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_478_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_479_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_480_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_481_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% zero_less_diff
tff(fact_482_diff__Suc__1,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,Na)),one_one(nat)) = Na ).

% diff_Suc_1
tff(fact_483_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_484_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% diff_is_0_eq
tff(fact_485_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_486_Suc__diff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_487_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D2) )
         => ( ( A2 = B2 )
          <=> ( C3 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_488_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_489_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_490_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),Xa)
         => ~ ! [M8: A] :
                ( ( Xa = aa(A,option(A),some(A),M8) )
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),M8) ) ) ) ).

% le_some_optE
tff(fact_491_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_492_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_mono
tff(fact_493_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_left_mono
tff(fact_494_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_right_mono
tff(fact_495_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_496_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) ) ) ).

% diff_strict_right_mono
tff(fact_497_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_498_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_499_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_500_zero__induct__lemma,axiom,
    ! [P2: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P2,K)
     => ( ! [N: nat] :
            ( aa(nat,$o,P2,aa(nat,nat,suc,N))
           => aa(nat,$o,P2,N) )
       => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),I)) ) ) ).

% zero_induct_lemma
tff(fact_501_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) = zero_zero(nat) )
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M) = zero_zero(nat) )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_502_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_503_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_504_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M)) ) ) ).

% diff_less_mono2
tff(fact_505_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_506_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% le_diff_iff
tff(fact_507_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( 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),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_508_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),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),Na),L)) ) ).

% diff_le_mono
tff(fact_509_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),M) ).

% diff_le_self
tff(fact_510_le__diff__iff_H,axiom,
    ! [A2: nat,C3: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),A2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),C3),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_511_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M)) ) ).

% diff_le_mono2
tff(fact_512_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_513_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_514_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_515_Suc__to__right,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,suc,Na) = M )
     => ( Na = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_516_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_517_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_518_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),M) ) ) ).

% diff_less
tff(fact_519_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% Suc_diff_le
tff(fact_520_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,Na)) = 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))),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_521_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% less_diff_iff
tff(fact_522_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),A2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),C3)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),C3)) ) ) ).

% diff_less_mono
tff(fact_523_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_524_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_525_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xa)) ) ).

% frac_ge_0
tff(fact_526_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xa)),one_one(A)) ) ).

% frac_lt_1
tff(fact_527_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_528_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P4: fun(A,$o),Q2: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P2,X)
              <=> aa(A,$o,P4,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q2,X)
                <=> aa(A,$o,Q4,X) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q2,X3) )
                <=> ( aa(A,$o,P4,X3)
                    & aa(A,$o,Q4,X3) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_529_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P4: fun(A,$o),Q2: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
             => ( aa(A,$o,P2,X)
              <=> aa(A,$o,P4,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z4),X)
               => ( aa(A,$o,Q2,X)
                <=> aa(A,$o,Q4,X) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q2,X3) )
                <=> ( aa(A,$o,P4,X3)
                    | aa(A,$o,Q4,X3) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_530_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ( X3 != T2 ) ) ) ).

% pinf(3)
tff(fact_531_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ( X3 != T2 ) ) ) ).

% pinf(4)
tff(fact_532_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),T2) ) ) ).

% pinf(5)
tff(fact_533_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X3) ) ) ).

% pinf(7)
tff(fact_534_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ( F4 = F4 ) ) ) ).

% pinf(11)
tff(fact_535_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P4: fun(A,$o),Q2: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P2,X)
              <=> aa(A,$o,P4,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q2,X)
                <=> aa(A,$o,Q4,X) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( ( aa(A,$o,P2,X3)
                    & aa(A,$o,Q2,X3) )
                <=> ( aa(A,$o,P4,X3)
                    & aa(A,$o,Q4,X3) ) ) ) ) ) ) ).

% minf(1)
tff(fact_536_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o),P4: fun(A,$o),Q2: fun(A,$o),Q4: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
             => ( aa(A,$o,P2,X)
              <=> aa(A,$o,P4,X) ) )
         => ( ? [Z4: A] :
              ! [X: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z4)
               => ( aa(A,$o,Q2,X)
                <=> aa(A,$o,Q4,X) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( ( aa(A,$o,P2,X3)
                    | aa(A,$o,Q2,X3) )
                <=> ( aa(A,$o,P4,X3)
                    | aa(A,$o,Q4,X3) ) ) ) ) ) ) ).

% minf(2)
tff(fact_537_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ( X3 != T2 ) ) ) ).

% minf(3)
tff(fact_538_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ( X3 != T2 ) ) ) ).

% minf(4)
tff(fact_539_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),T2) ) ) ).

% minf(5)
tff(fact_540_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),X3) ) ) ).

% minf(7)
tff(fact_541_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F4: B] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ( F4 = F4 ) ) ) ).

% minf(11)
tff(fact_542_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_543_vebt__member_Osimps_I3_J,axiom,
    ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz)),Xa) ).

% vebt_member.simps(3)
tff(fact_544_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( ! [Uv: $o] : Xa != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xa != vEBT_Leaf((Uu),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_545_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va2,Vb),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_546_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_547_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_548_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_549_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_550_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_551_vebt__member_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xa) ).

% vebt_member.simps(4)
tff(fact_552_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary),Xa) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_553_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Na: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Na))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_554_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_555_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% less_option_Some
tff(fact_556_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),Xa)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% less_eq_option_Some
tff(fact_557_Suc__if__eq,axiom,
    ! [A: $tType,F3: fun(nat,A),H2: fun(nat,A),G3: A,Na: nat] :
      ( ! [N: nat] : aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(nat,A,H2,N)
     => ( ( aa(nat,A,F3,zero_zero(nat)) = G3 )
       => ( aa(nat,A,F3,Na) = $ite(Na = zero_zero(nat),G3,aa(nat,A,H2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_558_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
             => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_559_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% diff_shunt_var
tff(fact_560_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),K),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),M)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_561_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_562_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Uw)
     => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X)
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)),X)
           => ~ ! [V: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)),X) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_563_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))
     => ( ! [A4: $o,B4: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B4: $o,N: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))
         => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Uu)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),X)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),X)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_564_diff__diff__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na) ) ) ).

% diff_diff_less
tff(fact_565_bijective__Empty,axiom,
    ! [B: $tType,A: $tType] : bijective(A,B,bot_bot(set(product_prod(A,B)))) ).

% bijective_Empty
tff(fact_566_Diff__empty,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),bot_bot(set(A))) = A5 ).

% Diff_empty
tff(fact_567_empty__Diff,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),bot_bot(set(A))),A5) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_568_Diff__cancel,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),A5) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_569_finite__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff
tff(fact_570_finite__Diff2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
      <=> aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_Diff2
tff(fact_571_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% Diff_eq_empty_iff
tff(fact_572_lesseq__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_573_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xa) ) ).

% less_eq_option_None_code
tff(fact_574_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),none(A)) ) ).

% less_option_None
tff(fact_575_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V3: option(product_prod(A,B))] :
      ( ! [X5: A,Y5: B] : V3 != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y5))
    <=> ( V3 = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_576_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),Xa)),none(A)) ) ).

% less_eq_option_Some_None
tff(fact_577_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),aa(A,option(A),some(A),Xa)) ) ).

% less_option_None_Some_code
tff(fact_578_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xa) ) ).

% less_eq_option_None
tff(fact_579_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xa),none(A))
         => ( Xa = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_580_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),aa(A,option(A),some(A),Xa)) ) ).

% less_option_None_Some
tff(fact_581_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),none(A)),Xa)
         => ? [Z3: A] : Xa = aa(A,option(A),some(A),Z3) ) ) ).

% less_option_None_is_Some
tff(fact_582_Diff__infinite__finite,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),T3)
     => ( ~ aa(set(A),$o,finite_finite2(A),S2)
       => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3)) ) ) ).

% Diff_infinite_finite
tff(fact_583_Diff__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),D5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D5),B5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),D5)) ) ) ).

% Diff_mono
tff(fact_584_Diff__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),A5) ).

% Diff_subset
tff(fact_585_double__diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5)) = A5 ) ) ) ).

% double_diff
tff(fact_586_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
     => ? [B4: A] : member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) ) ).

% psubset_imp_ex_mem
tff(fact_587_subset__minus__empty,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_588_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu2: nat] : vEBT_vebt_delete(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu2) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ).

% vebt_delete.simps(4)
tff(fact_589_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_590_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xa) ).

% vebt_member.simps(2)
tff(fact_591_vebt__member__code_I2_J,axiom,
    ! [T2: nat,R2: list(vEBT_VEBT),E3: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),T2,R2,E3)),Xa) ).

% vebt_member_code(2)
tff(fact_592_bijective__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] :
      ( bijective(A,B,R3)
    <=> ( ! [X5: A,Y5: B,Z2: B] :
            ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y5),R3)
              & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Z2),R3) )
           => ( Y5 = Z2 ) )
        & ! [X5: A,Y5: A,Z2: B] :
            ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Z2),R3)
              & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y5),Z2),R3) )
           => ( X5 = Y5 ) ) ) ) ).

% bijective_def
tff(fact_593_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_594_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_595_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu2: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu2) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_596_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu2: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Uu2) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_597_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Y) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : Xa = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu: $o] : Xa = vEBT_Leaf((Uu),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_598_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,Uv: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))
     => ( ! [A4: $o,Uw: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B4: $o,Va: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))
         => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Vb2)
           => ( ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)),Vf)
             => ( ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Vj)
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_599_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu: $o,B4: $o] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))
     => ( ! [Uv: $o,Uw: $o,N: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))
       => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Va3)
         => ( ! [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)),Ve)
           => ( ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Vi)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_600_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S3)),X)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)),X)
         => ( ! [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)),X)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_601_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B4: $o,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),X)
     => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),X)
       => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)),X)
         => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X)
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT,X: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),X) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_602_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ! [A4: $o,B4: $o] : Xa != vEBT_Leaf((A4),(B4))
     => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_603_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_604_delete__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_605_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : Xa != vEBT_Leaf($true,(Uv))
       => ( ! [Uu: $o] : Xa != vEBT_Leaf((Uu),$true)
         => ( ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_606_insertCI,axiom,
    ! [A: $tType,A2: A,B5: set(A),B2: A] :
      ( ( ~ member(A,A2,B5)
       => ( A2 = B2 ) )
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) ) ).

% insertCI
tff(fact_607_insert__iff,axiom,
    ! [A: $tType,A2: A,B2: A,A5: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A5))
    <=> ( ( A2 = B2 )
        | member(A,A2,A5) ) ) ).

% insert_iff
tff(fact_608_insert__absorb2,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5) ).

% insert_absorb2
tff(fact_609_DiffI,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,A5)
     => ( ~ member(A,C3,B5)
       => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ) ).

% DiffI
tff(fact_610_Diff__iff,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
    <=> ( member(A,C3,A5)
        & ~ member(A,C3,B5) ) ) ).

% Diff_iff
tff(fact_611_Diff__idemp,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ).

% Diff_idemp
tff(fact_612_set__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_set_vebt(vEBT_vebt_delete(T2,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% set_vebt_delete
tff(fact_613_delete__correct_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(T2,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),vEBT_VEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_614_singletonI,axiom,
    ! [A: $tType,A2: A] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% singletonI
tff(fact_615_finite__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))
    <=> aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_insert
tff(fact_616_insert__subset,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),B5)
    <=> ( member(A,Xa,B5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ) ).

% insert_subset
tff(fact_617_Diff__insert0,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      ( ~ member(A,Xa,A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ) ) ).

% Diff_insert0
tff(fact_618_insert__Diff1,axiom,
    ! [A: $tType,Xa: A,B5: set(A),A5: set(A)] :
      ( member(A,Xa,B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ) ) ).

% insert_Diff1
tff(fact_619_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A5: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_620_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_621_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) ).

% insert_Diff_single
tff(fact_622_finite__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) ) ).

% finite_Diff_insert
tff(fact_623_DiffE,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => ~ ( member(A,C3,A5)
         => member(A,C3,B5) ) ) ).

% DiffE
tff(fact_624_DiffD1,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => member(A,C3,A5) ) ).

% DiffD1
tff(fact_625_DiffD2,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))
     => ~ member(A,C3,B5) ) ).

% DiffD2
tff(fact_626_insert__Diff__if,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),B5) = $ite(member(A,Xa,B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ).

% insert_Diff_if
tff(fact_627_insertE,axiom,
    ! [A: $tType,A2: A,B2: A,A5: set(A)] :
      ( member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),A5))
     => ( ( A2 != B2 )
       => member(A,A2,A5) ) ) ).

% insertE
tff(fact_628_insertI1,axiom,
    ! [A: $tType,A2: A,B5: set(A)] : member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) ).

% insertI1
tff(fact_629_insertI2,axiom,
    ! [A: $tType,A2: A,B5: set(A),B2: A] :
      ( member(A,A2,B5)
     => member(A,A2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) ) ).

% insertI2
tff(fact_630_Set_Oset__insert,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] :
      ( member(A,Xa,A5)
     => ~ ! [B8: set(A)] :
            ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B8) )
           => member(A,Xa,B8) ) ) ).

% Set.set_insert
tff(fact_631_insert__ident,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      ( ~ member(A,Xa,A5)
     => ( ~ member(A,Xa,B5)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5) )
        <=> ( A5 = B5 ) ) ) ) ).

% insert_ident
tff(fact_632_insert__absorb,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = A5 ) ) ).

% insert_absorb
tff(fact_633_insert__eq__iff,axiom,
    ! [A: $tType,A2: A,A5: set(A),B2: A,B5: set(A)] :
      ( ~ member(A,A2,A5)
     => ( ~ member(A,B2,B5)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5) )
        <=> $ite(
              A2 = B2,
              A5 = B5,
              ? [C5: set(A)] :
                ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),C5) )
                & ~ member(A,B2,C5)
                & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C5) )
                & ~ member(A,A2,C5) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_634_insert__commute,axiom,
    ! [A: $tType,Xa: A,Y: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),A5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) ).

% insert_commute
tff(fact_635_mk__disjoint__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => ? [B8: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B8) )
          & ~ member(A,A2,B8) ) ) ).

% mk_disjoint_insert
tff(fact_636_singleton__inject,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
     => ( A2 = B2 ) ) ).

% singleton_inject
tff(fact_637_insert__not__empty,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_638_doubleton__eq__iff,axiom,
    ! [A: $tType,A2: A,B2: A,C3: A,D2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),D2),bot_bot(set(A)))) )
    <=> ( ( ( A2 = C3 )
          & ( B2 = D2 ) )
        | ( ( A2 = D2 )
          & ( B2 = C3 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_639_singleton__iff,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( B2 = A2 ) ) ).

% singleton_iff
tff(fact_640_singletonD,axiom,
    ! [A: $tType,B2: A,A2: A] :
      ( member(A,B2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
     => ( B2 = A2 ) ) ).

% singletonD
tff(fact_641_insert__minus__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,A5: set(A)] :
      ( ( Xa != Y )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_642_set__minus__singleton__eq,axiom,
    ! [A: $tType,Xa: A,X4: set(A)] :
      ( ~ member(A,Xa,X4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = X4 ) ) ).

% set_minus_singleton_eq
tff(fact_643_Diff__insert__absorb,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] :
      ( ~ member(A,Xa,A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = A5 ) ) ).

% Diff_insert_absorb
tff(fact_644_Diff__insert2,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_645_insert__Diff,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = A5 ) ) ).

% insert_Diff
tff(fact_646_Diff__insert,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_647_finite_OinsertI,axiom,
    ! [A: $tType,A5: set(A),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) ) ).

% finite.insertI
tff(fact_648_subset__insertI2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) ) ).

% subset_insertI2
tff(fact_649_subset__insertI,axiom,
    ! [A: $tType,B5: set(A),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) ).

% subset_insertI
tff(fact_650_subset__insert,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      ( ~ member(A,Xa,A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ) ).

% subset_insert
tff(fact_651_insert__mono,axiom,
    ! [A: $tType,C4: set(A),D5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),D5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),D5)) ) ).

% insert_mono
tff(fact_652_insert__subsetI,axiom,
    ! [A: $tType,Xa: A,A5: set(A),X4: set(A)] :
      ( member(A,Xa,A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),A5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),X4)),A5) ) ) ).

% insert_subsetI
tff(fact_653_subset__Diff__insert,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),Xa: A,C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),C4)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4))
        & ~ member(A,Xa,A5) ) ) ).

% subset_Diff_insert
tff(fact_654_finite_Ocases,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
     => ( ( A2 != bot_bot(set(A)) )
       => ~ ! [A6: set(A)] :
              ( ? [A4: A] : A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),A6)
             => ~ aa(set(A),$o,finite_finite2(A),A6) ) ) ) ).

% finite.cases
tff(fact_655_finite_Osimps,axiom,
    ! [A: $tType,A2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A2)
    <=> ( ( A2 = bot_bot(set(A)) )
        | ? [A8: set(A),A7: A] :
            ( ( A2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A7),A8) )
            & aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.simps
tff(fact_656_finite__induct,axiom,
    ! [A: $tType,F4: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ member(A,X,F5)
               => ( aa(set(A),$o,P2,F5)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) )
         => aa(set(A),$o,P2,F4) ) ) ) ).

% finite_induct
tff(fact_657_finite__ne__induct,axiom,
    ! [A: $tType,F4: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( F4 != bot_bot(set(A)) )
       => ( ! [X: A] : aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))
         => ( ! [X: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( ( F5 != bot_bot(set(A)) )
                 => ( ~ member(A,X,F5)
                   => ( aa(set(A),$o,P2,F5)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_ne_induct
tff(fact_658_infinite__finite__induct,axiom,
    ! [A: $tType,P2: fun(set(A),$o),A5: set(A)] :
      ( ! [A6: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A6)
         => aa(set(A),$o,P2,A6) )
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [X: A,F5: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F5)
             => ( ~ member(A,X,F5)
               => ( aa(set(A),$o,P2,F5)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),F5)) ) ) )
         => aa(set(A),$o,P2,A5) ) ) ) ).

% infinite_finite_induct
tff(fact_659_infinite__remove,axiom,
    ! [A: $tType,S2: set(A),A2: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_660_infinite__coinduct,axiom,
    ! [A: $tType,X4: fun(set(A),$o),A5: set(A)] :
      ( aa(set(A),$o,X4,A5)
     => ( ! [A6: set(A)] :
            ( aa(set(A),$o,X4,A6)
           => ? [X3: A] :
                ( member(A,X3,A6)
                & ( aa(set(A),$o,X4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% infinite_coinduct
tff(fact_661_finite__empty__induct,axiom,
    ! [A: $tType,A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,P2,A5)
       => ( ! [A4: A,A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( member(A,A4,A6)
               => ( aa(set(A),$o,P2,A6)
                 => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P2,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_662_subset__singleton__iff,axiom,
    ! [A: $tType,X4: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))
    <=> ( ( X4 = bot_bot(set(A)) )
        | ( X4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_663_subset__singletonD,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))
     => ( ( A5 = bot_bot(set(A)) )
        | ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_664_subset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xa: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5))
    <=> $ite(member(A,Xa,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ).

% subset_insert_iff
tff(fact_665_Diff__single__insert,axiom,
    ! [A: $tType,A5: set(A),Xa: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5)) ) ).

% Diff_single_insert
tff(fact_666_remove__subset,axiom,
    ! [A: $tType,Xa: A,S2: set(A)] :
      ( member(A,Xa,S2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),S2) ) ).

% remove_subset
tff(fact_667_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),P2: fun(set(A),$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [X: A,S4: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S4)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S4)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X)) )
                   => ( aa(set(A),$o,P2,S4)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S4)) ) ) )
             => aa(set(A),$o,P2,S2) ) ) ) ) ).

% finite_ranking_induct
tff(fact_668_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P2: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [B4: A,A6: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A6)
                 => ( ! [X3: A] :
                        ( member(A,X3,A6)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B4) )
                   => ( aa(set(A),$o,P2,A6)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A6)) ) ) )
             => aa(set(A),$o,P2,A5) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_669_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),P2: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [B4: A,A6: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A6)
                 => ( ! [X3: A] :
                        ( member(A,X3,A6)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),X3) )
                   => ( aa(set(A),$o,P2,A6)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),A6)) ) ) )
             => aa(set(A),$o,P2,A5) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_670_finite__subset__induct,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A5)
       => ( aa(set(A),$o,P2,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A5)
                 => ( ~ member(A,A4,F5)
                   => ( aa(set(A),$o,P2,F5)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_subset_induct
tff(fact_671_finite__subset__induct_H,axiom,
    ! [A: $tType,F4: set(A),A5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A5)
       => ( aa(set(A),$o,P2,bot_bot(set(A)))
         => ( ! [A4: A,F5: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F5)
               => ( member(A,A4,A5)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A5)
                   => ( ~ member(A,A4,F5)
                     => ( aa(set(A),$o,P2,F5)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A4),F5)) ) ) ) ) )
           => aa(set(A),$o,P2,F4) ) ) ) ) ).

% finite_subset_induct'
tff(fact_672_remove__induct,axiom,
    ! [A: $tType,P2: fun(set(A),$o),B5: set(A)] :
      ( aa(set(A),$o,P2,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B5)
         => aa(set(A),$o,P2,B5) )
       => ( ! [A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( ( A6 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),B5)
                 => ( ! [X3: A] :
                        ( member(A,X3,A6)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P2,A6) ) ) ) )
         => aa(set(A),$o,P2,B5) ) ) ) ).

% remove_induct
tff(fact_673_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [A6: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A6)
             => ( ( A6 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),B5)
                 => ( ! [X3: A] :
                        ( member(A,X3,A6)
                       => aa(set(A),$o,P2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P2,A6) ) ) ) )
         => aa(set(A),$o,P2,B5) ) ) ) ).

% finite_remove_induct
tff(fact_674_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P2: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(A),$o,P2,bot_bot(set(A)))
       => ( ! [T5: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T5),S2)
             => ( aa(set(A),$o,P2,T5)
               => ? [X3: A] :
                    ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T5))
                    & aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),T5)) ) ) )
         => aa(set(A),$o,P2,S2) ) ) ) ).

% finite_induct_select
tff(fact_675_psubset__insert__iff,axiom,
    ! [A: $tType,A5: set(A),Xa: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),B5))
    <=> $ite(
          member(A,Xa,B5),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5),
          $ite(member(A,Xa,A5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)) ) ) ).

% psubset_insert_iff
tff(fact_676_finite__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A5 ) ).

% finite_list
tff(fact_677_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B5)
    <=> ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => member(A,X5,B5) ) ) ).

% subset_code(1)
tff(fact_678_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ma),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_679_pred__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,Xa) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(T2),Xa,Px) ) ) ).

% pred_corr
tff(fact_680_pred__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),Xa,Sx) ) ) ).

% pred_correct
tff(fact_681_set__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Px: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,Xa) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(T2),Xa,Px) ) ) ).

% set_vebt_pred
tff(fact_682_succ__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(T2),Xa,Sx) ) ) ).

% succ_corr
tff(fact_683_succ__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Sx: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,Xa) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(T2),Xa,Sx) ) ) ).

% succ_correct
tff(fact_684_maxt__sound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),Xa)
       => ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Xa) ) ) ) ).

% maxt_sound
tff(fact_685_maxt__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Xa) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(T2),Xa) ) ) ).

% maxt_corr
tff(fact_686_set__vebt__maxt,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Xa) )
      <=> vEBT_VEBT_max_in_set(vEBT_set_vebt(T2),Xa) ) ) ).

% set_vebt_maxt
tff(fact_687_mint__sound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),Xa)
       => ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Xa) ) ) ) ).

% mint_sound
tff(fact_688_maxbmo,axiom,
    ! [T2: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Xa) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Xa) ) ).

% maxbmo
tff(fact_689_minNullmin,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T2)
     => ( vEBT_vebt_mint(T2) = none(nat) ) ) ).

% minNullmin
tff(fact_690_minminNull,axiom,
    ! [T2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(T2) = none(nat) )
     => vEBT_VEBT_minNull(T2) ) ).

% minminNull
tff(fact_691_vebt__minNull__mint,axiom,
    ! [T2: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(T2)
    <=> ( vEBT_vebt_mint(T2) = none(nat) ) ) ).

% vebt_minNull_mint
tff(fact_692_mint__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T2),Maxi) ) ) ).

% mint_member
tff(fact_693_maxt__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Maxi: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(T2),Maxi) ) ) ).

% maxt_member
tff(fact_694_mint__corr__help,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Mini: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(T2),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),Xa) ) ) ) ).

% mint_corr_help
tff(fact_695_maxt__corr__help,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Maxi: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(T2),Xa)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_696_set__vebt__mint_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Xa) )
      <=> ( member(nat,Xa,vEBT_set_vebt(T2))
          & ! [X5: nat] :
              ( member(nat,X5,vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),X5) ) ) ) ) ).

% set_vebt_mint'
tff(fact_697_set__vebt__maxt_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = aa(nat,option(nat),some(nat),Xa) )
      <=> ( member(nat,Xa,vEBT_set_vebt(T2))
          & ! [X5: nat] :
              ( member(nat,X5,vEBT_set_vebt(T2))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Xa) ) ) ) ) ).

% set_vebt_maxt'
tff(fact_698_mint__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_699_maxt__corr__help__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_maxt(T2) = none(nat) )
       => ( vEBT_VEBT_set_vebt(T2) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_700_set__vebt__mint,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Xa) )
      <=> vEBT_VEBT_min_in_set(vEBT_set_vebt(T2),Xa) ) ) ).

% set_vebt_mint
tff(fact_701_mint__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_mint(T2) = aa(nat,option(nat),some(nat),Xa) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(T2),Xa) ) ) ).

% mint_corr
tff(fact_702_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,Na)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_703_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_704_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Va2: nat] : vEBT_vebt_succ(vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz),Va2) = none(nat) ).

% vebt_succ.simps(3)
tff(fact_705_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz,Va2),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_706_vebt__succ_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2),Ve2) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_707_vebt__pred_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2),Vf2) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_708_vebt__succ_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_709_vebt__pred_Osimps_I6_J,axiom,
    ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2),Vj2) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_710_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_711_vebt__succ__code_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_vebt_succ(vEBT_Leaf((A2),(B2)),Xa) = $ite(
        ( (B2)
        & ( Xa = zero_zero(nat) ) ),
        aa(nat,option(nat),some(nat),one_one(nat)),
        none(nat) ) ).

% vebt_succ_code(1)
tff(fact_712_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_713_vebt__pred__code,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),Xa) = $ite(
        Xa = zero_zero(nat),
        none(nat),
        $ite(
          Xa = one_one(nat),
          $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)),
          $ite(
            (B2),
            aa(nat,option(nat),some(nat),one_one(nat)),
            $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) ).

% vebt_pred_code
tff(fact_714_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va2: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_715_vebt__maxt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_716_vebt__mint_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_717_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A2),(B2))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_718_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A2),(B2))) = $ite(
        (A2),
        aa(nat,option(nat),some(nat),zero_zero(nat)),
        $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ).

% vebt_mint.simps(1)
tff(fact_719_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_720_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_721_set__vebt__mint_H_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_mint(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),aa(nat,option(nat),some(nat),aa(set(nat),nat,lattic643756798350308766er_Min(nat),vEBT_set_vebt(T2)))) ) ) ).

% set_vebt_mint''
tff(fact_722_set__vebt__maxt_H_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_maxt(T2) = $ite(vEBT_set_vebt(T2) = bot_bot(set(nat)),none(nat),aa(nat,option(nat),some(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),vEBT_set_vebt(T2)))) ) ) ).

% set_vebt_maxt''
tff(fact_723_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_724_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_725_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Max_singleton
tff(fact_726_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Min_singleton
tff(fact_727_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xa)
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_728_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xa)
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Xa) ) ) ) ) ) ).

% Max_less_iff
tff(fact_729_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X5) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_730_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X5) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_731_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),A2) ) ) ) ).

% Min.coboundedI
tff(fact_732_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [Y3: A] :
                ( member(A,Y3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y3) )
           => ( member(A,Xa,A5)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = Xa ) ) ) ) ) ).

% Min_eqI
tff(fact_733_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xa) ) ) ) ).

% Min_le
tff(fact_734_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5) ) ) ) ).

% Min_in
tff(fact_735_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max.coboundedI
tff(fact_736_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X: A] :
                  ( member(A,X,A5)
                 => ? [Xa2: A] :
                      ( member(A,Xa2,B5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) )
             => ( ! [X: A] :
                    ( member(A,X,B5)
                   => ? [Xa2: A] :
                        ( member(A,Xa2,A5)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa2) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,lattic643756798349783984er_Max(A),B5) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_737_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [Y3: A] :
                ( member(A,Y3,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( member(A,Xa,A5)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = Xa ) ) ) ) ) ).

% Max_eqI
tff(fact_738_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ).

% Max_ge
tff(fact_739_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5) ) ) ) ).

% Max_in
tff(fact_740_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = M )
            <=> ( member(A,M,A5)
                & ! [X5: A] :
                    ( member(A,X5,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X5) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_741_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xa)
            <=> ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa) ) ) ) ) ) ).

% Min_le_iff
tff(fact_742_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A5) )
            <=> ( member(A,M,A5)
                & ! [X5: A] :
                    ( member(A,X5,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X5) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_743_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5))
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A9) ) ) ) ) ) ).

% Min.boundedE
tff(fact_744_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.boundedI
tff(fact_745_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),Xa)
            <=> ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Xa) ) ) ) ) ) ).

% Min_less_iff
tff(fact_746_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B4) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_747_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = M )
            <=> ( member(A,M,A5)
                & ! [X5: A] :
                    ( member(A,X5,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_748_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X5) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_749_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A5) )
            <=> ( member(A,M,A5)
                & ! [X5: A] :
                    ( member(A,X5,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_750_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xa)
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xa) ) ) ) ) ) ).

% Max.boundedE
tff(fact_751_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),Xa) ) ) ) ) ).

% Max.boundedI
tff(fact_752_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5))
            <=> ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X5) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_753_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),A2) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_754_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N6)
         => ( ( M7 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N6)),aa(set(A),A,lattic643756798350308766er_Min(A),M7)) ) ) ) ) ).

% Min_antimono
tff(fact_755_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min.subset_imp
tff(fact_756_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M7: set(A),N6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M7),N6)
         => ( ( M7 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N6)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M7)),aa(set(A),A,lattic643756798349783984er_Max(A),N6)) ) ) ) ) ).

% Max_mono
tff(fact_757_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ).

% Max.subset_imp
tff(fact_758_greater__shift,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
    <=> vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% greater_shift
tff(fact_759_less__shift,axiom,
    ! [Xa: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
    <=> vEBT_VEBT_less(aa(nat,option(nat),some(nat),Xa),aa(nat,option(nat),some(nat),Y)) ) ).

% less_shift
tff(fact_760_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_761_the__elem__eq,axiom,
    ! [A: $tType,Xa: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ).

% the_elem_eq
tff(fact_762_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xa) = Y )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv: $o] : Xa = vEBT_Leaf($true,(Uv))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu: $o] : Xa = vEBT_Leaf((Uu),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_763_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_764_is__singletonI,axiom,
    ! [A: $tType,Xa: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_765_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2),Vi2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_766_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2),Vj2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_767_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2),Vj2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_768_is__singleton__the__elem,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),the_elem(A,A5)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_769_is__singletonI_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [X: A,Y3: A] :
            ( member(A,X,A5)
           => ( member(A,Y3,A5)
             => ( X = Y3 ) ) )
       => is_singleton(A,A5) ) ) ).

% is_singletonI'
tff(fact_770_minNull__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(T2)),one_one(nat)) ).

% minNull_bound
tff(fact_771_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($false,$false)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_772_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv2: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($true,(Uv2))) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_773_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu2: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf((Uu2),$true)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_774_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A2),(B2)),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_775_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_776_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : vEBT_T_s_u_c_c(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,Na)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_777_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va2: nat] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_778_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((Uu2),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_779_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d(vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz,Va2),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_780_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Na: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,Na)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_781_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy2: nat,Uz: list(vEBT_VEBT),Va2: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d2(vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz,Va2),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_782_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Va2: nat] : vEBT_T_s_u_c_c(vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz),Va2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_783_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_784_is__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
     => ~ ! [X: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_785_is__singleton__def,axiom,
    ! [A: $tType,A5: set(A)] :
      ( is_singleton(A,A5)
    <=> ? [X5: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_786_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Va2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz),Va2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_787_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_788_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_789_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_790_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2),Vf2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_791_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_792_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_793_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2),Vf2) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_794_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2),Ve2) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_795_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz),Xa) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_796_set__removeAll,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),removeAll(A,Xa,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_797_vebt__maxt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (B4),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_798_vebt__mint_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi2) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_799_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary))) ) ).

% height_node
tff(fact_800_insertsimp_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,L: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_801_insersimp_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X_1)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_802_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: A] :
          ( ( archimedean_frac(A,Xa) = A2 )
        <=> ( member(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_803_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_804_remove__def,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xa),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% remove_def
tff(fact_805_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,Sa),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_806_member__remove,axiom,
    ! [A: $tType,Xa: A,Y: A,A5: set(A)] :
      ( member(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Y),A5))
    <=> ( member(A,Xa,A5)
        & ( Xa != Y ) ) ) ).

% member_remove
tff(fact_807_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = zero_zero(A) )
        <=> member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_808_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xa))
        <=> ~ member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_809_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A2),(B2))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_810_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A2),(B2)),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_811_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts2,Sa),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_812_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv)) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xa = vEBT_Leaf((Uu),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_813_delete__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% delete_bound_height'
tff(fact_814_height__compose__child,axiom,
    ! [T2: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summary: vEBT_VEBT] :
      ( member(vEBT_VEBT,T2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ) ).

% height_compose_child
tff(fact_815_height__compose__list,axiom,
    ! [T2: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( member(vEBT_VEBT,T2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ).

% height_compose_list
tff(fact_816_height__compose__summary,axiom,
    ! [Summary: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeList,Summary))) ).

% height_compose_summary
tff(fact_817_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Y) )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu: $o] :
                  ( ( Xa = vEBT_Leaf((Uu),$true) )
                 => ( ~ (Y)
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) ) )
             => ( ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) )
                   => ( (Y)
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_818_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_819_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv)) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu: $o] :
                ( ( Xa = vEBT_Leaf((Uu),$true) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_820_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_821_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat))),bot_bot(set(A))))),Na) ) ).

% enumerate_Suc'
tff(fact_822_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)
     => ~ ! [N: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) ) ).

% even_odd_cases
tff(fact_823_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add_right_cancel
tff(fact_824_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add_left_cancel
tff(fact_825_image__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),Xa: B,A5: set(B)] :
      ( ( B2 = aa(B,A,F3,Xa) )
     => ( member(B,Xa,A5)
       => member(A,B2,aa(set(B),set(A),image(B,A,F3),A5)) ) ) ).

% image_eqI
tff(fact_826_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_827_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_828_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_829_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_830_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_831_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_832_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_833_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_834_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_835_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_836_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_837_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_838_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_839_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_840_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: 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),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_841_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C3: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_right
tff(fact_842_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: 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),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_843_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ).

% add_diff_cancel_left
tff(fact_844_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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),A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_845_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: 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),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_846_image__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A)] : aa(set(B),set(A),image(B,A,F3),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_847_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image(B,A,F3),A5) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_848_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F3),A5) = bot_bot(set(A)) )
    <=> ( A5 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_849_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc_right
tff(fact_850_finite__imageI,axiom,
    ! [B: $tType,A: $tType,F4: set(A),H2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,H2),F4)) ) ).

% finite_imageI
tff(fact_851_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Na = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_852_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_853_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_854_insert__image,axiom,
    ! [B: $tType,A: $tType,Xa: A,A5: set(A),F3: fun(A,B)] :
      ( member(A,Xa,A5)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),aa(A,B,F3,Xa)),aa(set(A),set(B),image(A,B,F3),A5)) = aa(set(A),set(B),image(A,B,F3),A5) ) ) ).

% insert_image
tff(fact_855_image__insert,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: B,B5: set(B)] : aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A2)),aa(set(B),set(A),image(B,A,F3),B5)) ).

% image_insert
tff(fact_856_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_857_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_858_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_859_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_860_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_861_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_862_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_863_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_864_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_865_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_866_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_867_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_868_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_869_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_870_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_871_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_872_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_873_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_874_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% add_gr_0
tff(fact_875_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_876_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_877_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_878_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_879_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_880_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),Na))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% enumerate_mono_iff
tff(fact_881_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) )
         => ( B2 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_882_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
         => ( B2 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_883_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_884_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_885_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) )
        <=> ( B2 = C3 ) ) ) ).

% add.right_cancel
tff(fact_886_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
        <=> ( B2 = C3 ) ) ) ).

% add.left_cancel
tff(fact_887_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% add.assoc
tff(fact_888_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_889_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: A,K: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( 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),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_890_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_891_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A5: set(A),B2: B,F3: fun(A,B)] :
      ( member(A,Xa,A5)
     => ( ( B2 = aa(A,B,F3,Xa) )
       => member(B,B2,aa(set(A),set(B),image(A,B,F3),A5)) ) ) ).

% rev_image_eqI
tff(fact_892_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(set(B),set(A),image(B,A,F3),A5))
         => aa(A,$o,P2,X) )
     => ! [X3: B] :
          ( member(B,X3,A5)
         => aa(A,$o,P2,aa(B,A,F3,X3)) ) ) ).

% ball_imageD
tff(fact_893_image__cong,axiom,
    ! [B: $tType,A: $tType,M7: set(A),N6: set(A),F3: fun(A,B),G3: fun(A,B)] :
      ( ( M7 = N6 )
     => ( ! [X: A] :
            ( member(A,X,N6)
           => ( aa(A,B,F3,X) = aa(A,B,G3,X) ) )
       => ( aa(set(A),set(B),image(A,B,F3),M7) = aa(set(A),set(B),image(A,B,G3),N6) ) ) ) ).

% image_cong
tff(fact_894_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(set(B),set(A),image(B,A,F3),A5))
          & aa(A,$o,P2,X3) )
     => ? [X: B] :
          ( member(B,X,A5)
          & aa(A,$o,P2,aa(B,A,F3,X)) ) ) ).

% bex_imageD
tff(fact_895_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F3: fun(B,A),A5: set(B)] :
      ( member(A,Z,aa(set(B),set(A),image(B,A,F3),A5))
    <=> ? [X5: B] :
          ( member(B,X5,A5)
          & ( Z = aa(B,A,F3,X5) ) ) ) ).

% image_iff
tff(fact_896_imageI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A5: set(A),F3: fun(A,B)] :
      ( member(A,Xa,A5)
     => member(B,aa(A,B,F3,Xa),aa(set(A),set(B),image(A,B,F3),A5)) ) ).

% imageI
tff(fact_897_image__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,F3),B5)) ) ).

% image_mono
tff(fact_898_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,B),B5: set(B)] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => member(B,aa(A,B,F3,X),B5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),B5) ) ).

% image_subsetI
tff(fact_899_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F3),A5))
     => ~ ! [C6: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C6),A5)
           => ( B5 != aa(set(B),set(A),image(B,A,F3),C6) ) ) ) ).

% subset_imageE
tff(fact_900_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),B5)
    <=> ! [X5: B] :
          ( member(B,X5,A5)
         => member(A,aa(B,A,F3,X5),B5) ) ) ).

% image_subset_iff
tff(fact_901_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F3),A5))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A5)
          & ( B5 = aa(set(B),set(A),image(B,A,F3),AA) ) ) ) ).

% subset_image_iff
tff(fact_902_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ! [B9: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A5))
         => aa(set(A),$o,P2,B9) )
    <=> ! [B9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B9),A5)
         => aa(set(A),$o,P2,aa(set(B),set(A),image(B,A,F3),B9)) ) ) ).

% all_subset_image
tff(fact_903_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_904_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_905_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_906_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_907_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_908_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_909_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ? [C7: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C7) ) ) ).

% le_iff_add
tff(fact_910_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_right_mono
tff(fact_911_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ~ ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ) ).

% less_eqE
tff(fact_912_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_left_mono
tff(fact_913_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_mono
tff(fact_914_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_915_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_916_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_917_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_918_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_919_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_920_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_921_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% add_strict_left_mono
tff(fact_922_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ) ).

% add_strict_right_mono
tff(fact_923_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_924_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_925_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) ) ).

% diff_diff_eq
tff(fact_926_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) = A2 )
         => ( C3 = aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_927_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_928_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_add_eq
tff(fact_929_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ).

% diff_diff_eq2
tff(fact_930_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ).

% add_diff_eq
tff(fact_931_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_932_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = C3 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2) ) ) ) ).

% diff_eq_eq
tff(fact_933_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K: A,A2: A,B2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( 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),A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_934_nat__arith_Osuc1,axiom,
    ! [A5: nat,K: nat,A2: nat] :
      ( ( A5 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_935_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc
tff(fact_936_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) ).

% add_Suc_shift
tff(fact_937_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Na) = Na ).

% plus_nat.add_0
tff(fact_938_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = M )
     => ( Na = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_939_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_940_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),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_941_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_942_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_943_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_944_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_945_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_946_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% less_add_eq_less
tff(fact_947_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_948_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_949_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_950_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_951_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),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_952_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) ) ).

% le_Suc_ex
tff(fact_953_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% add_leD2
tff(fact_954_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% add_leD1
tff(fact_955_le__add2,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% le_add2
tff(fact_956_le__add1,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) ).

% le_add1
tff(fact_957_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% add_leE
tff(fact_958_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_959_diff__add__inverse,axiom,
    ! [Na: 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),Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_960_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% diff_cancel2
tff(fact_961_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na) ).

% Nat.diff_cancel
tff(fact_962_enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => member(A,aa(nat,A,infini527867602293511546merate(A,S2),Na),S2) ) ) ).

% enumerate_in_set
tff(fact_963_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_964_enumerate__Ex,axiom,
    ! [S2: set(nat),Sa: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ( member(nat,Sa,S2)
       => ? [N: nat] : aa(nat,nat,infini527867602293511546merate(nat,S2),N) = Sa ) ) ).

% enumerate_Ex
tff(fact_965_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ! [B9: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B9)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A5)) )
         => aa(set(A),$o,P2,B9) )
    <=> ! [B9: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B9),A5) )
         => aa(set(A),$o,P2,aa(set(B),set(A),image(B,A,F3),B9)) ) ) ).

% all_finite_subset_image
tff(fact_966_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),P2: fun(set(A),$o)] :
      ( ? [B9: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A5))
          & aa(set(A),$o,P2,B9) )
    <=> ? [B9: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B9),A5)
          & aa(set(A),$o,P2,aa(set(B),set(A),image(B,A,F3),B9)) ) ) ).

% ex_finite_subset_image
tff(fact_967_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F3),A5))
       => ? [C6: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C6),A5)
            & aa(set(B),$o,finite_finite2(B),C6)
            & ( B5 = aa(set(B),set(A),image(B,A,F3),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_968_finite__surj,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),aa(set(A),set(B),image(A,B,F3),A5))
       => aa(set(B),$o,finite_finite2(B),B5) ) ) ).

% finite_surj
tff(fact_969_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),aa(set(B),set(A),image(B,A,F3),B5))),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),B5))) ).

% image_diff_subset
tff(fact_970_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_971_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_972_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_973_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_974_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_increasing2
tff(fact_975_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ) ) ).

% add_decreasing2
tff(fact_976_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_increasing
tff(fact_977_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),B2) ) ) ) ).

% add_decreasing
tff(fact_978_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% pos_add_strict
tff(fact_979_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ ! [C2: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
               => ( C2 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_980_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_981_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_982_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_983_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_984_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_985_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_986_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_987_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A))) ) ) ).

% add_mono1
tff(fact_988_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% less_add_one
tff(fact_989_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_990_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_991_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_992_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_993_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),C3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_994_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_995_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_996_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_997_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C3)),A2)) ) ) ).

% le_add_diff
tff(fact_998_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_999_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% le_diff_eq
tff(fact_1000_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_le_eq
tff(fact_1001_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1002_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,aa(A,fun(A,A),minus_minus(A),Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1003_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ) ).

% diff_less_eq
tff(fact_1004_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C3) ) ) ).

% less_diff_eq
tff(fact_1005_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1006_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,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)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_1007_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1008_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1009_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ~ ! [Q5: nat] : Na != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ) ).

% less_natE
tff(fact_1010_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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_1011_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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_1012_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1013_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1014_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),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_1015_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Na) ).

% Suc_eq_plus1_left
tff(fact_1016_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1017_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1018_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P2: fun(A,$o),K: A,F3: fun(A,nat),Na: nat] :
      ( aa(A,$o,P2,K)
     => ( ! [X: A] :
            ( aa(A,$o,P2,X)
           => ? [Y4: A] :
                ( aa(A,$o,P2,Y4)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Y4)),aa(A,nat,F3,X)) ) )
       => ? [Y3: A] :
            ( aa(A,$o,P2,Y3)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Y3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1019_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),M: nat,K: nat] :
      ( ! [M3: nat,N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F3,M3)),aa(nat,nat,F3,N)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,M)),K)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_1020_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1021_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1022_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1023_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1024_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1025_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1026_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1027_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1028_le__enumerate,axiom,
    ! [S2: set(nat),Na: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,infini527867602293511546merate(nat,S2),Na)) ) ).

% le_enumerate
tff(fact_1029_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B),Xa: A] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [Y3: A] :
            ( member(A,Y3,A5)
           => ( aa(A,B,F3,Y3) = aa(A,B,F3,Xa) ) )
       => ( the_elem(B,aa(set(A),set(B),image(A,B,F3),A5)) = aa(A,B,F3,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_1030_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),A2)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_1031_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing2
tff(fact_1032_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3)) ) ) ) ).

% add_strict_increasing
tff(fact_1033_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1034_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1035_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1036_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1037_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_1038_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_1039_nat__diff__split__asm,axiom,
    ! [P2: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
            & ~ aa(nat,$o,P2,zero_zero(nat)) )
          | ? [D6: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
              & ~ aa(nat,$o,P2,D6) ) ) ) ).

% nat_diff_split_asm
tff(fact_1040_nat__diff__split,axiom,
    ! [P2: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
         => aa(nat,$o,P2,zero_zero(nat)) )
        & ! [D6: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D6) )
           => aa(nat,$o,P2,D6) ) ) ) ).

% nat_diff_split
tff(fact_1041_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1042_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu2,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_1043_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Na)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na))) ) ) ).

% enumerate_step
tff(fact_1044_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( ~ aa(set(A),$o,finite_finite2(A),S2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),Na)) ) ) ) ).

% enumerate_mono
tff(fact_1045_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = $ite(M = zero_zero(nat),Na,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))),Na))) ).

% add_eq_if
tff(fact_1046_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_1047_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_1048_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Va2: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_1049_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu2),(B2)),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_1050_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A2),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_1051_pred__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d2(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% pred_bound_height'
tff(fact_1052_succ_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% succ'_bound_height
tff(fact_1053_insert_H__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% insert'_bound_height
tff(fact_1054_member__bound__height_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))) ) ).

% member_bound_height'
tff(fact_1055_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = zero_zero(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Summary2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Node(Uu,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_1056_max__ins__scaled,axiom,
    ! [Na: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))))) ).

% max_ins_scaled
tff(fact_1057_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_epsilon
tff(fact_1058_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( ! [A4: $o] :
            ( ? [B4: $o] : Xa = vEBT_Leaf((A4),(B4))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_1059_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_1060_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_1061_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Leaf((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_1062_Euclid__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),B4)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P2,B4),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),zero_zero(nat))
       => ( ! [A4: nat,B4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),B4)
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B4)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P2,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_1063_triangle__Suc,axiom,
    ! [Na: nat] : nat_triangle(aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Na)),aa(nat,nat,suc,Na)) ).

% triangle_Suc
tff(fact_1064_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_1065_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_1066_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_1067_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_1068_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_1069_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_1070_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_1071_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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),Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_1072_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_1073_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_1074_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_1075_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1076_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1077_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_1078_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_1079_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_1080_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_1081_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B2: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_1082_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1083_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1084_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% mult_less_cancel2
tff(fact_1085_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_1086_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc_right
tff(fact_1087_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Na) ) ) ).

% one_le_mult_iff
tff(fact_1088_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% mult_le_cancel2
tff(fact_1089_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% mult.assoc
tff(fact_1090_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1091_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1092_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_1093_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_1094_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_1095_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_1096_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_1097_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1098_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_1099_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_1100_mult__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% mult_0
tff(fact_1101_add__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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),Na),K)) ).

% add_mult_distrib
tff(fact_1102_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% add_mult_distrib2
tff(fact_1103_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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_1104_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_1105_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),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_1106_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_1107_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),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_1108_nat__mult__1,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Na) = Na ).

% nat_mult_1
tff(fact_1109_nat__mult__1__right,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),one_one(nat)) = Na ).

% nat_mult_1_right
tff(fact_1110_diff__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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),Na),K)) ).

% diff_mult_distrib
tff(fact_1111_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% diff_mult_distrib2
tff(fact_1112_zero__notin__Suc__image,axiom,
    ! [A5: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),A5)) ).

% zero_notin_Suc_image
tff(fact_1113_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1114_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1115_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1116_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1117_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1118_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1119_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1120_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1121_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono
tff(fact_1122_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1123_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono
tff(fact_1124_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1125_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1126_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_1127_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)) ) ).

% zero_le_square
tff(fact_1128_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1129_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_1130_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_1131_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_1132_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1133_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1134_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1135_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_1136_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1137_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1138_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_1139_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1140_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1141_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1142_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1143_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1144_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1145_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1146_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1147_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1148_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1149_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),Na)) ) ) ) ).

% less_1_mult
tff(fact_1150_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_1151_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),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_1152_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),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_1153_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc
tff(fact_1154_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_1155_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
     => ( ( Na = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1156_mlex__snd__decrI,axiom,
    ! [A2: nat,A3: nat,B2: nat,B3: nat,N6: nat] :
      ( ( A2 = A3 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),B3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N6)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N6)),B3)) ) ) ).

% mlex_snd_decrI
tff(fact_1157_mlex__fst__decrI,axiom,
    ! [A2: nat,A3: nat,B2: nat,N6: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N6)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),N6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N6)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N6)),B3)) ) ) ) ).

% mlex_fst_decrI
tff(fact_1158_mlex__bound,axiom,
    ! [A2: nat,A5: nat,B2: nat,N6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),N6)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N6)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N6)) ) ) ).

% mlex_bound
tff(fact_1159_mlex__leI,axiom,
    ! [A2: nat,A3: nat,B2: nat,B3: nat,N6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),A3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),B3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),N6)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N6)),B3)) ) ) ).

% mlex_leI
tff(fact_1160_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xa)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1161_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),Xa: A,A5: set(A)] :
      ( ! [C6: set(A)] :
          ( member(set(A),C6,B5)
         => ~ member(A,Xa,C6) )
     => ( member(set(A),A5,aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa)),B5))
      <=> ( member(A,Xa,A5)
          & member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_1162_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1163_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1164_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1165_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1166_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1167_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1168_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1169_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1170_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1171_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1172_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1173_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1174_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1175_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1176_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1177_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),A2) ) ) ) ).

% mult_left_le
tff(fact_1178_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1179_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y)),Xa) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1180_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)),Xa) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1181_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1182_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C3)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1183_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1184_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)),E3)),D2)) ) ) ).

% less_add_iff2
tff(fact_1185_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),E3)),C3)),D2) ) ) ).

% less_add_iff1
tff(fact_1186_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% one_less_mult
tff(fact_1187_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_1188_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1189_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),X_1) ) ).

% linordered_field_no_ub
tff(fact_1190_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y3),X3) ) ).

% linordered_field_no_lb
tff(fact_1191_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1192_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1193_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1194_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1195_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1196_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1197_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1198_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1199_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xa: A,A2: A,Y: A,U: A,V3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V3)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V3) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V3),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1200_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),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))),Na))) ).

% mult_eq_if
tff(fact_1201_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_1202_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_1203_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_1204_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xa: A,A2: A,Y: A,U: A,V3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V3)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V3) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V3),Y))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1205_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_1206_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A2),(B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_1207_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_1208_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1209_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1210_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1211_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_1212_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_1213_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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)),Na)) = 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)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_1214_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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)),Na)) = 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)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_1215_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_1216_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_1217_star__false__left,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P2) = bot_bot(assn) ).

% star_false_left
tff(fact_1218_star__false__right,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_1219_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),Xa: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,P3),Xa)),aa(list(A),assn,snga_assn(A,P3),Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_1220_sngr__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: ref(A),Xa: A,Y: A] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),sngr_assn(A,P3,Xa)),sngr_assn(A,P3,Y)) = bot_bot(assn) ) ).

% sngr_same_false
tff(fact_1221_mod__h__bot__iff_I5_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_iff(5)
tff(fact_1222_mod__starE,axiom,
    ! [A2: assn,B2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2)),H2)
     => ~ ( ? [X_1: product_prod(heap_ext(product_unit),set(nat))] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(A2),X_1)
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(B2),H_2) ) ) ).

% mod_starE
tff(fact_1223_mod__starD,axiom,
    ! [A5: assn,B5: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B5)),H2)
     => ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(A5),H1)
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(B5),H22) ) ) ).

% mod_starD
tff(fact_1224_assn__one__left,axiom,
    ! [P2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P2) = P2 ).

% assn_one_left
tff(fact_1225_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1226_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_1227_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1228_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1229_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1230_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_1231_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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)),Na) )
      <=> ( 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) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_1232_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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)),Na) )
      <=> ( 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)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1233_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1234_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1235_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A2: A,B2: A,C3: A,D2: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C3 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1236_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% mult_less_iff1
tff(fact_1237_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_1238_mul__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_1239_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S2)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_1240_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S2)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_1241_add__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_1242_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A2)) = A2 ) ).

% add.inverse_inverse
tff(fact_1243_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,uminus_uminus(A),B2) )
        <=> ( A2 = B2 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1244_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5) ) ).

% Compl_subset_Compl_iff
tff(fact_1245_Compl__anti__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A5)) ) ).

% Compl_anti_mono
tff(fact_1246_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = A2 )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1247_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1248_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1249_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A2) )
        <=> ( zero_zero(A) = A2 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1250_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_1251_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_1252_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) ) ) ).

% compl_le_compl_iff
tff(fact_1253_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_1254_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ) ).

% compl_less_compl_iff
tff(fact_1255_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_1256_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_1257_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_1258_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% minus_diff_eq
tff(fact_1259_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_1260_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_1261_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_1262_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_1263_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_1264_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_1265_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_1266_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_1267_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_1268_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_1269_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_1270_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_1271_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_1272_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_1273_subset__Compl__singleton,axiom,
    ! [A: $tType,A5: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A5) ) ).

% subset_Compl_singleton
tff(fact_1274_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_1275_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_1276_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_1277_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_1278_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_1279_assn__times__comm,axiom,
    ! [P2: assn,Q2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),P2) ).

% assn_times_comm
tff(fact_1280_assn__times__assoc,axiom,
    ! [P2: assn,Q2: assn,R3: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),R3) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),R3)) ).

% assn_times_assoc
tff(fact_1281_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% equation_minus_iff
tff(fact_1282_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,uminus_uminus(A),B2) = A2 ) ) ) ).

% minus_equation_iff
tff(fact_1283_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_1284_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_1285_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_imp_neg_le
tff(fact_1286_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% compl_le_swap2
tff(fact_1287_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_1288_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xa)) ) ) ).

% compl_mono
tff(fact_1289_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_1290_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% less_minus_iff
tff(fact_1291_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1292_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Y)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% compl_less_swap2
tff(fact_1293_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_1294_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A5: A,K: A,A2: A] :
          ( ( A5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( 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),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_1295_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),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),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_1296_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_1297_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_1298_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_1299_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_1300_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_1301_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_1302_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_1303_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_1304_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_1305_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_1306_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_1307_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B5) = 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),A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1308_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_1309_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1310_subset__Compl__self__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5))
    <=> ( A5 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1311_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1312_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1313_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_1314_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_1315_Compl__insert,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_1316_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xa)) = $ite(member(A,Xa,ring_1_Ints(A)),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),archimedean_frac(A,Xa))) ) ).

% frac_neg
tff(fact_1317_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_1318_foldr__one,axiom,
    ! [D2: nat,Ys: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D2),foldr(nat,nat,plus_plus(nat),Ys,D2)) ).

% foldr_one
tff(fact_1319_Gcd__remove0__nat,axiom,
    ! [M7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M7)
     => ( gcd_Gcd(nat,M7) = gcd_Gcd(nat,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),M7),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_1320_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( ( gcd_Gcd(A,A5) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_1321_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P3: A,M: nat] :
      power2(A,One,Times,P3,M) = $ite(M = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ).

% power.power_eq_if
tff(fact_1322_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_1323_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,Sa),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_1324_the__dflt__None__empty,axiom,
    ! [A: $tType] : dflt_None_set(A,bot_bot(set(A))) = none(set(A)) ).

% the_dflt_None_empty
tff(fact_1325_ComplI,axiom,
    ! [A: $tType,C3: A,A5: set(A)] :
      ( ~ member(A,C3,A5)
     => member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A5)) ) ).

% ComplI
tff(fact_1326_Compl__iff,axiom,
    ! [A: $tType,C3: A,A5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A5))
    <=> ~ member(A,C3,A5) ) ).

% Compl_iff
tff(fact_1327_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),uminus_uminus(set(A)),B5) )
    <=> ( A5 = B5 ) ) ).

% Compl_eq_Compl_iff
tff(fact_1328_the__dflt__None__nonempty,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( S2 != bot_bot(set(A)) )
     => ( dflt_None_set(A,S2) = aa(set(A),option(set(A)),some(set(A)),S2) ) ) ).

% the_dflt_None_nonempty
tff(fact_1329_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_1330_ComplD,axiom,
    ! [A: $tType,C3: A,A5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A5))
     => ~ member(A,C3,A5) ) ).

% ComplD
tff(fact_1331_double__complement,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = A5 ).

% double_complement
tff(fact_1332_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A,Na: nat] : power2(A,One,Times,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),Times,A2),power2(A,One,Times,A2,Na)) ).

% power.power.power_Suc
tff(fact_1333_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A] : power2(A,One,Times,A2,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_1334_dflt__None__set__def,axiom,
    ! [A: $tType,S2: set(A)] :
      dflt_None_set(A,S2) = $ite(S2 = bot_bot(set(A)),none(set(A)),aa(set(A),option(set(A)),some(set(A)),S2)) ).

% dflt_None_set_def
tff(fact_1335_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts2,Sa),Xa) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_1336_the__dflt__None__set,axiom,
    ! [A: $tType,Xa: set(A)] : the_default(set(A),bot_bot(set(A)),dflt_None_set(A,Xa)) = Xa ).

% the_dflt_None_set
tff(fact_1337_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X: nat,Y3: nat] :
          ( member(nat,X,aa(list(nat),set(nat),set2(nat),Xs))
         => ( member(nat,Y3,aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: nat] :
            ( member(nat,X,aa(list(nat),set(nat),set2(nat),Xs))
           => ( X = Y ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_1338_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ( semiring_gcd_Gcd_fin(A,A5) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite2(A),A5) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_1339_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1340_card__insert__le__m1,axiom,
    ! [A: $tType,Na: nat,Y: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),Y))),Na) ) ) ).

% card_insert_le_m1
tff(fact_1341_count__notin,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xa) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_1342_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F3)),finite_Fpow(B,A5))),finite_Fpow(A,B5)) ) ).

% image_Fpow_mono
tff(fact_1343_set__to__map__empty,axiom,
    ! [A: $tType,B: $tType,X3: A] : aa(A,option(B),set_to_map(A,B,bot_bot(set(product_prod(A,B)))),X3) = none(B) ).

% set_to_map_empty
tff(fact_1344_power__shift,axiom,
    ! [Xa: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),Xa)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_1345_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_1346_power__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_1347_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xa = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1348_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xa),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
        | ( Na = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1349_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_1350_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,Na)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_1351_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_1352_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_1353_card_Oinfinite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_1354_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_1355_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( semiring_gcd_Gcd_fin(A,A5) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_1356_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( semiring_gcd_Gcd_fin(A,A5) = gcd_Gcd(A,A5) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_1357_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1358_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% power_eq_0_iff
tff(fact_1359_card__0__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,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_1360_card__insert__disjoint,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ~ member(A,Xa,A5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% card_insert_disjoint
tff(fact_1361_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1362_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1363_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xa: nat,Y: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y) ) ) ) ).

% power_increasing_iff
tff(fact_1364_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( member(A,A2,A5)
     => ( ~ member(A,A2,B5)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5))) = 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),B5))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_1365_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),M: nat,Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(set(A),nat,finite_card(A),S2))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),Na))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_1366_Fpow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_1367_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xa) != aa(A,nat,size_size(A),Y) )
         => ( Xa != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_1368_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),Xa)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_1369_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_1370_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_1371_length__induct,axiom,
    ! [A: $tType,P2: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P2,Ys2) )
         => aa(list(A),$o,P2,Xs2) )
     => aa(list(A),$o,P2,Xs) ) ).

% length_induct
tff(fact_1372_finite__maxlen,axiom,
    ! [A: $tType,M7: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M7)
     => ? [N: nat] :
        ! [X3: list(A)] :
          ( member(list(A),X3,M7)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X3)),N) ) ) ).

% finite_maxlen
tff(fact_1373_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_le_power
tff(fact_1374_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono
tff(fact_1375_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_less_power
tff(fact_1376_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% one_le_power
tff(fact_1377_card__subset__eq,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,finite_card(A),B5) )
         => ( A5 = B5 ) ) ) ) ).

% card_subset_eq
tff(fact_1378_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ? [B8: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B8)
          & ( aa(set(A),nat,finite_card(A),B8) = Na )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B8),A5) ) ) ).

% infinite_arbitrarily_large
tff(fact_1379_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A5: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( ! [A4: B] :
            ( member(B,A4,A5)
           => ? [B10: A] :
                ( member(A,B10,B5)
                & aa(A,$o,aa(B,fun(A,$o),R2,A4),B10) ) )
       => ( ! [A1: B,A22: B,B4: A] :
              ( member(B,A1,A5)
             => ( member(B,A22,A5)
               => ( member(A,B4,B5)
                 => ( aa(A,$o,aa(B,fun(A,$o),R2,A1),B4)
                   => ( aa(A,$o,aa(B,fun(A,$o),R2,A22),B4)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_1380_empty__in__Fpow,axiom,
    ! [A: $tType,A5: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,A5)) ).

% empty_in_Fpow
tff(fact_1381_card__insert__le,axiom,
    ! [A: $tType,A5: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5))) ).

% card_insert_le
tff(fact_1382_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ).

% power_Suc
tff(fact_1383_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),A2) ) ).

% power_Suc2
tff(fact_1384_Fpow__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),finite_Fpow(A,B5)) ) ).

% Fpow_mono
tff(fact_1385_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1386_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_1387_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xa,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_1388_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_1389_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_1390_card__ge__0__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
     => aa(set(A),$o,finite_finite2(A),A5) ) ).

% card_ge_0_finite
tff(fact_1391_card__eq__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = zero_zero(nat) )
    <=> ( ( A5 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% card_eq_0_iff
tff(fact_1392_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1393_card__insert__if,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(member(A,Xa,A5),aa(set(A),nat,finite_card(A),A5),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A5))) ) ) ).

% card_insert_if
tff(fact_1394_card__Suc__eq__finite,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B9: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B7),B9) )
          & ~ member(A,B7,B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & aa(set(A),$o,finite_finite2(A),B9) ) ) ).

% card_Suc_eq_finite
tff(fact_1395_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Na)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_1396_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Na)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_1397_card__image__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(A),nat,finite_card(A),A5)) ) ).

% card_image_le
tff(fact_1398_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))) ) ) ).

% power_gt1_lemma
tff(fact_1399_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))) ) ) ).

% power_less_power_Suc
tff(fact_1400_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F4: set(A),C4: nat] :
      ( ! [G4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G4),F4)
         => ( aa(set(A),$o,finite_finite2(A),G4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G4)),C4) ) )
     => ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F4)),C4) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_1401_card__seteq,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B5)),aa(set(A),nat,finite_card(A),A5))
         => ( A5 = B5 ) ) ) ) ).

% card_seteq
tff(fact_1402_card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% card_mono
tff(fact_1403_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Na: nat,S2: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),S2))
     => ~ ! [T5: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T5),S2)
           => ( ( aa(set(A),nat,finite_card(A),T5) = Na )
             => ~ aa(set(A),$o,finite_finite2(A),T5) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_1404_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))) ) ) ).

% power_gt1
tff(fact_1405_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_1406_card__1__singletonE,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A5) = one_one(nat) )
     => ~ ! [X: A] : A5 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_1407_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N6: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N6)) ) ) ) ).

% power_strict_increasing
tff(fact_1408_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1409_card__less__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_less_sym_Diff
tff(fact_1410_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Na) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1411_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N6: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N6)) ) ) ) ).

% power_increasing
tff(fact_1412_card__le__sym__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) ) ) ) ).

% card_le_sym_Diff
tff(fact_1413_psubset__card__mono,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ).

% psubset_card_mono
tff(fact_1414_length__pos__if__in__set,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_1415_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),K)) ) ).

% power_gt_expt
tff(fact_1416_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Na)) ) ).

% nat_one_le_power
tff(fact_1417_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X4: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X4))
             => ( aa(nat,A,infini527867602293511546merate(A,X4),I2) = aa(nat,A,infini527867602293511546merate(A,Y6),I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X4)
           => ( aa(set(A),$o,finite_finite2(A),Y6)
             => ( ( aa(set(A),nat,finite_card(A),X4) = aa(set(A),nat,finite_card(A),Y6) )
               => ( X4 = Y6 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_1418_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Sa: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( member(A,Sa,S2)
           => ? [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(set(A),nat,finite_card(A),S2))
                & ( aa(nat,A,infini527867602293511546merate(A,S2),N) = Sa ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_1419_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
           => member(A,aa(nat,A,infini527867602293511546merate(A,S2),Na),S2) ) ) ) ).

% finite_enumerate_in_set
tff(fact_1420_length__removeAll__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xa,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_1421_card__gt__0__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
    <=> ( ( A5 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% card_gt_0_iff
tff(fact_1422_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => ! [Xa3: A] :
                ( member(A,Xa3,A5)
               => ( X5 = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_1423_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,B9: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B7),B9) )
          & ~ member(A,B7,B9)
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_1424_card__eq__SucD,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,K) )
     => ? [B4: A,B8: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B4),B8) )
          & ~ member(A,B4,B8)
          & ( aa(set(A),nat,finite_card(A),B8) = K )
          & ( ( K = zero_zero(nat) )
           => ( B8 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_1425_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)) )
    <=> ? [X5: A] : A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_1426_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% power_Suc_less
tff(fact_1427_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_1428_card__le__Suc__iff,axiom,
    ! [A: $tType,Na: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),A5))
    <=> ? [A7: A,B9: set(A)] :
          ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A7),B9) )
          & ~ member(A,A7,B9)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),B9))
          & aa(set(A),$o,finite_finite2(A),B9) ) ) ).

% card_le_Suc_iff
tff(fact_1429_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1430_set__to__map__empty__iff_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( ! [X5: A] : aa(A,option(B),set_to_map(A,B,S2),X5) = none(B)
    <=> ( S2 = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(1)
tff(fact_1431_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),aa(set(A),set(B),image(A,B,F3),A5))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B5)),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% surj_card_le
tff(fact_1432_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N6: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1433_card__1__singletonI,axiom,
    ! [A: $tType,S2: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( ( aa(set(A),nat,finite_card(A),S2) = one_one(nat) )
       => ( member(A,Xa,S2)
         => ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% card_1_singletonI
tff(fact_1434_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N6: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N6)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_1435_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1436_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1437_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1438_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% self_le_power
tff(fact_1439_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ).

% one_less_power
tff(fact_1440_card__Diff1__le,axiom,
    ! [A: $tType,A5: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff1_le
tff(fact_1441_card__Diff__subset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),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),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5)) ) ) ) ).

% card_Diff_subset
tff(fact_1442_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))),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),B5))) ) ).

% diff_card_le_card_Diff
tff(fact_1443_card__psubset,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),B5))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5) ) ) ) ).

% card_psubset
tff(fact_1444_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),M)),aa(nat,A,infini527867602293511546merate(A,S2),Na)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_1445_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),M))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1446_card__le__Suc__Max,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S2)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S2))) ) ).

% card_le_Suc_Max
tff(fact_1447_finite__le__enumerate,axiom,
    ! [S2: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(nat),nat,finite_card(nat),S2))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,infini527867602293511546merate(nat,S2),Na)) ) ) ).

% finite_le_enumerate
tff(fact_1448_card__insert__disjoint_H,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ~ member(A,Xa,A5)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5))),aa(nat,nat,suc,zero_zero(nat))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_insert_disjoint'
tff(fact_1449_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_1450_card_Oremove,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xa,A5)
       => ( aa(set(A),nat,finite_card(A),A5) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_1451_card_Oinsert__remove,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_1452_card__Suc__Diff1,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xa,A5)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% card_Suc_Diff1
tff(fact_1453_card__Diff1__less,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xa,A5)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% card_Diff1_less
tff(fact_1454_card__Diff2__less,axiom,
    ! [A: $tType,A5: set(A),Xa: A,Y: A] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( member(A,Xa,A5)
       => ( member(A,Y,A5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5)) ) ) ) ).

% card_Diff2_less
tff(fact_1455_card__Diff1__less__iff,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A5))
    <=> ( aa(set(A),$o,finite_finite2(A),A5)
        & member(A,Xa,A5) ) ) ).

% card_Diff1_less_iff
tff(fact_1456_card__Diff__singleton__if,axiom,
    ! [A: $tType,A5: set(A),Xa: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) = $ite(member(A,Xa,A5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),one_one(nat)),aa(set(A),nat,finite_card(A),A5)) ).

% card_Diff_singleton_if
tff(fact_1457_card__Diff__singleton,axiom,
    ! [A: $tType,Xa: A,A5: set(A)] :
      ( member(A,Xa,A5)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),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_1458_the__default_Osimps_I1_J,axiom,
    ! [A: $tType,Uu2: A,Xa: A] : the_default(A,Uu2,aa(A,option(A),some(A),Xa)) = Xa ).

% the_default.simps(1)
tff(fact_1459_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,M: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,aa(A,fun(nat,A),power_power(A),P3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1460_the__default_Osimps_I2_J,axiom,
    ! [A: $tType,Xa: A] : the_default(A,Xa,none(A)) = Xa ).

% the_default.simps(2)
tff(fact_1461_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na) ) ) ) ).

% power_minus_mult
tff(fact_1462_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S2),Na)),aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na))) ) ) ) ).

% finite_enumerate_step
tff(fact_1463_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X4: set(A),Y6: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X4))
             => ( aa(nat,A,infini527867602293511546merate(A,X4),I2) = aa(nat,A,infini527867602293511546merate(A,Y6),I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X4)
           => ( aa(set(A),$o,finite_finite2(A),Y6)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X4)),aa(set(A),nat,finite_card(A),Y6))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Y6) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_1464_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( 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),Na),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),Na),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1465_inthall,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Na: nat] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P2,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% inthall
tff(fact_1466_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From) ) ) ) ).

% slice_len
tff(fact_1467_card__Min__le__sum,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image(A,nat,F3),A5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)) ) ).

% card_Min_le_sum
tff(fact_1468_realpow__pos__nth,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R),Na) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1469_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X),Na) = A2 )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),Na) = A2 ) )
               => ( Y4 = X ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1470_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Na: nat] :
      ( ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X) = Na ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),Na) ) ) ).

% length_mul_elem
tff(fact_1471_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_1472_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa))
    <=> ( Xa = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1473_slice__complete,axiom,
    ! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).

% slice_complete
tff(fact_1474_real__add__le__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_le_0_iff
tff(fact_1475_real__0__le__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),Y) ) ).

% real_0_le_add_iff
tff(fact_1476_less__eq__real__def,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_real_def
tff(fact_1477_real__arch__pow,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),N)) ) ).

% real_arch_pow
tff(fact_1478_real__arch__pow__inv,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),N)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1479_real__add__less__0__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_less_0_iff
tff(fact_1480_real__0__less__add__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xa)),Y) ) ).

% real_0_less_add_iff
tff(fact_1481_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1482_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P2: fun(nat,fun(A,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P2,I3),X_13) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K)
             => aa(A,$o,aa(nat,fun(A,$o),P2,I3),aa(nat,A,nth(A,Xs3),I3)) ) ) ) ).

% Skolem_list_nth
tff(fact_1483_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1484_obtain__list__from__elements,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(nat,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P2,Li),I2) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = Na )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
                 => aa(nat,$o,aa(A,fun(nat,$o),P2,aa(nat,A,nth(A,L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_1485_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_1486_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P2,X5) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P2,aa(nat,A,nth(A,L),I3)) ) ) ).

% all_set_conv_nth
tff(fact_1487_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Na),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_1488_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),P2: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P2,X) )
       => aa(A,$o,P2,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_1489_in__set__conv__nth,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I3) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_1490_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o),Xa: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P2,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_1491_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P2,X5) )
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3)) ) ) ).

% all_set_conv_all_nth
tff(fact_1492_realpow__pos__nth2,axiom,
    ! [A2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ? [R: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R),aa(nat,nat,suc,Na)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1493_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_1494_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),F3: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F3,aa(nat,A,nth(A,L),I2)) = aa(A,B,F3,aa(nat,A,nth(A,L3),I2)) ) )
       => ( aa(set(A),set(B),image(A,B,F3),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image(A,B,F3),aa(list(A),set(A),set2(A),L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_1495_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xa: B,L: list(B)] :
      ( member(A,aa(B,A,F3,Xa),aa(set(B),set(A),image(B,A,F3),aa(list(B),set(B),set2(B),L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F3,aa(nat,B,nth(B,L),I3)) = aa(B,A,F3,Xa) ) ) ) ).

% in_set_image_conv_nth
tff(fact_1496_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X4)
     => ( aa(set(A),$o,finite_finite2(A),X4)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X4) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_1497_sum__le__card__Max,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image(A,nat,F3),A5)))) ) ).

% sum_le_card_Max
tff(fact_1498_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xa: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum.insert
tff(fact_1499_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_1500_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F4: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),F4) = zero_zero(B) )
          <=> ! [X5: A] :
                ( member(A,X5,F4)
               => ( aa(A,B,F3,X5) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_1501_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_1502_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X)) )
           => ( aa(set(A),$o,finite_finite2(A),A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ) ) ).

% member_le_sum
tff(fact_1503_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A5: set(A),B2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( member(A,B2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,B2))
               => ( ! [X: A] :
                      ( member(A,X,B5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_1504_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),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),F3),A5)),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ) ).

% sum_diff1
tff(fact_1505_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_1506_complete__real,axiom,
    ! [S2: set(real)] :
      ( ? [X3: real] : member(real,X3,S2)
     => ( ? [Z4: real] :
          ! [X: real] :
            ( member(real,X,S2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
       => ? [Y3: real] :
            ( ! [X3: real] :
                ( member(real,X3,S2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Y3) )
            & ! [Z4: real] :
                ( ! [X: real] :
                    ( member(real,X,S2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Z4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),Z4) ) ) ) ) ).

% complete_real
tff(fact_1507_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_1508_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A5: set(nat),F3: fun(nat,A),G3: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A5)
         => ( ! [X: nat] :
                ( member(nat,aa(nat,nat,suc,X),A5)
               => ( aa(nat,A,F3,aa(nat,nat,suc,X)) = aa(nat,A,G3,aa(nat,nat,suc,X)) ) )
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A5) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),A5) ) ) ) ) ).

% sum_cong_Suc
tff(fact_1509_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => ( aa(A,B,G3,X) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_1510_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) != zero_zero(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A5)
               => ( aa(B,A,G3,A4) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_1511_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_1512_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ).

% sum_nonneg
tff(fact_1513_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(B,A),I5: set(B),G3: fun(B,A),I: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,I2)),aa(B,A,G3,I2)) )
           => ( member(B,I,I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F3,I) = aa(B,A,G3,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_1514_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5) = zero_zero(B) )
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => ( aa(A,B,F3,X5) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_1515_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [Sa: set(A),T2: set(B),G3: fun(B,C),I: fun(B,A),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( aa(set(B),$o,finite_finite2(B),T2)
           => ( ! [X: B] :
                  ( member(B,X,T2)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G3,X)) )
             => ( ! [X: A] :
                    ( member(A,X,Sa)
                   => ? [Xa2: B] :
                        ( member(B,Xa2,T2)
                        & ( aa(B,A,I,Xa2) = X )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X)),aa(B,C,G3,Xa2)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F3),Sa)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),T2)) ) ) ) ) ) ).

% sum_le_included
tff(fact_1516_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
           => ( ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G3,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_1517_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R3: fun(A,fun(A,$o)),S2: set(B),H2: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R3,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R3,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X: B] :
                    ( member(B,X,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R3,aa(B,A,H2,X)),aa(B,A,G3,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H2),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),S2)) ) ) ) ) ) ).

% sum.related
tff(fact_1518_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( member(A,X,A5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5)) ) ) ) ) ).

% sum_strict_mono
tff(fact_1519_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(member(A,Xa,A5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5))) ) ) ) ).

% sum.insert_if
tff(fact_1520_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S5: set(A),T6: set(B),S2: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G3: fun(A,C),H2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
                     => ( aa(A,B,J,aa(B,A,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G3,A4) = zero_zero(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T6)
                           => ( aa(B,C,H2,B4) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H2,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G3),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H2),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_1521_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),Na: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = aa(nat,nat,suc,Na) )
     => ? [X: A] :
          ( member(A,X,A5)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X)) ) ) ).

% sum_SucD
tff(fact_1522_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X5: A] :
            ( member(A,X5,A5)
            & ( aa(A,nat,F3,X5) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A5)
               => ( ( X5 != Xa3 )
                 => ( aa(A,nat,F3,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_1523_sum__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5) = one_one(nat) )
      <=> ? [X5: A] :
            ( member(A,X5,A5)
            & ( aa(A,nat,F3,X5) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A5)
               => ( ( X5 != Xa3 )
                 => ( aa(A,nat,F3,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_1524_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_1525_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I5)) ) ) ) ) ).

% sum_pos
tff(fact_1526_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C4: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),B5))
                     => ( aa(A,B,H2,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),B5) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),C4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),C4) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_1527_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C4: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5))
                   => ( aa(A,B,G3,A4) = zero_zero(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),B5))
                     => ( aa(A,B,H2,B4) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),C4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),C4) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_1528_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_1529_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_1530_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H2,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_1531_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = zero_zero(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_1532_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_1533_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = 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),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ).

% sum_diff
tff(fact_1534_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B5)) ) ) ) ).

% sum_diff_nat
tff(fact_1535_sum__diff1__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A),A2: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(A,nat,F3,A2)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)) ).

% sum_diff1_nat
tff(fact_1536_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,B4)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5)) ) ) ) ) ).

% sum_mono2
tff(fact_1537_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),Xa: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_1538_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Na: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13)))))))) ) ).

% max_idx_list
tff(fact_1539_cnt__non__neg,axiom,
    ! [T2: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)) ).

% cnt_non_neg
tff(fact_1540_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,size_size(word(A)),W2)) ) ).

% word_size_gt_0
tff(fact_1541_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Foo),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13))))) ) ).

% height_i_max
tff(fact_1542_Bolzano,axiom,
    ! [A2: real,B2: real,P2: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B4: real,C2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P2,A4),B4)
           => ( aa(real,$o,aa(real,fun(real,$o),P2,B4),C2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B4)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B4),C2)
                 => aa(real,$o,aa(real,fun(real,$o),P2,A4),C2) ) ) ) )
       => ( ! [X: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
               => ? [D4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                    & ! [A4: real,B4: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B4)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B4),A4)),D4) )
                       => aa(real,$o,aa(real,fun(real,$o),P2,A4),B4) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P2,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1543_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_1544_option_Osize_I4_J,axiom,
    ! [A: $tType,X2: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_1545_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_1546_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xa) = Xa ) ).

% max_bot
tff(fact_1547_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),bot_bot(A)) = Xa ) ).

% max_bot2
tff(fact_1548_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)) ).

% max_Suc_Suc
tff(fact_1549_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1550_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_1551_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1552_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_1553_max__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Na) = Na ).

% max_0L
tff(fact_1554_max__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),zero_zero(nat)) = Na ).

% max_0R
tff(fact_1555_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_1556_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_1557_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M7) = zero_zero(nat) )
    <=> ( M7 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_1558_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ).

% Max_insert
tff(fact_1559_union__diff__assoc,axiom,
    ! [A: $tType,C4: multiset(A),B5: multiset(A),A5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),B5) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5)),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C4)) ) ) ).

% union_diff_assoc
tff(fact_1560_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_1561_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Xa ) ) ) ).

% max_absorb1
tff(fact_1562_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_1563_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_1564_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: 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),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_1565_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: 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),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_1566_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).

% nat_add_max_right
tff(fact_1567_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Q3)) ).

% nat_add_max_left
tff(fact_1568_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_max_right
tff(fact_1569_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ).

% nat_mult_max_left
tff(fact_1570_nat__minus__add__max,axiom,
    ! [Na: 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),Na),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M) ).

% nat_minus_add_max
tff(fact_1571_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ).

% Max.in_idem
tff(fact_1572_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Leaf((A2),(B2))) = one_one(real) ).

% VEBT_internal.cnt.simps(1)
tff(fact_1573_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,Xa: A,A2: word(B),C3: word(B),B2: word(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C3)),B2)
           => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),A2),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C3))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),C3),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),minus_minus(word(B)),B2),A2)) ) ) ) ) ).

% le_minus
tff(fact_1574_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),V3: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W2) = zero_zero(nat) )
         => ( V3 = W2 ) ) ) ).

% size_0_eq
tff(fact_1575_lens__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(word(A),nat,size_size(word(A)),W2) != zero_zero(nat) ) ).

% lens_not_0
tff(fact_1576_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),V3: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W2) = zero_zero(nat) )
         => ( W2 = V3 ) ) ) ).

% size_0_same'
tff(fact_1577_option_Osize__neq,axiom,
    ! [A: $tType,Xa: option(A)] : aa(option(A),nat,size_size(option(A)),Xa) != zero_zero(nat) ).

% option.size_neq
tff(fact_1578_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),multiset(B),aa(fun(A,multiset(B)),fun(set(A),multiset(B)),groups7311177749621191930dd_sum(A,multiset(B)),F3),A5) = zero_zero(multiset(B)) )
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => ( aa(A,multiset(B),F3,X5) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_1579_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N6: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H2,X)),aa(A,A,H2,Y3))
         => ( aa(set(A),$o,finite_finite2(A),N6)
           => ( ( N6 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic643756798349783984er_Max(A),N6)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image(A,A,H2),N6)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_1580_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B5)),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) = aa(set(A),A,lattic643756798349783984er_Max(A),A5) ) ) ) ) ) ).

% Max.subset
tff(fact_1581_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A5),A5) ) ) ) ) ).

% Max.closed
tff(fact_1582_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A5)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_1583_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_1584_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_1585_nonempty__has__size,axiom,
    ! [A: $tType,S2: multiset(A)] :
      ( ( S2 != zero_zero(multiset(A)) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S2)) ) ).

% nonempty_has_size
tff(fact_1586_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M7: multiset(A),M9: multiset(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(multiset(A),nat,size_size(multiset(A)),M9))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M7),M9))) ).

% diff_size_le_size_Diff
tff(fact_1587_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_1588_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_1589_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_1590_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.bounded_iff
tff(fact_1591_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1592_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_1593_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X2: A] : size_option(A,Xa,aa(A,option(A),some(A),X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_1594_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_option(A,Xa,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1595_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
        <=> ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_1596_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),zero_zero(word(A)))
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_1597_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_order.extremum
tff(fact_1598_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2)
        <=> ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_1599_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_n1_ge
tff(fact_1600_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% max_word_max
tff(fact_1601_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),A2) ) ).

% word_coorder.extremum
tff(fact_1602_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
         => ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1603_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_1604_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> ( ( Xa = Y )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_le_less_eq
tff(fact_1605_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_1606_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_1607_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_1608_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ) ).

% sub_wrap
tff(fact_1609_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y7: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y7),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1610_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_1611_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),I)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A))))
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_1612_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C3))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),Ab)
           => ( ( C3 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C3) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_1613_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C3))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C3)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1614_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_1615_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Z))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_1616_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),Xb: word(A),Xa: word(A),W2: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1617_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),Xb: word(A),W2: word(A),Xa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),Xc)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xc)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1618_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1619_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1620_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)) ) ) ) ).

% word_diff_ls(1)
tff(fact_1621_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)) ) ) ) ).

% word_diff_ls(2)
tff(fact_1622_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1623_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C3))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),Ab)
           => ( ( C3 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C3) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1624_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P2,zero_zero(word(A)))
         => ( ! [N: word(A)] :
                ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),N),M)
               => ( aa(word(A),$o,P2,N)
                 => aa(word(A),$o,P2,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) ) )
           => aa(word(A),$o,P2,M) ) ) ) ).

% word_induct_less
tff(fact_1625_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y7: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y7)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y7),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1626_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A5: multiset(A),C4: multiset(A),B5: multiset(A),D5: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),A5),C4)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B5),D5)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D5)) ) ) ) ).

% union_less_mono
tff(fact_1627_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D5: multiset(A),C4: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B5),D5)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C4),D5)) ) ) ).

% union_le_mono2
tff(fact_1628_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D5: multiset(A),C4: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),B5),D5)
         => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D5),C4)) ) ) ).

% union_le_mono1
tff(fact_1629_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),Xa)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_1630_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Ab: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Ab),C3))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),C3)) ) ) ) ).

% plus_minus_no_overflow_ab
tff(fact_1631_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C3: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3)),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),A2)) ) ) ) ).

% le_minus'
tff(fact_1632_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),A2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3)),B2) ) ) ) ).

% le_plus'
tff(fact_1633_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C3: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),A2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3)),B2) ) ) ) ).

% le_plus
tff(fact_1634_word__plus__mcs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),Xb: word(A),Xa: word(A),W2: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)) ) ) ) ).

% word_plus_mcs(3)
tff(fact_1635_word__plus__mcs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),Xb: word(A),W2: word(A),Xa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xb)),Xc)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xc)) ) ) ) ).

% word_plus_mcs(4)
tff(fact_1636_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_1637_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(4)
tff(fact_1638_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xaa)) ) ) ) ).

% word_diff_ls(3)
tff(fact_1639_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xb)) ) ) ) ).

% word_diff_ls(4)
tff(fact_1640_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3)) ) ) ) ).

% word_plus_mono_right2
tff(fact_1641_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)) ) ) ) ).

% word_plus_mono_right
tff(fact_1642_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z)) ) ) ) ).

% word_le_plus_either
tff(fact_1643_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) ) ) ) ).

% le_no_overflow
tff(fact_1644_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa)) ) ) ).

% olen_add_eqv
tff(fact_1645_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_le_minus_mono_left
tff(fact_1646_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z) ) ) ) ).

% word_le_minus_cancel
tff(fact_1647_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C3: word(A),D2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),C3)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C3),D2)),C3)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C3),D2)) ) ) ) ) ) ).

% word_le_minus_mono
tff(fact_1648_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Xa)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa) ) ) ).

% word_sub_le_iff
tff(fact_1649_word__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Xa) ) ) ).

% word_sub_le
tff(fact_1650_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2)
         => ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_1651_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Z) ) ) ).

% sub_wrap_lt
tff(fact_1652_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y))
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ).

% word_sub_less_iff
tff(fact_1653_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C3: word(A),D2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),C3)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),D2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C3),D2)),C3)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),C3),D2)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_1654_mset__distrib,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A),M7: multiset(A),N6: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M7),N6) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M7 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N6 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_1655_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ) ).

% max.mono
tff(fact_1656_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_1657_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_1658_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% max.boundedE
tff(fact_1659_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2) ) ) ) ).

% max.boundedI
tff(fact_1660_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.order_iff
tff(fact_1661_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded1
tff(fact_1662_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% max.cobounded2
tff(fact_1663_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_1664_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_1665_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1666_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_1667_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_1668_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1669_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1670_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1671_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_1672_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_1673_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N2: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),N2),Na)
         => ( ( N2 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),N2)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_1674_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1675_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1676_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1677_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1678_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1679_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W2) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1680_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1681_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),one_one(word(A)))
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1682_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xa)
        <=> ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_1683_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),W2)
        <=> ( W2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_1684_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N6: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less_eq(multiset(A)),M7),N6)
        <=> ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N6)
            | ( M7 = N6 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_1685_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xa) ) ).

% max_word_not_less
tff(fact_1686_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),M7) ) ).

% mset_le_not_refl
tff(fact_1687_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N6: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N6)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N6),M7) ) ) ).

% mset_le_not_sym
tff(fact_1688_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A)] : ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),M7) ) ).

% mset_le_irrefl
tff(fact_1689_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K4: multiset(A),M7: multiset(A),N6: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K4),M7)
         => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N6)
           => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),K4),N6) ) ) ) ).

% mset_le_trans
tff(fact_1690_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M7: multiset(A),N6: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),M7),N6)
         => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less(multiset(A)),N6),M7) ) ) ).

% mset_le_asym
tff(fact_1691_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1692_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1693_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1694_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( ( W2 != zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),W2) ) ) ).

% word_neq_0_conv
tff(fact_1695_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Na)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na) ) ) ).

% word_gt_a_gt_0
tff(fact_1696_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A2)
        <=> ( A2 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1697_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Y) ) ).

% word_not_simps(3)
tff(fact_1698_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2) ) ).

% word_order.extremum_strict
tff(fact_1699_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( A2 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_1700_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
        <=> ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ).

% word_le_not_less
tff(fact_1701_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ).

% word_not_le
tff(fact_1702_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1703_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_1704_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))))
          <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa)
              | ( Y = Xa ) ) ) ) ) ).

% less_x_plus_1
tff(fact_1705_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),K)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_1706_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),M)
         => ? [N: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_1707_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1708_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Xa) ) ) ).

% gt0_iff_gem1
tff(fact_1709_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A))) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_less_cases
tff(fact_1710_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_1711_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_le_make_less
tff(fact_1712_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Xa: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),K) ) ) ) ).

% word_Suc_leq
tff(fact_1713_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),K: word(A)] :
          ( ( Xa != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),K)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),K) ) ) ) ).

% word_Suc_le
tff(fact_1714_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1715_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Na) ) ) ).

% plus_one_helper
tff(fact_1716_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Na))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),Na)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A2)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_1717_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),C3),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C3)) ) ) ) ).

% word_le_plus
tff(fact_1718_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),Xa)) ) ) ) ).

% word_plus_mono_left
tff(fact_1719_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),W2)),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),W2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),Xa) ) ) ) ).

% word_add_increasing
tff(fact_1720_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),V3: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),V3)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),W2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),V3))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V3),W2) ) ) ) ).

% word_plus_mcs_4'
tff(fact_1721_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),J),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff2
tff(fact_1722_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),Xa: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V3),W2) ) ) ) ).

% word_plus_mcs_4
tff(fact_1723_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V3),W2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V3),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)) ) ) ) ).

% word_plus_mcs_3
tff(fact_1724_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: word(A),D2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C3),D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),A2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C3),D2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),D2) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_1725_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),X6: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),X6))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),X6)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),Xa)) ) ) ) ).

% word_random
tff(fact_1726_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),Xa)
        <=> ( Xa = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_1727_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),P3)),P3)
        <=> ( P3 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_1728_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Na)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_1729_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1730_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( ( Xa != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1731_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),K)),M) ) ) ) ).

% word_le_imp_diff_le
tff(fact_1732_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Xa)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Z)) ) ) ) ) ).

% word_le_minus_mono_right
tff(fact_1733_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_1734_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),Na)
         => ( ( I != Na )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A)))) ) ) ) ).

% le_step_down_word
tff(fact_1735_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1736_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( ( Xa != zero_zero(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1737_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),M),Na)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1738_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),Xa) ) ) ) ).

% le_m1_iff_lt
tff(fact_1739_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),M)
            & ( Na != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1740_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_1741_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Y)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_1742_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),K)),M) ) ) ) ).

% word_less_imp_diff_less
tff(fact_1743_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A)))),Xa) ) ) ).

% word_sub_1_le
tff(fact_1744_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Na),one_one(word(A))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Na),Xa)
           => ( Na = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1745_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_1746_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Z: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_1747_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W2) ) ) ) ).

% word_diff_ls'(4)
tff(fact_1748_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),W2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)) ) ) ) ).

% word_diff_ls'(3)
tff(fact_1749_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% word_l_diffs'(4)
tff(fact_1750_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_1751_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W2: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa)) ) ) ) ).

% word_diff_ls''(4)
tff(fact_1752_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A),W2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Y),Xa)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)) ) ) ) ).

% word_diff_ls''(3)
tff(fact_1753_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Z)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1754_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W2),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Z),Xa)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1755_sorted__in__between,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: nat,J: nat,L: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
             => ( sorted_wrt(A,ord_less_eq(A),L)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),Xa)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),J))
                   => ~ ! [K2: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K2)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),J)
                           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),K2)),Xa)
                             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),one_one(nat)))) ) ) ) ) ) ) ) ) ) ) ).

% sorted_in_between
tff(fact_1756_udvd__incr2__K,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),A2: word(A),Sa: word(A),K4: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa))
           => ( udvd(A,K4,Sa)
             => ( udvd(A,K4,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),P3),A2))
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),P3)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K4)
                   => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4))
                      & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),Sa)) ) ) ) ) ) ) ) ) ).

% udvd_incr2_K
tff(fact_1757_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Na,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),Na),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_1758_word__order_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ordering_top(word(A),ord_less_eq(word(A)),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_order.ordering_top_axioms
tff(fact_1759_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A5: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_1760_nth__step__trancl,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),Na: nat,M: nat] :
      ( ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
         => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N))),aa(nat,A,nth(A,Xs),N)),R3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => 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),Na)),aa(nat,A,nth(A,Xs),M)),transitive_trancl(A,R3)) ) ) ) ).

% nth_step_trancl
tff(fact_1761_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_1762_length__nth__simps_I4_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xs),Na) ).

% length_nth_simps(4)
tff(fact_1763_nth__Cons__Suc,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),aa(nat,nat,suc,Na)) = aa(nat,A,nth(A,Xs),Na) ).

% nth_Cons_Suc
tff(fact_1764_length__nth__simps_I3_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),zero_zero(nat)) = Xa ).

% length_nth_simps(3)
tff(fact_1765_nth__Cons__0,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),zero_zero(nat)) = Xa ).

% nth_Cons_0
tff(fact_1766_trancl__single,axiom,
    ! [A: $tType,A2: A,B2: A] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),bot_bot(set(product_prod(A,A))))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),bot_bot(set(product_prod(A,A)))) ).

% trancl_single
tff(fact_1767_listI__assn__finite,axiom,
    ! [B: $tType,A: $tType,I5: set(nat),A5: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),I5)
     => ( vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) = bot_bot(assn) ) ) ).

% listI_assn_finite
tff(fact_1768_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] : enumerate(A,Na,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),cons(product_prod(nat,A)),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Na),Xa)),enumerate(A,aa(nat,nat,suc,Na),Xs)) ).

% enumerate_simps(2)
tff(fact_1769_list__tail__coinc,axiom,
    ! [A: $tType,N1: A,R1: list(A),N22: A,R22: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N1),R1) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N22),R22) )
     => ( ( N1 = N22 )
        & ( R1 = R22 ) ) ) ).

% list_tail_coinc
tff(fact_1770_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_1771_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))
        <=> ( ! [X5: A] :
                ( member(A,X5,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X5) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_1772_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys))
        <=> ( ! [X5: A] :
                ( member(A,X5,aa(list(A),set(A),set2(A),Ys))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X5) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_1773_trancl__mono__mp,axiom,
    ! [A: $tType,U2: set(product_prod(A,A)),V2: set(product_prod(A,A)),Xa: product_prod(A,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),U2),V2)
     => ( member(product_prod(A,A),Xa,transitive_trancl(A,U2))
       => member(product_prod(A,A),Xa,transitive_trancl(A,V2)) ) ) ).

% trancl_mono_mp
tff(fact_1774_trancl__sub,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R3),transitive_trancl(A,R3)) ).

% trancl_sub
tff(fact_1775_trancl__sub__insert__trancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xa: product_prod(A,A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),Xa),R3))) ).

% trancl_sub_insert_trancl
tff(fact_1776_strict__sorted__imp__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% strict_sorted_imp_sorted
tff(fact_1777_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_1778_length__nth__simps_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_nth_simps(2)
tff(fact_1779_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Na) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Na ) ) ) ).

% length_Suc_conv
tff(fact_1780_Suc__length__conv,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Na) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Na ) ) ) ).

% Suc_length_conv
tff(fact_1781_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys) ) ) ).

% impossible_Cons
tff(fact_1782_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs))) ).

% set_subset_Cons
tff(fact_1783_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A5: fun(A,fun(B,assn)),A10: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I2: nat] :
                ( member(nat,I2,I5)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                   => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                      & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Xsi),I2)) = aa(B,assn,aa(A,fun(B,assn),A10,aa(nat,A,nth(A,Xs4),I2)),aa(nat,B,nth(B,Xsi2),I2)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A10,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_1784_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A5: fun(A,fun(B,assn)),A10: fun(A,fun(B,assn)),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A5 = A10 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I2: nat] :
                  ( member(nat,I2,I5)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                     => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                        & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A10,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_1785_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P2,Xs) ) ).

% sorted_wrt01
tff(fact_1786_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P2,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_1787_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P2,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_1788_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_1789_Suc__le__length__iff,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Ys3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_1790_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_1791_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_1792_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_1793_nth__Cons_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),Na) = $ite(Na = zero_zero(nat),Xa,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ).

% nth_Cons'
tff(fact_1794_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A5: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),I5),A5,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_1795_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_1796_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_1797_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_1798_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ~ ! [L2: list(A)] :
                ( sorted_wrt(A,ord_less(A),L2)
               => ( ( aa(list(A),set(A),set2(A),L2) = A5 )
                 => ( aa(list(A),nat,size_size(list(A)),L2) != aa(set(A),nat,finite_card(A),A5) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_1799_udvd__minus__le_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xy: word(A),K: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xy),K)
         => ( udvd(A,Z,Xy)
           => ( udvd(A,Z,K)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xy),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),K),Z)) ) ) ) ) ).

% udvd_minus_le'
tff(fact_1800_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Y: A,Xs: list(A),Na: nat] :
      ( ( Xa != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),Na) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) = Y )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_1801_nth__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),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),Xa),Xs)),Na) = Xa )
        <=> ( Na = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_1802_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xa: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),End) ),
        aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),slice(A,Begin,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs)),
        slice(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Begin),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),End),one_one(nat)),Xs) ) ).

% slice_Cons
tff(fact_1803_length__Cons,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_1804_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P2: assn,A5: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,Q2: assn] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi)),F4),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_1805_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( linord4507533701916653071of_set(A,A5) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_1806_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A5 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A5) ) )
          <=> ( linord4507533701916653071of_set(A,A5) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_1807_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A)),Sa: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P3,transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),Sa)
       => member(product_prod(A,A),P3,transitive_trancl(A,Sa)) ) ) ).

% trancl_mono
tff(fact_1808_converse__trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),B2),R2)
           => aa(A,$o,P2,Y3) )
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2),transitive_trancl(A,R2))
               => ( aa(A,$o,P2,Z3)
                 => aa(A,$o,P2,Y3) ) ) )
         => aa(A,$o,P2,A2) ) ) ) ).

% converse_trancl_induct
tff(fact_1809_trancl__trans__induct,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),P2: fun(A,fun(A,$o))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R2))
     => ( ! [X: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),R2)
           => aa(A,$o,aa(A,fun(A,$o),P2,X),Y3) )
       => ( ! [X: A,Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),transitive_trancl(A,R2))
             => ( aa(A,$o,aa(A,fun(A,$o),P2,X),Y3)
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),transitive_trancl(A,R2))
                 => ( aa(A,$o,aa(A,fun(A,$o),P2,Y3),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),P2,X),Z3) ) ) ) )
         => aa(A,$o,aa(A,fun(A,$o),P2,Xa),Y) ) ) ) ).

% trancl_trans_induct
tff(fact_1810_groupy,axiom,
    ! [A5: assn,B5: assn,C4: assn,D5: assn,X4: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),D5)),X4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B5)),C4)),D5),X4) ) ).

% groupy
tff(fact_1811_midextr,axiom,
    ! [P2: assn,Q2: assn,Q4: assn,R3: assn,X4: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),Q4)),R3),X4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),R3)),Q2)),Q4),X4) ) ).

% midextr
tff(fact_1812_swappa,axiom,
    ! [B5: assn,A5: assn,C4: assn,X4: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),A5)),C4),X4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B5)),C4),X4) ) ).

% swappa
tff(fact_1813_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A5)) = A5 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_1814_ent__false__iff,axiom,
    ! [P2: assn] :
      ( entails(P2,bot_bot(assn))
    <=> ! [H4: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H4) ) ).

% ent_false_iff
tff(fact_1815_ent__trans,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( entails(P2,Q2)
     => ( entails(Q2,R3)
       => entails(P2,R3) ) ) ).

% ent_trans
tff(fact_1816_ent__refl,axiom,
    ! [P2: assn] : entails(P2,P2) ).

% ent_refl
tff(fact_1817_ent__iffI,axiom,
    ! [A5: assn,B5: assn] :
      ( entails(A5,B5)
     => ( entails(B5,A5)
       => ( A5 = B5 ) ) ) ).

% ent_iffI
tff(fact_1818_ent__star__mono,axiom,
    ! [P2: assn,P4: assn,Q2: assn,Q4: assn] :
      ( entails(P2,P4)
     => ( entails(Q2,Q4)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P4),Q4)) ) ) ).

% ent_star_mono
tff(fact_1819_entails__def,axiom,
    ! [P2: assn,Q2: assn] :
      ( entails(P2,Q2)
    <=> ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H4)
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H4) ) ) ).

% entails_def
tff(fact_1820_entailsI,axiom,
    ! [P2: assn,Q2: assn] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H)
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H) )
     => entails(P2,Q2) ) ).

% entailsI
tff(fact_1821_entailsD,axiom,
    ! [P2: assn,Q2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( entails(P2,Q2)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H2) ) ) ).

% entailsD
tff(fact_1822_ent__fwd,axiom,
    ! [P2: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Q2: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
     => ( entails(P2,Q2)
       => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H2) ) ) ).

% ent_fwd
tff(fact_1823_ent__false,axiom,
    ! [P2: assn] : entails(bot_bot(assn),P2) ).

% ent_false
tff(fact_1824_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ( linord4507533701916653071of_set(A,A5) = linord4507533701916653071of_set(A,B5) )
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( A5 = B5 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_1825_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A5)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_1826_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A5)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_1827_trancl__induct2,axiom,
    ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set(product_prod(product_prod(A,B),product_prod(A,B))),P2: fun(A,fun(B,$o))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)),transitive_trancl(product_prod(A,B),R2))
     => ( ! [A4: A,B4: B] :
            ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),R2)
           => aa(B,$o,aa(A,fun(B,$o),P2,A4),B4) )
       => ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
              ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),transitive_trancl(product_prod(A,B),R2))
             => ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),R2)
               => ( aa(B,$o,aa(A,fun(B,$o),P2,A4),B4)
                 => aa(B,$o,aa(A,fun(B,$o),P2,Aa2),Ba) ) ) )
         => aa(B,$o,aa(A,fun(B,$o),P2,Bx),By) ) ) ) ).

% trancl_induct2
tff(fact_1828_trancl_Ocases,axiom,
    ! [A: $tType,A12: A,A23: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23),R2)
       => ~ ! [B4: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),B4),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A23),R2) ) ) ) ).

% trancl.cases
tff(fact_1829_trancl_Osimps,axiom,
    ! [A: $tType,A12: A,A23: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A12),A23),transitive_trancl(A,R2))
    <=> ( ? [A7: A,B7: A] :
            ( ( A12 = A7 )
            & ( A23 = B7 )
            & 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),R2) )
        | ? [A7: A,B7: A,C7: A] :
            ( ( A12 = A7 )
            & ( A23 = C7 )
            & 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,R2))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B7),C7),R2) ) ) ) ).

% trancl.simps
tff(fact_1830_trancl_Or__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2)) ) ).

% trancl.r_into_trancl
tff(fact_1831_tranclE,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
       => ~ ! [C2: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C2),transitive_trancl(A,R2))
             => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2),R2) ) ) ) ).

% tranclE
tff(fact_1832_trancl__trans,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Z: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z),transitive_trancl(A,R2)) ) ) ).

% trancl_trans
tff(fact_1833_trancl__induct,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),P2: fun(A,$o)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( ! [Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),R2)
           => aa(A,$o,P2,Y3) )
       => ( ! [Y3: A,Z3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),Y3),transitive_trancl(A,R2))
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
               => ( aa(A,$o,P2,Y3)
                 => aa(A,$o,P2,Z3) ) ) )
         => aa(A,$o,P2,B2) ) ) ) ).

% trancl_induct
tff(fact_1834_r__r__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R3)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R3)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R3)) ) ) ).

% r_r_into_trancl
tff(fact_1835_converse__tranclE,axiom,
    ! [A: $tType,Xa: A,Z: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z),transitive_trancl(A,R2))
     => ( ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Z),R2)
       => ~ ! [Y3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y3),R2)
             => ~ 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,R2)) ) ) ) ).

% converse_tranclE
tff(fact_1836_irrefl__trancl__rD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Y: A] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
       => ( Xa != Y ) ) ) ).

% irrefl_trancl_rD
tff(fact_1837_Transitive__Closure_Otrancl__into__trancl,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),transitive_trancl(A,R2))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% Transitive_Closure.trancl_into_trancl
tff(fact_1838_trancl__into__trancl2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),C3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3),transitive_trancl(A,R2))
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),C3),transitive_trancl(A,R2)) ) ) ).

% trancl_into_trancl2
tff(fact_1839_local_Oext,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_1840_mod__frame__fwd,axiom,
    ! [Ps: assn,H2: product_prod(heap_ext(product_unit),set(nat)),P2: assn,R3: assn,F4: assn] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Ps),H2)
     => ( entails(P2,R3)
       => ( entails(Ps,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),F4))
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R3),F4)),H2) ) ) ) ).

% mod_frame_fwd
tff(fact_1841_assnle,axiom,
    ! [TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] : entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ).

% assnle
tff(fact_1842_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: assn,A5: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F4: assn,C3: heap_Time_Heap(C),Q2: fun(C,assn)] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi)),F4),C3,Q2)
           => hoare_hoare_triple(C,P2,C3,Q2) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_1843_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P2: assn,A5: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F4: assn,Q2: assn] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A5,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F4),Q2)
           => entails(P2,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_1844_recomp,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% recomp
tff(fact_1845_repack,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),Rest: assn,X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),Rest)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Rest),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% repack
tff(fact_1846_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U)) ).

% finite_atLeastLessThan
tff(fact_1847_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_1848_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_1849_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_1850_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_1851_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_1852_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Na) ) ) ) ) ).

% ivl_subset
tff(fact_1853_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Na)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Na)) = set_or7035219750837199246ssThan(A,Na,M) ) ) ) ).

% ivl_diff
tff(fact_1854_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_1855_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_1856_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list(vEBT_VEBT),TreeList3: list(A),Y: vEBT_VEBT,Xa: vEBT_VEBTi,X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(list(A),nat,size_size(list(A)),TreeList3) )
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Y),Xa)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xa)))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y),list_update(vEBT_VEBTi,Tree_is,I,Xa))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y)),list_update(vEBT_VEBTi,Tree_is,I,Xa)))) ) ) ).

% tcd
tff(fact_1857_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,Xa: A] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,list_update(A,L,J,Xa)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_1858_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_1859_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_1860_txe,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% txe
tff(fact_1861_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_1862_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G3,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_1863_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_1864_deleti_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_V1365221501068881998eletei(T2,Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(T2,Xa))) ) ).

% deleti'_rf_abstr
tff(fact_1865_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D2) )
            <=> ( ( A2 = C3 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_1866_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
             => ( A2 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_1867_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C3,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_1868_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs: list(A),A5: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ( Na = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na),A5,Xs,Xsi) = aa(list(B),assn,vEBT_List_list_assn(A,B,A5,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_1869_list__assn__conv__idx,axiom,
    ! [A: $tType,B: $tType,A5: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,A5,Xs),Xsi) = vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs)),A5,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_1870_list__update__code_I3_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_1871_list__update__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),zero_zero(nat),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).

% list_update_code(2)
tff(fact_1872_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A),Xa: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5)
     => ( member(A,Xa,A5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),A5) ) ) ).

% set_update_subsetI
tff(fact_1873_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_1874_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_1875_ex__nat__less__eq,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
          & aa(nat,$o,P2,M5) )
    <=> ? [X5: nat] :
          ( member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P2,X5) ) ) ).

% ex_nat_less_eq
tff(fact_1876_all__nat__less__eq,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),Na)
         => aa(nat,$o,P2,M5) )
    <=> ! [X5: nat] :
          ( member(nat,X5,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P2,X5) ) ) ).

% all_nat_less_eq
tff(fact_1877_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_1878_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs: list(A),A5: fun(A,fun(B,assn)),Xsi: list(B),F4: assn] :
      ( ( Na = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na),A5,Xs,Xsi)),F4) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A5,Xs),Xsi)),F4) ) ) ).

% listI_assn_conv'
tff(fact_1879_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N6) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_1880_subset__card__intvl__is__intvl,axiom,
    ! [A5: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),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_1881_extract__pre__list__assn__lengthD,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(list(B),assn,vEBT_List_list_assn(A,B,A5,Xs),Xsi)),H2)
     => ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_1882_set__update__memI,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_1883_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ( member(A,Xa,aa(list(A),set(A),set2(A),L))
            & ! [Y5: A] : member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_1884_in__set__upd__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),I: nat,Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xa != Y ) )
       => member(A,Xa,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_1885_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A,Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xa = Y )
          | ! [Y5: A] : member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,L,I,Y5))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_1886_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xa: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,Xa)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L)) ),
        Xa,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_1887_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xa) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_1888_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),J) = $ite(I = J,Xa,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_1889_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_1890_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C3: A,B2: A,D2: A,G3: fun(A,B),H2: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D2 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),set_or7035219750837199246ssThan(A,C3,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_1891_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,P3: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_1892_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,P3: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M,P3))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,Na,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_1893_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A5: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X2: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A5,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X2)) = vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_1894_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_1895_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N6),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N6)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_1896_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: assn,A5: fun(A,fun(B,assn)),Xa: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F4: assn,C3: heap_Time_Heap(C),Q2: fun(C,assn)] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,Xa),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A5,list_update(A,Xs,I,Xa),list_update(B,Xsi,I,Xi))),F4),C3,Q2)
           => hoare_hoare_triple(C,P2,C3,Q2) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_1897_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_1898_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G3,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_1899_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_1900_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_1901_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_1902_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_1903_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,L),I)),aa(list(A),set(A),set2(A),list_update(A,L,I,Xa))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_1904_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A5: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X2: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),I5),A5,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,X1),X2)),vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_1905_big__assn__simp_H,axiom,
    ! [H2: nat,TreeList: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xa: vEBT_VEBTi,Xb: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),H2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( Xaa = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xaa),Xa)),pure_assn(Xb = vEBT_vebt_mint(Xaa)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H2,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),H2),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H2,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xb = vEBT_vebt_mint(Xaa)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H2,Xaa)),list_update(vEBT_VEBTi,Tree_is,H2,Xa)))) ) ) ).

% big_assn_simp'
tff(fact_1906_big__assn__simp,axiom,
    ! [H2: nat,TreeList: list(vEBT_VEBT),L: nat,Xa: vEBT_VEBTi,Xaa: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),H2),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L)),Xa)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L))))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H2,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),H2),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H2,Xa))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L))))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H2,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L))),list_update(vEBT_VEBTi,Tree_is,H2,Xa)))) ) ).

% big_assn_simp
tff(fact_1907_listI__assn__wrap__insert,axiom,
    ! [A: $tType,P2: assn,Uu2: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set(nat),I: nat,Xs: list(vEBT_VEBT),Xsi: list(vEBT_VEBTi),F4: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Uu2,Uua)),Xi)),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),I5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,Xs,Xsi))),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,I5,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,Xs,I,vEBT_vebt_insert(Uu2,Uua)),list_update(vEBT_VEBTi,Xsi,I,Xi))),F4),C3,Q2)
           => hoare_hoare_triple(A,P2,C3,Q2) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_1908_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Na: nat,Xa: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Na,Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),Na)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_1909_hoare__triple__preI,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn)] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat))] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H)
         => hoare_hoare_triple(A,P2,C3,Q2) )
     => hoare_hoare_triple(A,P2,C3,Q2) ) ).

% hoare_triple_preI
tff(fact_1910_subset__mset_Osum__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(multiset(A)),Xa: multiset(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Xs))
     => ( groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),list_update(multiset(A),Xs,K,Xa)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Xs)),Xa)),aa(nat,multiset(A),nth(multiset(A),Xs),K)) ) ) ).

% subset_mset.sum_list_update
tff(fact_1911_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(set(nat),set(A),image(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_1912_pure__assn__eq__conv,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( pure_assn((P2)) = pure_assn((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% pure_assn_eq_conv
tff(fact_1913_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        & (B2) )) ).

% merge_pure_star
tff(fact_1914_take__update,axiom,
    ! [A: $tType,Na: nat,L: list(A),I: nat,Xa: A] : take(A,Na,list_update(A,L,I,Xa)) = list_update(A,take(A,Na,L),I,Xa) ).

% take_update
tff(fact_1915_pure__assn__eq__emp__iff,axiom,
    ! [P2: $o] :
      ( ( pure_assn((P2)) = one_one(assn) )
    <=> (P2) ) ).

% pure_assn_eq_emp_iff
tff(fact_1916_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_1917_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_1918_pure__assn__eq__false__iff,axiom,
    ! [P2: $o] :
      ( ( pure_assn((P2)) = bot_bot(assn) )
    <=> ~ (P2) ) ).

% pure_assn_eq_false_iff
tff(fact_1919_take__Suc__Cons,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] : take(A,aa(nat,nat,suc,Na),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),take(A,Na,Xs)) ).

% take_Suc_Cons
tff(fact_1920_take__all,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na)
     => ( take(A,Na,Xs) = Xs ) ) ).

% take_all
tff(fact_1921_take__all__iff,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( take(A,Na,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% take_all_iff
tff(fact_1922_nth__take,axiom,
    ! [A: $tType,I: nat,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,take(A,Na,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_1923_take__update__cancel,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( take(A,Na,list_update(A,Xs,M,Y)) = take(A,Na,Xs) ) ) ).

% take_update_cancel
tff(fact_1924_ent__pure__pre__iff,axiom,
    ! [P2: assn,B2: $o,Q2: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn((B2))),Q2)
    <=> ( (B2)
       => entails(P2,Q2) ) ) ).

% ent_pure_pre_iff
tff(fact_1925_mod__pure__star__dist,axiom,
    ! [P2: assn,B2: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),pure_assn((B2)))),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
        & (B2) ) ) ).

% mod_pure_star_dist
tff(fact_1926_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q2: assn] :
      ( entails(pure_assn((B2)),Q2)
    <=> ( (B2)
       => entails(one_one(assn),Q2) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_1927_ent__pure__post__iff,axiom,
    ! [P2: assn,Q2: assn,B2: $o] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),pure_assn((B2))))
    <=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H4)
           => (B2) )
        & entails(P2,Q2) ) ) ).

% ent_pure_post_iff
tff(fact_1928_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(pure_assn((B2))),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> (B2) ) ).

% mod_h_bot_iff(1)
tff(fact_1929_ent__pure__post__iff__sng,axiom,
    ! [P2: assn,B2: $o] :
      ( entails(P2,pure_assn((B2)))
    <=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H4)
           => (B2) )
        & entails(P2,one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_1930_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( distinct(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_1931_VEBT__internal_OperInsTrans_Ointros_I2_J,axiom,
    ! [T2: vEBT_VEBT,Sa: vEBT_VEBT,Xa: nat,U: vEBT_VEBT] :
      ( ( T2 = vEBT_vebt_insert(Sa,Xa) )
     => ( vEBT_V6289311342943941716sTrans(T2,U)
       => vEBT_V6289311342943941716sTrans(Sa,U) ) ) ).

% VEBT_internal.perInsTrans.intros(2)
tff(fact_1932_VEBT__internal_OperInsTrans_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(A12,A23)
     => ( ( A23 != A12 )
       => ~ ! [T7: vEBT_VEBT] :
              ( ? [X: nat] : T7 = vEBT_vebt_insert(A12,X)
             => ~ vEBT_V6289311342943941716sTrans(T7,A23) ) ) ) ).

% VEBT_internal.perInsTrans.cases
tff(fact_1933_VEBT__internal_OperInsTrans_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: vEBT_VEBT] :
      ( vEBT_V6289311342943941716sTrans(A12,A23)
    <=> ( ? [T4: vEBT_VEBT] :
            ( ( A12 = T4 )
            & ( A23 = T4 ) )
        | ? [T4: vEBT_VEBT,S6: vEBT_VEBT,X5: nat,U3: vEBT_VEBT] :
            ( ( A12 = S6 )
            & ( A23 = U3 )
            & ( T4 = vEBT_vebt_insert(S6,X5) )
            & vEBT_V6289311342943941716sTrans(T4,U3) ) ) ) ).

% VEBT_internal.perInsTrans.simps
tff(fact_1934_set__take__subset,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Na,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_1935_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Na: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,Na,Xs)) ) ) ).

% sorted_take
tff(fact_1936_finite__distinct__list,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A5 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_1937_finite__set__image,axiom,
    ! [A: $tType,A5: set(list(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),A5))
     => ( ! [Xs2: list(A)] :
            ( member(list(A),Xs2,A5)
           => distinct(A,Xs2) )
       => aa(set(list(A)),$o,finite_finite2(list(A)),A5) ) ) ).

% finite_set_image
tff(fact_1938_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_1939_strict__sorted__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less(A),L)
        <=> ( sorted_wrt(A,ord_less_eq(A),L)
            & distinct(A,L) ) ) ) ).

% strict_sorted_iff
tff(fact_1940_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I3 != J3 )
               => ( aa(nat,A,nth(A,Xs),I3) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_1941_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_1942_distinct__length__le,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( distinct(A,Ys)
     => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% distinct_length_le
tff(fact_1943_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_1944_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] : vEBT_VEBT_insert(vEBT_Leaf((A2),(B2)),Xa) = vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xa) ).

% VEBT_internal.insert'.simps(1)
tff(fact_1945_vebt__insert__code_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xa) = $ite(
        Xa = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xa = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert_code(1)
tff(fact_1946_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_1947_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ? [X: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X) = A5 )
              & sorted_wrt(A,ord_less_eq(A),X)
              & distinct(A,X)
              & ! [Y4: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y4) = A5 )
                    & sorted_wrt(A,ord_less_eq(A),Y4)
                    & distinct(A,Y4) )
                 => ( Y4 = X ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_1948_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
       => ? [X: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X) = Xa )
            & ! [Y4: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y4),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y4) = Xa ) )
               => ( Y4 = X ) ) ) ) ) ).

% distinct_Ex1
tff(fact_1949_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( linord4507533701916653071of_set(A,aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_1950_distinct__sorted__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( distinct(A,L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J)) ) ) ) ) ) ).

% distinct_sorted_mono
tff(fact_1951_distinct__sorted__strict__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_strict_mono_iff
tff(fact_1952_distinct__sorted__mono__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),I: nat,J: nat] :
          ( distinct(A,L)
         => ( sorted_wrt(A,ord_less_eq(A),L)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),L))
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,L),I)),aa(nat,A,nth(A,L),J))
                <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J) ) ) ) ) ) ) ).

% distinct_sorted_mono_iff
tff(fact_1953_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ member(A,A2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A2)) ) ) ).

% distinct_list_update
tff(fact_1954_vebt__insert_Osimps_I4_J,axiom,
    ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_1955_heaphelp,axiom,
    ! [A: $tType,Xa: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xb: vEBT_VEBTi,Na: nat,Xc: vEBT_VEBTi,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,
          rep_assn(aa(assn,assn,
              aa(assn,fun(assn,assn),times_times(assn),
                aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Xa),Tree_is)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Xb))),
                  pure_assn(( ( none(A) = none(A) )
                    & ( Na = Na ) )))),
              pure_assn(Xc = vEBT_Nodei(none(product_prod(nat,nat)),Na,Xa,Xb)))),
          H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(none(product_prod(nat,nat)),Na,TreeList,Summary)),Xc)),H2) ) ).

% heaphelp
tff(fact_1956_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,Sa),Xa) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,Sa) ).

% vebt_insert.simps(3)
tff(fact_1957_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts2,Sa),Xa) = vEBT_Node(Info,zero_zero(nat),Ts2,Sa) ).

% vebt_insert.simps(2)
tff(fact_1958_mergesort__remdups__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( distinct(A,mergesort_remdups(A,L))
          & sorted_wrt(A,ord_less_eq(A),mergesort_remdups(A,L))
          & ( aa(list(A),set(A),set2(A),mergesort_remdups(A,L)) = aa(list(A),set(A),set2(A),L) ) ) ) ).

% mergesort_remdups_correct
tff(fact_1959_subset__mset_Oelem__le__sum__list,axiom,
    ! [A: $tType,K: nat,Ns: list(multiset(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(multiset(A)),nat,size_size(list(multiset(A))),Ns))
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),nth(multiset(A),Ns),K)),groups4543113879258116180m_list(multiset(A),plus_plus(multiset(A)),zero_zero(multiset(A)),Ns)) ) ).

% subset_mset.elem_le_sum_list
tff(fact_1960_last__take__nth__conv,axiom,
    ! [A: $tType,Na: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),L))
     => ( ( Na != zero_zero(nat) )
       => ( last(A,take(A,Na,L)) = aa(nat,A,nth(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ) ).

% last_take_nth_conv
tff(fact_1961_VEBTi_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: array(vEBT_VEBTi),Y14: vEBT_VEBTi] :
      ( ( vEBT_Nodei(X11,X12,X13,X14) = vEBT_Nodei(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_1962_mset__le__incr__right2,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),B2)) ) ).

% mset_le_incr_right2
tff(fact_1963_mset__le__incr__right1,axiom,
    ! [A: $tType,A2: multiset(A),B2: multiset(A),C3: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B2),C3)) ) ).

% mset_le_incr_right1
tff(fact_1964_mset__le__decr__left2,axiom,
    ! [A: $tType,C3: multiset(A),A2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C3),A2)),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),B2) ) ).

% mset_le_decr_left2
tff(fact_1965_mset__le__decr__left1,axiom,
    ! [A: $tType,A2: multiset(A),C3: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A2),C3)),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),B2) ) ).

% mset_le_decr_left1
tff(fact_1966_mset__union__subset,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5)),C4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),C4)
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),C4) ) ) ).

% mset_union_subset
tff(fact_1967_mset__le__distrib,axiom,
    ! [A: $tType,X4: multiset(A),A5: multiset(A),B5: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),X4),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5))
     => ~ ! [Xa4: multiset(A),Xb2: multiset(A)] :
            ( ( X4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xa4),Xb2) )
           => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xa4),A5)
             => ~ aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xb2),B5) ) ) ) ).

% mset_le_distrib
tff(fact_1968_mset__le__addE,axiom,
    ! [A: $tType,Xs: multiset(A),Ys: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Xs),Ys)
     => ~ ! [Zs2: multiset(A)] : Ys != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xs),Zs2) ) ).

% mset_le_addE
tff(fact_1969_mset__le__subtract,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),B5)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),A5),C4)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),C4)) ) ).

% mset_le_subtract
tff(fact_1970_subset__mset_Olift__Suc__mono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),Na: nat,N2: nat] :
      ( ! [N: nat] : aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),F3,N)),aa(nat,multiset(A),F3,aa(nat,nat,suc,N)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),F3,Na)),aa(nat,multiset(A),F3,N2)) ) ) ).

% subset_mset.lift_Suc_mono_le
tff(fact_1971_subset__mset_Olift__Suc__antimono__le,axiom,
    ! [A: $tType,F3: fun(nat,multiset(A)),Na: nat,N2: nat] :
      ( ! [N: nat] : aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),F3,aa(nat,nat,suc,N))),aa(nat,multiset(A),F3,N))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
       => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(nat,multiset(A),F3,N2)),aa(nat,multiset(A),F3,Na)) ) ) ).

% subset_mset.lift_Suc_antimono_le
tff(fact_1972_mset__le__subtract__right,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A),X4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5)),X4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X4),B5))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),X4) ) ) ).

% mset_le_subtract_right
tff(fact_1973_mset__le__subtract__left,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A),X4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A5),B5)),X4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X4),A5))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),X4) ) ) ).

% mset_le_subtract_left
tff(fact_1974_size__mset__mono,axiom,
    ! [A: $tType,A5: multiset(A),B5: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A5),B5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),A5)),aa(multiset(A),nat,size_size(multiset(A)),B5)) ) ).

% size_mset_mono
tff(fact_1975_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd2: $o,Ve2: $o,V3: option(product_prod(nat,nat)),Va2: nat,Vb: array(vEBT_VEBTi),Vc: vEBT_VEBTi] : aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Leaf((Vd2),(Ve2))),vEBT_Nodei(V3,Va2,Vb,Vc)) = bot_bot(assn) ).

% vebt_assn_raw.simps(4)
tff(fact_1976_subset__mset_Osum__mono2,axiom,
    ! [B: $tType,A: $tType,B5: set(A),A5: set(A),F3: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => ( ! [B4: A] :
              ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
             => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F3,B4)) )
         => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,A5)),groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,B5)) ) ) ) ).

% subset_mset.sum_mono2
tff(fact_1977_subset__subseqs,axiom,
    ! [A: $tType,X4: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X4,aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_1978_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xa,Xs)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_1979_distinct__foldl__invar,axiom,
    ! [B: $tType,A: $tType,S2: list(A),I5: fun(set(A),fun(B,$o)),Sigma_0: B,F3: fun(B,fun(A,B))] :
      ( distinct(A,S2)
     => ( aa(B,$o,aa(set(A),fun(B,$o),I5,aa(list(A),set(A),set2(A),S2)),Sigma_0)
       => ( ! [X: A,It: set(A),Sigma: B] :
              ( member(A,X,It)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),It),aa(list(A),set(A),set2(A),S2))
               => ( aa(B,$o,aa(set(A),fun(B,$o),I5,It),Sigma)
                 => aa(B,$o,aa(set(A),fun(B,$o),I5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),It),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(A,B,aa(B,fun(A,B),F3,Sigma),X)) ) ) )
         => aa(B,$o,aa(set(A),fun(B,$o),I5,bot_bot(set(A))),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),Sigma_0),S2)) ) ) ) ).

% distinct_foldl_invar
tff(fact_1980_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys),I2)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_1981_take__butlast__conv,axiom,
    ! [A: $tType,L: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,L) ).

% take_butlast_conv
tff(fact_1982_take__minus__one__conv__butlast,axiom,
    ! [A: $tType,Na: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),L))
     => ( take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat))),L) = butlast(A,take(A,Na,L)) ) ) ).

% take_minus_one_conv_butlast
tff(fact_1983_subset__mset_Osum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType,F4: set(A),F3: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,F4) = zero_zero(multiset(B)) )
      <=> ! [X5: A] :
            ( member(A,X5,F4)
           => ( aa(A,multiset(B),F3,X5) = zero_zero(multiset(B)) ) ) ) ) ).

% subset_mset.sum_eq_0_iff
tff(fact_1984_Cons__in__lex,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xa = Y )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_1985_foldl__A1__eq,axiom,
    ! [A: $tType,F3: fun(A,fun(A,A)),Na: A,I: A,Ww: list(A)] :
      ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,Na),E2) = E2
     => ( ! [E2: A] : aa(A,A,aa(A,fun(A,A),F3,E2),Na) = E2
       => ( ! [A4: A,B4: A,C2: A] : aa(A,A,aa(A,fun(A,A),F3,A4),aa(A,A,aa(A,fun(A,A),F3,B4),C2)) = aa(A,A,aa(A,fun(A,A),F3,aa(A,A,aa(A,fun(A,A),F3,A4),B4)),C2)
         => ( aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),I),Ww) = aa(A,A,aa(A,fun(A,A),F3,I),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,F3),Na),Ww)) ) ) ) ) ).

% foldl_A1_eq
tff(fact_1986_foldl__foldl__conv__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,A)),A2: A,Xs: list(list(B))] : aa(list(list(B)),A,aa(A,fun(list(list(B)),A),foldl(A,list(B),foldl(A,B,F3)),A2),Xs) = aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),A2),concat(B,Xs)) ).

% foldl_foldl_conv_concat
tff(fact_1987_butlast__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xa: A] : list_update(A,butlast(A,L),I,Xa) = butlast(A,list_update(A,L,I,Xa)) ).

% butlast_update'
tff(fact_1988_foldl__absorb1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xa: A,Zs: list(A)] : aa(A,A,aa(A,fun(A,A),times_times(A),Xa),aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),one_one(A)),Zs)) = aa(list(A),A,aa(A,fun(list(A),A),foldl(A,A,times_times(A)),Xa),Zs) ) ).

% foldl_absorb1
tff(fact_1989_set__remove1__subset,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xa,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_1990_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_1991_nth__butlast,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),butlast(A,Xs)))
     => ( aa(nat,A,nth(A,butlast(A,Xs)),Na) = aa(nat,A,nth(A,Xs),Na) ) ) ).

% nth_butlast
tff(fact_1992_take__butlast,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,Na,butlast(A,Xs)) = take(A,Na,Xs) ) ) ).

% take_butlast
tff(fact_1993_distinct__butlast__swap,axiom,
    ! [A: $tType,Pq: list(A),I: nat] :
      ( distinct(A,Pq)
     => distinct(A,butlast(A,list_update(A,Pq,I,last(A,Pq)))) ) ).

% distinct_butlast_swap
tff(fact_1994_butlast__take,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( butlast(A,take(A,Na,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_1995_subset__mset_Osum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: set(A),T2: set(B),G3: fun(B,multiset(C)),I: fun(B,A),F3: fun(A,multiset(C))] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X: B] :
              ( member(B,X,T2)
             => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),zero_zero(multiset(C))),aa(B,multiset(C),G3,X)) )
         => ( ! [X: A] :
                ( member(A,X,Sa)
               => ? [Xa2: B] :
                    ( member(B,Xa2,T2)
                    & ( aa(B,A,I,Xa2) = X )
                    & aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),aa(A,multiset(C),F3,X)),aa(B,multiset(C),G3,Xa2)) ) )
           => aa(multiset(C),$o,aa(multiset(C),fun(multiset(C),$o),subseteq_mset(C),groups3894954378712506084id_sum(multiset(C),A,plus_plus(multiset(C)),zero_zero(multiset(C)),F3,Sa)),groups3894954378712506084id_sum(multiset(C),B,plus_plus(multiset(C)),zero_zero(multiset(C)),G3,T2)) ) ) ) ) ).

% subset_mset.sum_le_included
tff(fact_1996_subset__mset_Osum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,multiset(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F3,X)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,A5) = zero_zero(multiset(B)) )
        <=> ! [X5: A] :
              ( member(A,X5,A5)
             => ( aa(A,multiset(B),F3,X5) = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_eq_0_iff
tff(fact_1997_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) = remove1(A,Xa,linord4507533701916653071of_set(A,A5)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_1998_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),Na: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Na),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),Na),R2) )
        | ( ( M = Na )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_1999_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
    <=> ? [Y5: A,N5: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xs),N5)),Y5),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys = list_update(A,Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_2000_total__on__singleton,axiom,
    ! [A: $tType,Xa: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_2001_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys),I)) ) ) ) ).

% nth_zip
tff(fact_2002_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P2: assn,A5: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F4: assn,I: nat,C3: heap_Time_Heap(C),Q4: fun(C,assn),F6: fun(C,assn)] :
      ( entails(P2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A5,Xs),Xsi)),F4))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),F4),C3,Q4)
         => ( ! [R: C] : entails(aa(C,assn,Q4,R),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A5,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),I),bot_bot(set(nat)))),A5,Xs,Xsi))),aa(C,assn,F6,R)))
           => hoare_hoare_triple(C,P2,C3,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_aa(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A5),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_2003_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_lessThan(nat,Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_2004_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_2005_set__vebt_H__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_VEBT_set_vebt(T2) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(T2)) ).

% set_vebt'_def
tff(fact_2006_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( M = Na ) ) ) ).

% of_nat_eq_iff
tff(fact_2007_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ab(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_2008_finite__Collect__disjI,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ac(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q2)) ) ) ).

% finite_Collect_disjI
tff(fact_2009_finite__Collect__conjI,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q2)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2))) ) ).

% finite_Collect_conjI
tff(fact_2010_succ__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,Xa) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ae(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),Xa)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_2011_pred__empty,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,Xa) = none(nat) )
      <=> ( aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),Xa)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_2012_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C3: nat,D2: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,nth(nat,Xs),I2)),aa(nat,nat,nth(nat,Ys),I2)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),D2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),foldr(nat,nat,plus_plus(nat),Xs,C3)),aa(list(nat),nat,size_size(list(nat)),Ys))),foldr(nat,nat,plus_plus(nat),Ys,D2)) ) ) ) ).

% foldr_mono
tff(fact_2013_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_lessThan(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_2014_zip__eq__zip__same__len,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),A3: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A2,B2) = zip(A,B,A3,B3) )
        <=> ( ( A2 = A3 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_2015_finite__lessThan,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),set_ord_lessThan(nat,K)) ).

% finite_lessThan
tff(fact_2016_foldr__zero,axiom,
    ! [Xs: list(nat),D2: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),foldr(nat,nat,plus_plus(nat),Xs,D2)),D2)) ) ).

% foldr_zero
tff(fact_2017_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_ag(B,A)),A5) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_2018_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2019_singleton__conv,axiom,
    ! [A: $tType,A2: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ah(A,fun(A,$o),A2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2020_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_2021_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_2022_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Na))) = Na ).

% card_Collect_less_nat
tff(fact_2023_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_2024_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( zero_zero(nat) = Na ) ) ) ).

% of_nat_0_eq_iff
tff(fact_2025_semiring__1__class_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% semiring_1_class.of_nat_0
tff(fact_2026_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_iff
tff(fact_2027_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% of_nat_le_iff
tff(fact_2028_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = 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),Na)) ) ).

% of_nat_add
tff(fact_2029_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_2030_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_2031_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Na) = one_one(A) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_2032_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = 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),Na)) ) ).

% of_nat_mult
tff(fact_2033_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_lessThan(A,Xa)),set_ord_lessThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% lessThan_subset_iff
tff(fact_2034_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_2035_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( 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_ak(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_2036_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( 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_al(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_2037_foldl__length,axiom,
    ! [A: $tType,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_am(nat,fun(A,nat))),zero_zero(nat)),L) = aa(list(A),nat,size_size(list(A)),L) ).

% foldl_length
tff(fact_2038_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),Na))) = aa(nat,nat,suc,Na) ).

% card_Collect_le_nat
tff(fact_2039_finite__Collect__subsets,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_an(set(A),fun(set(A),$o),A5))) ) ).

% finite_Collect_subsets
tff(fact_2040_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2041_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_2042_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_lessThan(nat,Na))),aa(nat,A,G3,Na)) ) ).

% sum.lessThan_Suc
tff(fact_2043_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_2044_Cons__listrel1__Cons,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
          & ( Xs = Ys ) )
        | ( ( Xa = Y )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_2045_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_2046_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A5: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aTP_Lamp_ao(B,fun(A,B),C3)),A5)) = C3 ) ) ) ) ).

% Max_const
tff(fact_2047_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A5: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,aTP_Lamp_ao(B,fun(A,B),C3)),A5)) = C3 ) ) ) ) ).

% Min_const
tff(fact_2048_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_ap(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_2049_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% of_nat_0_less_iff
tff(fact_2050_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_2051_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),Xa) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_2052_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W2: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)),Xa) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_2053_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B2: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W2)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_2054_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_2055_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_aq(fun(nat,A),fun(nat,A),C3)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            aa(nat,A,C3,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_2056_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ar(set(A),fun(nat,fun(list(A),$o)),A5),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_2057_vebt__memberi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_V854960066525838166emberi(T2,Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T2),Tia),Xa)) ).

% vebt_memberi'_rf_abstr
tff(fact_2058_vebt__succi_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_VEBT_vebt_succi(T2,Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_at(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T2),Tia),Xa)) ) ).

% vebt_succi'_rf_abstr
tff(fact_2059_vebt__pred_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_VEBT_vebt_predi(T2,Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_au(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T2),Tia),Xa)) ) ).

% vebt_pred'_rf_abstr
tff(fact_2060_vebt__inserti_H__rf__abstr,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_V3964819847710782039nserti(T2,Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T2,Xa))) ).

% vebt_inserti'_rf_abstr
tff(fact_2061_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( ! [X5: A,Xa3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3),R3)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3),S2) )
    <=> ( R3 = S2 ) ) ).

% pred_equals_eq2
tff(fact_2062_constraint__expand,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Lower: word(A),Upper: word(A)] :
          ( member(word(A),Xa,aa(fun(word(A),$o),set(word(A)),collect(word(A)),aa(word(A),fun(word(A),$o),aTP_Lamp_av(word(A),fun(word(A),fun(word(A),$o)),Lower),Upper)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Lower),Xa)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Upper) ) ) ) ).

% constraint_expand
tff(fact_2063_pred__subset__eq,axiom,
    ! [A: $tType,R3: set(A),S2: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),R3)),aTP_Lamp_a(set(A),fun(A,$o),S2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R3),S2) ) ).

% pred_subset_eq
tff(fact_2064_less__eq__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5)) ) ).

% less_eq_set_def
tff(fact_2065_Collect__subset,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))),A5) ).

% Collect_subset
tff(fact_2066_prop__restrict,axiom,
    ! [A: $tType,Xa: A,Z5: set(A),X4: set(A),P2: fun(A,$o)] :
      ( member(A,Xa,Z5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),X4),P2)))
       => aa(A,$o,P2,Xa) ) ) ).

% prop_restrict
tff(fact_2067_Collect__restrict,axiom,
    ! [A: $tType,X4: set(A),P2: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),X4),P2))),X4) ).

% Collect_restrict
tff(fact_2068_subset__CollectI,axiom,
    ! [A: $tType,B5: set(A),A5: set(A),Q2: fun(A,$o),P2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
     => ( ! [X: A] :
            ( member(A,X,B5)
           => ( aa(A,$o,Q2,X)
             => aa(A,$o,P2,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),B5),Q2))),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))) ) ) ).

% subset_CollectI
tff(fact_2069_subset__Collect__iff,axiom,
    ! [A: $tType,B5: set(A),A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2)))
      <=> ! [X5: A] :
            ( member(A,X5,B5)
           => aa(A,$o,P2,X5) ) ) ) ).

% subset_Collect_iff
tff(fact_2070_insert__compr,axiom,
    ! [A: $tType,A2: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ax(A,fun(set(A),fun(A,$o)),A2),B5)) ).

% insert_compr
tff(fact_2071_insert__Collect,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(fun(A,$o),set(A),collect(A),P2)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ay(A,fun(fun(A,$o),fun(A,$o)),A2),P2)) ).

% insert_Collect
tff(fact_2072_Compl__eq,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(set(A),fun(A,$o),A5)) ).

% Compl_eq
tff(fact_2073_Collect__neg__eq,axiom,
    ! [A: $tType,P2: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ba(fun(A,$o),fun(A,$o),P2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2)) ).

% Collect_neg_eq
tff(fact_2074_uminus__set__def,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5))) ).

% uminus_set_def
tff(fact_2075_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bb(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_2076_less__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5)) ) ).

% less_set_def
tff(fact_2077_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bc(A,$o)) ).

% Set.empty_def
tff(fact_2078_minus__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),minus_minus(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5))) ).

% minus_set_def
tff(fact_2079_set__diff__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_bd(set(A),fun(set(A),fun(A,$o)),A5),B5)) ).

% set_diff_eq
tff(fact_2080_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),R3: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ! [X: A] :
              ( member(A,X,A5)
             => ? [Xa2: B] :
                  ( member(B,Xa2,B5)
                  & aa(B,$o,aa(A,fun(B,$o),R3,X),Xa2) ) )
         => ? [X: B] :
              ( member(B,X,B5)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A5),R3),X))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_2081_not__finite__existsD,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ? [X_1: A] : aa(A,$o,P2,X_1) ) ).

% not_finite_existsD
tff(fact_2082_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),A5: set(B)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F3),A5))
     => ~ ! [X: B] :
            ( ( B2 = aa(B,A,F3,X) )
           => ~ member(B,X,A5) ) ) ).

% imageE
tff(fact_2083_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G3: fun(C,B),A5: set(C)] : aa(set(B),set(A),image(B,A,F3),aa(set(C),set(B),image(C,B,G3),A5)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_bf(fun(B,A),fun(fun(C,B),fun(C,A)),F3),G3)),A5) ).

% image_image
tff(fact_2084_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),P2: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_bg(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F3),A5),P2)) = aa(set(B),set(A),image(B,A,F3),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_bh(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F3),A5),P2))) ).

% Compr_image_eq
tff(fact_2085_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_bi(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_2086_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),G3),Na)),set_ord_lessThan(nat,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_lessThan(nat,Na)) ) ).

% sum.nat_diff_reindex
tff(fact_2087_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K4: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),K4)) ) ) ).

% sum_mono
tff(fact_2088_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,F3),A5))
       => ? [X: A] :
            ( member(A,X,A5)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_bk(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A5),F3),X))) ) ) ) ).

% pigeonhole_infinite
tff(fact_2089_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F3: fun(A,B),B5: set(B)] :
      ( ! [X: A] :
          ( aa(A,$o,P2,X)
         => member(B,aa(A,B,F3,X),B5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),aa(fun(A,$o),set(A),collect(A),P2))),B5) ) ).

% image_Collect_subsetI
tff(fact_2090_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bl(A,fun(fun(A,$o),fun(A,$o)),A2),P2)) = $ite(aa(A,$o,P2,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_2091_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P2: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bm(A,fun(fun(A,$o),fun(A,$o)),A2),P2)) = $ite(aa(A,$o,P2,A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_2092_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A5: set(A),B5: set(B),G3: fun(A,fun(B,C)),R3: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),$o,finite_finite2(B),B5)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_bo(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G3),R3)),A5) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_bq(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A5),G3),R3)),B5) ) ) ) ) ).

% sum.swap_restrict
tff(fact_2093_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q2: fun(A,nat),P2: fun(A,nat),Na: A] :
          ( ! [X: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q2,X)),aa(A,nat,P2,X))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P2),set_ord_lessThan(A,Na))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q2),set_ord_lessThan(A,Na))) = 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_br(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q2),P2)),set_ord_lessThan(A,Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_2094_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2),Xa2,X3) ) ).

% max_def_raw
tff(fact_2095_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),S2))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R3),S2) ) ).

% pred_subset_eq2
tff(fact_2096_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X3),Xa2)
    <=> 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),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_2097_finite__M__bounded__by__nat,axiom,
    ! [P2: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bt(fun(nat,$o),fun(nat,fun(nat,$o)),P2),I))) ).

% finite_M_bounded_by_nat
tff(fact_2098_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),aa(nat,nat,F3,N))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bu(fun(nat,nat),fun(nat,fun(nat,$o)),F3),U))) ) ).

% finite_less_ub
tff(fact_2099_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(T2)) ).

% set_vebt_def
tff(fact_2100_zip__inj,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),A3: list(A),B3: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
       => ( ( zip(A,B,A2,B2) = zip(A,B,A3,B3) )
         => ( ( A2 = A3 )
            & ( B2 = B3 ) ) ) ) ) ).

% zip_inj
tff(fact_2101_pair__list__split,axiom,
    ! [A: $tType,B: $tType,L: list(product_prod(A,B))] :
      ~ ! [L1: list(A),L22: list(B)] :
          ( ( L = zip(A,B,L1,L22) )
         => ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L22) )
           => ( aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),L) != aa(list(B),nat,size_size(list(B)),L22) ) ) ) ).

% pair_list_split
tff(fact_2102_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% real_arch_simple
tff(fact_2103_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% reals_Archimedean2
tff(fact_2104_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xa)) ) ).

% mult_of_nat_commute
tff(fact_2105_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] : set_ord_lessThan(A,Xa) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_2106_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),set_ord_lessThan(A,A2)) ) ).

% infinite_Iio
tff(fact_2107_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_2108_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bw(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_2109_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bx(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,M)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,M)) ) ).

% sum_lessThan_telescope'
tff(fact_2110_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bz(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_2111_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xa: fun(A,B),Y: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_cb(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_2112_filter__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cc(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_cd(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M7),P2))) ) ).

% filter_preserves_multiset
tff(fact_2113_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_ce(fun(A,B),fun(fun(A,$o),fun(A,B)),G3),P2)),A5) ) ) ) ).

% sum.inter_filter
tff(fact_2114_image__constant,axiom,
    ! [A: $tType,B: $tType,Xa: A,A5: set(A),C3: B] :
      ( member(A,Xa,A5)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_cf(B,fun(A,B),C3)),A5) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),C3),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_2115_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C3: A,A5: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_cg(A,fun(B,A)),C3)),A5) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_2116_foldl__length__aux,axiom,
    ! [A: $tType,A2: nat,L: list(A)] : aa(list(A),nat,aa(nat,fun(list(A),nat),foldl(nat,A,aTP_Lamp_am(nat,fun(A,nat))),A2),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldl_length_aux
tff(fact_2117_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),H2: fun(A,B),G3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),S2) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ci(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H2),G3)),aa(set(A),set(C),image(A,C,G3),S2)) ) ) ) ).

% sum.image_gen
tff(fact_2118_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A5: set(A),F3: fun(nat,A),Na: nat] :
      ( ( A5 = aa(set(nat),set(A),image(nat,A,F3),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Na))) )
     => aa(set(A),$o,finite_finite2(A),A5) ) ).

% nat_seg_image_imp_finite
tff(fact_2119_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
    <=> ? [N5: nat,F7: fun(nat,A)] : A5 = aa(set(nat),set(A),image(nat,A,F7),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N5))) ) ).

% finite_conv_nat_seg_image
tff(fact_2120_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_cj(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_2121_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_2122_listrel1__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Sa: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),Sa)
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,Sa)) ) ).

% listrel1_mono
tff(fact_2123_sum__subtractf__nat,axiom,
    ! [A: $tType,A5: set(A),G3: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G3,X)),aa(A,nat,F3,X)) )
     => ( 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_ck(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G3),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G3),A5)) ) ) ).

% sum_subtractf_nat
tff(fact_2124_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),T2: set(B),R3: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( aa(set(B),$o,finite_finite2(B),T2)
       => ( ! [X: B] :
              ( member(B,X,T2)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Sa),R3),X))) = aa(B,nat,K,X) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_cl(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T2),R3)),Sa) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),T2) ) ) ) ) ).

% sum_multicount_gen
tff(fact_2125_total__onI,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,A5)
         => ( member(A,Y3,A5)
           => ( ( X != Y3 )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),R2)
                | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),X),R2) ) ) ) )
     => total_on(A,A5,R2) ) ).

% total_onI
tff(fact_2126_total__on__def,axiom,
    ! [A: $tType,A5: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A5,R2)
    <=> ! [X5: A] :
          ( member(A,X5,A5)
         => ! [Xa3: A] :
              ( member(A,Xa3,A5)
             => ( ( X5 != Xa3 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3),R2)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_2127_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S2: set(A),S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_cm(set(A),fun(set(A),fun(set(A),$o)),S2),S))) ) ).

% finite_if_eq_beyond_finite
tff(fact_2128_total__on__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).

% total_on_empty
tff(fact_2129_Fpow__def,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_cn(set(A),fun(set(A),$o),A5)) ).

% Fpow_def
tff(fact_2130_distinct__finite__set,axiom,
    ! [A: $tType,Xa: set(A)] : aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_co(set(A),fun(list(A),$o),Xa))) ).

% distinct_finite_set
tff(fact_2131_set__to__map__empty__iff_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( ( aTP_Lamp_cp(A,option(B)) = set_to_map(A,B,S2) )
    <=> ( S2 = bot_bot(set(product_prod(A,B))) ) ) ).

% set_to_map_empty_iff(2)
tff(fact_2132_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cq(A,fun(nat,fun(A,fun(nat,A))),Xa),Na),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Na)))) ) ).

% diff_power_eq_sum
tff(fact_2133_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat,Y: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cr(A,fun(nat,fun(A,fun(nat,A))),Xa),Na),Y)),set_ord_lessThan(nat,Na))) ) ).

% power_diff_sumr2
tff(fact_2134_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F3: fun(A,B),B5: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( ! [I2: A] :
                ( member(A,I2,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),Sa) = B5 )
             => ( member(A,I,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),B5) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_2135_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [Sa: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),Sa)
         => ( ! [I2: A] :
                ( member(A,I2,Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),Sa) = zero_zero(B) )
             => ( member(A,I,Sa)
               => ( aa(A,B,F3,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_2136_add__mset__in__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),A2: A] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cc(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_cs(fun(A,nat),fun(A,fun(A,$o)),M7),A2))) ) ).

% add_mset_in_multiset
tff(fact_2137_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R3: set(B),G3: fun(A,B),F3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),R3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G3),S2)),R3)
             => ( 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_ct(fun(A,B),fun(fun(B,C),fun(A,C)),G3),F3)),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_cv(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G3),F3)),R3) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_2138_diff__preserves__multiset,axiom,
    ! [A: $tType,M7: fun(A,nat),N6: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cc(fun(A,nat),fun(A,$o),M7)))
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,nat),fun(A,$o),aTP_Lamp_cw(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M7),N6))) ) ).

% diff_preserves_multiset
tff(fact_2139_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S2: set(A),T3: set(B),G3: fun(A,B),H2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G3),S2)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_cx(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G3),H2)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H2),S2) ) ) ) ) ) ).

% sum.group
tff(fact_2140_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cy(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_2141_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A5: set(A)] : aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_cz(fun(A,nat),fun(A,nat),F3)),A5) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,finite_card(A),A5)) ).

% sum_Suc
tff(fact_2142_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S2: set(A),T3: set(B),R3: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ! [X: B] :
              ( member(B,X,T3)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S2),R3),X))) = K ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_cl(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R3)),S2) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T3)) ) ) ) ) ).

% sum_multicount
tff(fact_2143_card__less,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_da(set(nat),fun(nat,fun(nat,$o)),M7),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_2144_card__less__Suc,axiom,
    ! [M7: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M7)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_db(set(nat),fun(nat,fun(nat,$o)),M7),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_da(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc
tff(fact_2145_card__less__Suc2,axiom,
    ! [M7: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M7)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_db(set(nat),fun(nat,fun(nat,$o)),M7),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_da(set(nat),fun(nat,fun(nat,$o)),M7),I))) ) ) ).

% card_less_Suc2
tff(fact_2146_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : foldr(A,nat,aTP_Lamp_ap(A,fun(nat,nat)),L,A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_2147_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_dc(A,fun(A,$o)),aTP_Lamp_dd(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_2148_card__sum__le__nat__sum,axiom,
    ! [S2: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_de(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_de(nat,nat)),S2)) ).

% card_sum_le_nat_sum
tff(fact_2149_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ).

% set_zip_rightD
tff(fact_2150_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,Xa: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => member(A,Xa,aa(list(A),set(A),set2(A),Xs)) ) ).

% set_zip_leftD
tff(fact_2151_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
     => ~ ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
         => ~ member(B,Y,aa(list(B),set(B),set2(B),Ys)) ) ) ).

% in_set_zipE
tff(fact_2152_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),aa(list(product_prod(A,A)),set(product_prod(A,A)),set2(product_prod(A,A)),zip(A,A,Xs,Xs)))
    <=> ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_2153_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_0_le_iff
tff(fact_2154_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_2155_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,Xa: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I,Xa),list_update(B,Ys,I,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)) ).

% zip_update
tff(fact_2156_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),Xa),Na)),set_ord_lessThan(nat,Na))) ) ).

% one_diff_power_eq'
tff(fact_2157_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Na)) != zero_zero(A) ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_2158_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_imp_less
tff(fact_2159_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ).

% less_imp_of_nat_less
tff(fact_2160_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),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_2161_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [Na: A] :
          ( ( set_ord_lessThan(A,Na) = bot_bot(set(A)) )
        <=> ( Na = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_2162_lessThan__atLeast0,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,Na) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ).

% lessThan_atLeast0
tff(fact_2163_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_ord_lessThan(A,M)),set_ord_lessThan(A,Na))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na) ) ) ).

% lessThan_strict_subset_iff
tff(fact_2164_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( set_ord_lessThan(nat,Na) = bot_bot(set(nat)) )
    <=> ( Na = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_2165_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K),set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_2166_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bw(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Na)),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_2167_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F3: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_dg(fun(A,B),fun(B,fun(A,B)),F3),K)),S2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,F3),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_2168_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F3: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_dg(fun(A,B),fun(B,fun(A,B)),F3),K)),S2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,F3),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_2169_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,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_dh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_2170_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_2171_word__coorder_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ordering_top(word(A),aTP_Lamp_di(word(A),fun(word(A),$o)),aTP_Lamp_dj(word(A),fun(word(A),$o)),zero_zero(word(A))) ) ).

% word_coorder.ordering_top_axioms
tff(fact_2172_finite__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dk(set(A),fun(nat,fun(list(A),$o)),A5),Na))) ) ).

% finite_lists_length_eq
tff(fact_2173_subset__mset_Osum__nonneg__0,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),F3: fun(A,multiset(B)),I: A] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( ! [I2: A] :
            ( member(A,I2,Sa)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F3,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,Sa) = zero_zero(multiset(B)) )
         => ( member(A,I,Sa)
           => ( aa(A,multiset(B),F3,I) = zero_zero(multiset(B)) ) ) ) ) ) ).

% subset_mset.sum_nonneg_0
tff(fact_2174_subset__mset_Osum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType,Sa: set(A),F3: fun(A,multiset(B)),B5: multiset(B),I: A] :
      ( aa(set(A),$o,finite_finite2(A),Sa)
     => ( ! [I2: A] :
            ( member(A,I2,Sa)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),zero_zero(multiset(B))),aa(A,multiset(B),F3,I2)) )
       => ( ( groups3894954378712506084id_sum(multiset(B),A,plus_plus(multiset(B)),zero_zero(multiset(B)),F3,Sa) = B5 )
         => ( member(A,I,Sa)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F3,I)),B5) ) ) ) ) ).

% subset_mset.sum_nonneg_leq_bound
tff(fact_2175_distinct__finite__subset,axiom,
    ! [A: $tType,Xa: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),Xa)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_dl(set(A),fun(list(A),$o),Xa))) ) ).

% distinct_finite_subset
tff(fact_2176_finite__nat__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [K2: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K2)) ) ).

% finite_nat_bounded
tff(fact_2177_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K3)) ) ).

% finite_nat_iff_bounded
tff(fact_2178_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_dm(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_2179_card__lists__length__eq,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dk(set(A),fun(nat,fun(list(A),$o)),A5),Na))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(A),nat,finite_card(A),A5)),Na) ) ) ).

% card_lists_length_eq
tff(fact_2180_finite__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dn(set(A),fun(nat,fun(list(A),$o)),A5),Na))) ) ).

% finite_lists_length_le
tff(fact_2181_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(B,Y,aa(list(B),set(B),set2(B),Ys))
       => ~ ! [X: A] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_2182_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xa: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
       => ~ ! [Y3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_2183_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),Xa)) ) ) ).

% ex_less_of_nat_mult
tff(fact_2184_of__nat__gt__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),K) != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K) ) ) ).

% of_nat_gt_0
tff(fact_2185_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = 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),Na)) ) ) ) ).

% of_nat_diff
tff(fact_2186_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) )
     => ~ ! [X: A,Xs5: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs5) )
           => ! [Y3: B,Ys4: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys4) )
               => ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
                 => ( Xys != zip(A,B,Xs5,Ys4) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_2187_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R2)) ) ).

% lenlex_irreflexive
tff(fact_2188_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ! [Y4: real] :
        ? [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xa)) ) ).

% reals_Archimedean3
tff(fact_2189_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Xa: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_do(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,Xa,Y)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C3),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),C3)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_2190_listrel1I1,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)),listrel1(A,R2)) ) ).

% listrel1I1
tff(fact_2191_Cons__listrel1E1,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)),Ys),listrel1(A,R2))
     => ( ! [Y3: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Xs) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y3),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E1
tff(fact_2192_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),listrel1(A,R2))
     => ( ! [X: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) )
           => ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R2) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys),listrel1(A,R2)) ) ) ) ).

% Cons_listrel1E2
tff(fact_2193_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),K4: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K4),aa(A,B,F3,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)) ) ) ).

% sum_bounded_below
tff(fact_2194_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),K4) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K4)) ) ) ).

% sum_bounded_above
tff(fact_2195_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Na)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_2196_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_2197_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_2198_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_2199_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
       => ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M3)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M3)),Xa)),C3) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2200_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A5: set(A),F3: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I2)),K4) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A5))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5))),K4)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_2201_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_2202_cnt__cnt__eq,axiom,
    ! [T2: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,T2)) ).

% cnt_cnt_eq
tff(fact_2203_minNulli__rule,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_VEBT_minNulli(Tia),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_dp(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),T2),Tia)) ).

% minNulli_rule
tff(fact_2204_builupi_Hcorr,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupi'corr
tff(fact_2205_builupicorr,axiom,
    ! [Na: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na))) ).

% builupicorr
tff(fact_2206_vebt__maxtilist,axiom,
    ! [I: nat,Ts2: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts2))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_maxti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_dq(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts2),Tsi)) ) ).

% vebt_maxtilist
tff(fact_2207_vebt__mintilist,axiom,
    ! [I: nat,Ts2: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts2))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts2),Tsi),vEBT_vebt_minti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_dr(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts2),Tsi)) ) ).

% vebt_mintilist
tff(fact_2208_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I2)),aa(nat,A,A2,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),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_ds(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_2209_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_2210_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2211_predicate2I,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Q2: fun(A,fun(B,$o))] :
      ( ! [X: A,Y3: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P2,X),Y3)
         => aa(B,$o,aa(A,fun(B,$o),Q2,X),Y3) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P2),Q2) ) ).

% predicate2I
tff(fact_2212_predicate1I,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] :
      ( ! [X: A] :
          ( aa(A,$o,P2,X)
         => aa(A,$o,Q2,X) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P2),Q2) ) ).

% predicate1I
tff(fact_2213_vebt__minti__h,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_minti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_dt(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T2),Tia)) ).

% vebt_minti_h
tff(fact_2214_vebt__maxti__h,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_maxti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_du(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T2),Tia)) ).

% vebt_maxti_h
tff(fact_2215_predicate1D,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o),Xa: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P2),Q2)
     => ( aa(A,$o,P2,Xa)
       => aa(A,$o,Q2,Xa) ) ) ).

% predicate1D
tff(fact_2216_predicate2D,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Q2: fun(A,fun(B,$o)),Xa: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P2),Q2)
     => ( aa(B,$o,aa(A,fun(B,$o),P2,Xa),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q2,Xa),Y) ) ) ).

% predicate2D
tff(fact_2217_rev__predicate1D,axiom,
    ! [A: $tType,P2: fun(A,$o),Xa: A,Q2: fun(A,$o)] :
      ( aa(A,$o,P2,Xa)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P2),Q2)
       => aa(A,$o,Q2,Xa) ) ) ).

% rev_predicate1D
tff(fact_2218_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Xa: A,Y: B,Q2: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P2,Xa),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P2),Q2)
       => aa(B,$o,aa(A,fun(B,$o),Q2,Xa),Y) ) ) ).

% rev_predicate2D
tff(fact_2219_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_2220_int__Suc,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ).

% int_Suc
tff(fact_2221_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_2222_negative__zless__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2223_negD,axiom,
    ! [Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),zero_zero(int))
     => ? [N: nat] : Xa = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% negD
tff(fact_2224_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))) ).

% not_zle_0_negative
tff(fact_2225_int__cases,axiom,
    ! [Z: int] :
      ( ! [N: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% int_cases
tff(fact_2226_int__of__nat__induct,axiom,
    ! [P2: fun(int,$o),Z: int] :
      ( ! [N: nat] : aa(int,$o,P2,aa(nat,int,semiring_1_of_nat(int),N))
     => ( ! [N: nat] : aa(int,$o,P2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))))
       => aa(int,$o,P2,Z) ) ) ).

% int_of_nat_induct
tff(fact_2227_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_2228_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2229_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(2)
tff(fact_2230_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% zle_int
tff(fact_2231_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_2232_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_2233_nat__less__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_less_as_int
tff(fact_2234_nat__leq__as__int,axiom,
    ! [X3: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_leq_as_int
tff(fact_2235_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),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_2236_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2237_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% pos_int_cases
tff(fact_2238_int__cases4,axiom,
    ! [M: int] :
      ( ! [N: nat] : M != aa(nat,int,semiring_1_of_nat(int),N)
     => ~ ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ) ) ).

% int_cases4
tff(fact_2239_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) )
       => ~ ! [N: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ) ).

% int_cases3
tff(fact_2240_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ~ ! [N: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N) ) ) ).

% neg_int_cases
tff(fact_2241_Abs__fnat__hom__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),zero_zero(nat)) ) ) ).

% Abs_fnat_hom_0
tff(fact_2242_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [N: nat] : Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ).

% word_le_add
tff(fact_2243_Abs__fnat__hom__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Abs_fnat_hom_1
tff(fact_2244_zdiff__int__split,axiom,
    ! [P2: fun(int,$o),Xa: nat,Y: nat] :
      ( aa(int,$o,P2,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
         => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),Xa)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y)
         => aa(int,$o,P2,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_2245_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Leaf((A2),(B2))) = one_one(nat) ).

% VEBT_internal.cnt'.simps(1)
tff(fact_2246_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Xa: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2) ) ) ) ).

% word_1_0
tff(fact_2247_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_aj(nat,fun(nat,$o)),aTP_Lamp_ai(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_2248_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I2)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I2)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_dv(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_2249_minNrulli__ruleT,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_VEBT_minNulli(Tia),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_dp(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),T2),Tia),one_one(nat)) ).

% minNrulli_ruleT
tff(fact_2250_T__vebt__buildupi,axiom,
    ! [Na: nat,H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Na),H2)),vEBT_V441764108873111860ildupi(Na)) ).

% T_vebt_buildupi
tff(fact_2251_vebt__maxti__hT,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_maxti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_du(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T2),Tia),one_one(nat)) ).

% vebt_maxti_hT
tff(fact_2252_vebt__minti__hT,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_minti(Tia),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_dt(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),T2),Tia),one_one(nat)) ).

% vebt_minti_hT
tff(fact_2253_htt__vebt__buildupi_H,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),vEBT_V441764108873111860ildupi(Na)) ).

% htt_vebt_buildupi'
tff(fact_2254_htt__vebt__buildupi,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),vEBT_V441764108873111860ildupi(Na)) ).

% htt_vebt_buildupi
tff(fact_2255_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dw(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_2256_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dx(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_2257_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dy(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_2258_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_dz(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_2259_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_2260_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ).

% zle_add1_eq_le
tff(fact_2261_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% zle_diff1_eq
tff(fact_2262_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nonpos_int_cases
tff(fact_2263_negative__zle__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),zero_zero(int)) ).

% negative_zle_0
tff(fact_2264_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ? [N: nat] : K = aa(nat,int,semiring_1_of_nat(int),N) ) ).

% zero_le_imp_eq_int
tff(fact_2265_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ~ ! [N: nat] : K != aa(nat,int,semiring_1_of_nat(int),N) ) ).

% nonneg_int_cases
tff(fact_2266_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z)
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).

% zle_iff_zadd
tff(fact_2267_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_2268_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( Na = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_2269_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(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_2270_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_2271_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_2272_int__gr__induct,axiom,
    ! [K: int,I: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P2,I2)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P2,I) ) ) ) ).

% int_gr_induct
tff(fact_2273_int__less__induct,axiom,
    ! [I: int,K: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P2,I2)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P2,I) ) ) ) ).

% int_less_induct
tff(fact_2274_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_2275_plusinfinity,axiom,
    ! [D2: int,P4: fun(int,$o),P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P4,X)
          <=> aa(int,$o,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
             => ( aa(int,$o,P2,X)
              <=> aa(int,$o,P4,X) ) )
         => ( ? [X_12: int] : aa(int,$o,P4,X_12)
           => ? [X_1: int] : aa(int,$o,P2,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_2276_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P1,X)
          <=> aa(int,$o,P1,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z4: int] :
            ! [X: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
             => ( aa(int,$o,P2,X)
              <=> aa(int,$o,P1,X) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_1: int] : aa(int,$o,P2,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_2277_verit__la__generic,axiom,
    ! [A2: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),Xa)
      | ( A2 = Xa )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),A2) ) ).

% verit_la_generic
tff(fact_2278_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_2279_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% add1_zle_eq
tff(fact_2280_int__ge__induct,axiom,
    ! [K: int,I: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P2,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P2,I2)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P2,I) ) ) ) ).

% int_ge_induct
tff(fact_2281_int__le__induct,axiom,
    ! [I: int,K: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P2,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P2,I2)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P2,I) ) ) ) ).

% int_le_induct
tff(fact_2282_int__induct,axiom,
    ! [P2: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P2,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P2,I2)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P2,I2)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),I2),one_one(int))) ) )
         => aa(int,$o,P2,I) ) ) ) ).

% int_induct
tff(fact_2283_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q6) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_2284_unique__quotient__lemma,axiom,
    ! [B2: int,Q6: int,R4: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q6)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q6),Q3) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_2285_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B3: int,Q6: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q6),Q3) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_2286_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R2: int,B3: int,Q6: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q3),Q6) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_2287_q__pos__lemma,axiom,
    ! [B3: int,Q6: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q6)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q6) ) ) ) ).

% q_pos_lemma
tff(fact_2288_incr__mult__lemma,axiom,
    ! [D2: int,P2: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int] :
            ( aa(int,$o,P2,X)
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P2,X3)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_2289_pos__mult__pos__ge,axiom,
    ! [Xa: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Na),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),Na),Xa)) ) ) ).

% pos_mult_pos_ge
tff(fact_2290_decr__mult__lemma,axiom,
    ! [D2: int,P2: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int] :
            ( aa(int,$o,P2,X)
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P2,X3)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_2291_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_2292_less__1__helper,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Na),one_one(int))),M) ) ).

% less_1_helper
tff(fact_2293_imp__le__cong,axiom,
    ! [Xa: int,X6: int,P2: $o,P4: $o] :
      ( ( Xa = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P2)
          <=> (P4) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => (P2) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
           => (P4) ) ) ) ) ).

% imp_le_cong
tff(fact_2294_conj__le__cong,axiom,
    ! [Xa: int,X6: int,P2: $o,P4: $o] :
      ( ( Xa = X6 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
         => ( (P2)
          <=> (P4) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & (P2) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X6)
            & (P4) ) ) ) ) ).

% conj_le_cong
tff(fact_2295_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_2296_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_2297_Gcd__int__greater__eq__0,axiom,
    ! [K4: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K4)) ).

% Gcd_int_greater_eq_0
tff(fact_2298_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_2299_time__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C3: nat,Na: nat,H2: heap_ext(product_unit)] :
      ( ! [H: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,Xa,H)),C3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(list(A),vEBT_VEBT_replicatei(A,Na,Xa),H2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C3)),Na))) ) ).

% time_replicate
tff(fact_2300_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F3: fun(nat,A),K4: A,K: nat] :
          ( ! [P5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P5),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,P5)),K4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),K4)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_2301_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P2: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_cp(A,option(B)),P2) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_2302_htt__def,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),T2: nat] :
      ( time_htt(A,P2,C3,Q2,T2)
    <=> ( hoare_hoare_triple(A,P2,C3,Q2)
        & ! [H4: heap_ext(product_unit),As2: set(nat)] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C3,H4)),T2) ) ) ) ).

% htt_def
tff(fact_2303_httI,axiom,
    ! [A: $tType,P2: assn,C3: heap_Time_Heap(A),Q2: fun(A,assn),T2: nat] :
      ( hoare_hoare_triple(A,P2,C3,Q2)
     => ( ! [H: heap_ext(product_unit),As: set(nat)] :
            ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C3,H)),T2) )
       => time_htt(A,P2,C3,Q2,T2) ) ) ).

% httI
tff(fact_2304_Tbuildupi__buildupi_H,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na)) = vEBT_V9176841429113362141ildupi(Na) ).

% Tbuildupi_buildupi'
tff(fact_2305_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V9176841429113362141ildupi(zero_zero(nat)) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_2306_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,zero_zero(nat))) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_2307_htt__cons__rule,axiom,
    ! [A: $tType,P4: assn,C3: heap_Time_Heap(A),Q4: fun(A,assn),T8: nat,P2: assn,Q2: fun(A,assn),T2: nat] :
      ( time_htt(A,P4,C3,Q4,T8)
     => ( entails(P2,P4)
       => ( ! [X: A] : entails(aa(A,assn,Q4,X),aa(A,assn,Q2,X))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T8),T2)
           => time_htt(A,P2,C3,Q2,T2) ) ) ) ) ).

% htt_cons_rule
tff(fact_2308_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ea(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_2309_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ea(nat,fun(A,$o),Na)))),Na) ) ) ).

% card_roots_unity
tff(fact_2310_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_2311_finite__nth__roots,axiom,
    ! [Na: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(set(complex),$o,finite_finite2(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_eb(nat,fun(complex,fun(complex,$o)),Na),C3))) ) ).

% finite_nth_roots
tff(fact_2312_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X: real,Y3: real] :
          ( member(real,X,aa(list(real),set(real),set2(real),Xs))
         => ( member(real,Y3,aa(list(real),set(real),set2(real),Xs))
           => ( X = Y3 ) ) )
     => ( ! [X: real] :
            ( member(real,X,aa(list(real),set(real),set2(real),Xs))
           => ( X = Y ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_2313_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,X)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F3),Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2314_vebt__maxti__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_maxti(Tia),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),aTP_Lamp_ec(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn)))),Na),Sa),Tia),Y),one_one(nat)) ).

% vebt_maxti_rule
tff(fact_2315_foldr0,axiom,
    ! [Xs: list(real),C3: real,D2: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C3),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D2)),C3) ).

% foldr0
tff(fact_2316_nth__map,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs: list(A),F3: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),Na) = aa(A,B,F3,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% nth_map
tff(fact_2317_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W2: list(A),F3: fun(B,A),Ww: list(B),A2: B] :
      ( ( W2 = aa(list(B),list(A),map(B,A,F3),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,A2)),W2) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Ww)) ) ) ).

% map_consI(1)
tff(fact_2318_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls: list(B),Fa: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
     => ~ ! [A4: B,L2: list(B)] :
            ( ( Ls = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),L2) )
           => ( ( aa(B,A,F3,A4) = Fa )
             => ( aa(list(B),list(A),map(B,A,F3),L2) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_2319_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),L3: list(B),I: nat] :
      ( ( aa(list(B),list(A),map(B,A,F3),L) = aa(list(B),list(A),map(B,A,F3),L3) )
     => ( aa(B,A,F3,aa(nat,B,nth(B,L),I)) = aa(B,A,F3,aa(nat,B,nth(B,L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_2320_distinct__mapI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => distinct(B,L) ) ).

% distinct_mapI
tff(fact_2321_distinct__map__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B),Xa: B,Y: B] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( ( aa(B,A,F3,Xa) = aa(B,A,F3,Y) )
       => ( member(B,Xa,aa(list(B),set(B),set2(B),L))
         => ( member(B,Y,aa(list(B),set(B),set2(B),L))
           => ( Xa = Y ) ) ) ) ) ).

% distinct_map_eq
tff(fact_2322_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_ed(fun(B,A),fun(B,fun(B,$o)),F3),Xs) ) ) ).

% sorted_map
tff(fact_2323_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,Xa,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_2324_distinct__idx,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),L: list(B),I: nat,J: nat] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),L))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),L))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(B),nat,size_size(list(B)),L))
         => ( ( aa(B,A,F3,aa(nat,B,nth(B,L),I)) = aa(B,A,F3,aa(nat,B,nth(B,L),J)) )
           => ( I = J ) ) ) ) ) ).

% distinct_idx
tff(fact_2325_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F3: fun(A,B),Xa: A] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F3,aa(nat,A,nth(A,L),I)) = aa(A,B,F3,Xa) ) )
     => ( aa(list(A),list(B),map(A,B,F3),list_update(A,L,I,Xa)) = aa(list(A),list(B),map(A,B,F3),L) ) ) ).

% map_upd_eq
tff(fact_2326_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_2327_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_2328_card__nth__roots,axiom,
    ! [C3: complex,Na: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_ee(complex,fun(nat,fun(complex,$o)),C3),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_2329_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_ef(nat,fun(complex,$o),Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_2330_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_eg(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_ef(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_2331_sum__nth__roots,axiom,
    ! [Na: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_eg(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_eb(nat,fun(complex,fun(complex,$o)),Na),C3))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_2332_vebt__buildupi__rule__basic,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat)))) ) ).

% vebt_buildupi_rule_basic
tff(fact_2333_vebt__minti__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_minti(Tia),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),aTP_Lamp_eh(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn)))),Na),Sa),Tia),Y),one_one(nat)) ).

% vebt_minti_rule
tff(fact_2334_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_2335_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,real),Bound: real,I: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F3,X)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F3),Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_2336_real__nat__list,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A),C3: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F3),Xs),C3)) = foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,aTP_Lamp_ei(fun(A,nat),fun(A,real),F3)),Xs),aa(nat,real,semiring_1_of_nat(real),C3)) ).

% real_nat_list
tff(fact_2337_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,real),C3: real,G3: fun(A,real),D2: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F3,X)),aa(real,real,aa(real,fun(real,real),times_times(real),C3),aa(A,real,G3,X))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F3),Xs),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C3),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,G3),Xs),zero_zero(real)))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_2338_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_2339_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_2340_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,real),Y: real] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F3,X)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F3),Xs),Y)) ) ).

% listsum_bound
tff(fact_2341_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_ej(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_2342_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary))),foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_2343_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_2344_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(real) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList2),zero_zero(real))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_2345_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(A,aa(list(A),assn,snga_assn(A,A2),Xs),array_nth(A,A2,I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_ek(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A2)) ) ) ).

% nth_rule
tff(fact_2346_same__fst__trancl,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),R3: fun(A,set(product_prod(B,B)))] : transitive_trancl(product_prod(A,B),same_fst(A,B,P2,R3)) = same_fst(A,B,P2,aTP_Lamp_el(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),R3)) ).

% same_fst_trancl
tff(fact_2347_accp__subset,axiom,
    ! [A: $tType,R12: fun(A,fun(A,$o)),R23: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R12),R23)
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),accp(A,R23)),accp(A,R12)) ) ).

% accp_subset
tff(fact_2348_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xa: heap_Time_Heap(A),A5: fun(B,fun(A,assn)),Y: B,Na: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xa,aa(B,fun(A,assn),A5,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,Na,Xa),vEBT_List_list_assn(B,A,A5,replicate(B,Na,Y))) ) ).

% repli_emp
tff(fact_2349_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_em(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_2350_intind,axiom,
    ! [A: $tType,I: nat,Na: nat,P2: fun(A,$o),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(A,$o,P2,Xa)
       => aa(A,$o,P2,aa(nat,A,nth(A,replicate(A,Na,Xa)),I)) ) ) ).

% intind
tff(fact_2351_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q2: assn,Xa: heap_Time_Heap(A),A5: fun(B,fun(A,assn)),Y: B,Na: nat] :
      ( hoare_hoare_triple(A,Q2,Xa,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_en(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Q2),A5),Y))
     => hoare_hoare_triple(list(A),Q2,vEBT_VEBT_replicatei(A,Na,Xa),aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_eo(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q2),A5),Y),Na)) ) ).

% repli_cons_repl
tff(fact_2352_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,Xa: A,Na: nat,Y: A] :
      ( ( replicate(A,M,Xa) = replicate(A,Na,Y) )
    <=> ( ( M = Na )
        & ( ( M != zero_zero(nat) )
         => ( Xa = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_2353_Ball__set__replicate,axiom,
    ! [A: $tType,Na: nat,A2: A,P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),replicate(A,Na,A2)))
         => aa(A,$o,P2,X5) )
    <=> ( aa(A,$o,P2,A2)
        | ( Na = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_2354_Bex__set__replicate,axiom,
    ! [A: $tType,Na: nat,A2: A,P2: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(list(A),set(A),set2(A),replicate(A,Na,A2)))
          & aa(A,$o,P2,X5) )
    <=> ( aa(A,$o,P2,A2)
        & ( Na != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_2355_in__set__replicate,axiom,
    ! [A: $tType,Xa: A,Na: nat,Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),replicate(A,Na,Y)))
    <=> ( ( Xa = Y )
        & ( Na != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_2356_nth__replicate,axiom,
    ! [A: $tType,I: nat,Na: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,replicate(A,Na,Xa)),I) = Xa ) ) ).

% nth_replicate
tff(fact_2357_last__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( last(A,replicate(A,Na,Xa)) = Xa ) ) ).

% last_replicate
tff(fact_2358_set__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Na,Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_2359_replicate__Suc,axiom,
    ! [A: $tType,Na: nat,Xa: A] : replicate(A,aa(nat,nat,suc,Na),Xa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),replicate(A,Na,Xa)) ).

% replicate_Suc
tff(fact_2360_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Na: nat,Xa: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Na,Xa)) ) ).

% sorted_replicate
tff(fact_2361_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xa: A,Ys: list(B)] : zip(A,B,replicate(A,Na,Xa),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),Xa)),take(B,Na,Ys)) ).

% zip_replicate1
tff(fact_2362_map__zip2,axiom,
    ! [A: $tType,B: $tType,K: A,L: list(B)] : aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L) = zip(A,B,replicate(A,aa(list(B),nat,size_size(list(B)),L),K),L) ).

% map_zip2
tff(fact_2363_map__zip1,axiom,
    ! [A: $tType,B: $tType,K: B,L: list(A)] : aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ep(B,fun(A,product_prod(A,B))),K)),L) = zip(A,B,L,replicate(B,aa(list(A),nat,size_size(list(A)),L),K)) ).

% map_zip1
tff(fact_2364_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Na: nat,Y: B] : zip(A,B,Xs,replicate(B,Na,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ep(B,fun(A,product_prod(A,B))),Y)),take(A,Na,Xs)) ).

% zip_replicate2
tff(fact_2365_set__replicate__Suc,axiom,
    ! [A: $tType,Na: nat,Xa: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Na),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_2366_set__replicate__conv__if,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      aa(list(A),set(A),set2(A),replicate(A,Na,Xa)) = $ite(Na = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_2367_Cons__replicate__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) = replicate(A,Na,Y) )
    <=> ( ( Xa = Y )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
        & ( Xs = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),Xa) ) ) ) ).

% Cons_replicate_eq
tff(fact_2368_accp__subset__induct,axiom,
    ! [A: $tType,D5: fun(A,$o),R3: fun(A,fun(A,$o)),Xa: A,P2: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),D5),accp(A,R3))
     => ( ! [X: A,Z3: A] :
            ( aa(A,$o,D5,X)
           => ( aa(A,$o,aa(A,fun(A,$o),R3,Z3),X)
             => aa(A,$o,D5,Z3) ) )
       => ( aa(A,$o,D5,Xa)
         => ( ! [X: A] :
                ( aa(A,$o,D5,X)
               => ( ! [Z4: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),R3,Z4),X)
                     => aa(A,$o,P2,Z4) )
                 => aa(A,$o,P2,X) ) )
           => aa(A,$o,P2,Xa) ) ) ) ) ).

% accp_subset_induct
tff(fact_2369_same__fstI,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),Xa: A,Y7: B,Y: B,R3: fun(A,set(product_prod(B,B)))] :
      ( aa(A,$o,P2,Xa)
     => ( member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y7),Y),aa(A,set(product_prod(B,B)),R3,Xa))
       => member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y7)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),same_fst(A,B,P2,R3)) ) ) ).

% same_fstI
tff(fact_2370_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xaa),aa(nat,nat,suc,Xa)))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_2371_in__measure,axiom,
    ! [A: $tType,Xa: A,Y: A,F3: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),measure(A,F3))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xa)),aa(A,nat,F3,Y)) ) ).

% in_measure
tff(fact_2372_in__finite__psubset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A5),B5),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
        & aa(set(A),$o,finite_finite2(A),B5) ) ) ).

% in_finite_psubset
tff(fact_2373_vebt__buildupi__refines,axiom,
    ! [Na: nat] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_buildupi(Na),vEBT_V739175172307565963ildupi(Na)) ).

% vebt_buildupi_refines
tff(fact_2374_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),Na) = semiri8178284476397505188at_aux(A,aTP_Lamp_eq(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_2375_refines__replicate,axiom,
    ! [A: $tType,F3: heap_Time_Heap(A),F8: heap_Time_Heap(A),Na: nat] :
      ( refine_Imp_refines(A,F3,F8)
     => refine_Imp_refines(list(A),vEBT_VEBT_replicatei(A,Na,F3),vEBT_VEBT_replicatei(A,Na,F8)) ) ).

% refines_replicate
tff(fact_2376_vebt__deletei__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T2: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_deletei(Tia,Xa),vEBT_V1365221501068881998eletei(T2,Tia,Xa)) ).

% vebt_deletei_refines
tff(fact_2377_vebt__inserti__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T2: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_inserti(Tia,Xa),vEBT_V3964819847710782039nserti(T2,Tia,Xa)) ).

% vebt_inserti_refines
tff(fact_2378_refines__case__VEBTi,axiom,
    ! [A: $tType,Tia: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun($o,fun($o,heap_Time_Heap(A))),F12: fun($o,fun($o,heap_Time_Heap(A))),F22: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),F23: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))))] :
      ( ( Tia = Ti )
     => ( ! [A4: $o,B4: $o] : refine_Imp_refines(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F1,(A4)),(B4)),aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F12,(A4)),(B4)))
       => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeArray: array(vEBT_VEBTi),Summary2: vEBT_VEBTi] : refine_Imp_refines(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F22,Info2),Deg2),TreeArray),Summary2),aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F23,Info2),Deg2),TreeArray),Summary2))
         => refine_Imp_refines(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),fun(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A))),vEBT_case_VEBTi(heap_Time_Heap(A)),F22),F1),Tia),aa(vEBT_VEBTi,heap_Time_Heap(A),aa(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),fun(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A))),vEBT_case_VEBTi(heap_Time_Heap(A)),F23),F12),Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_2379_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Na: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Na),I) = semiri8178284476397505188at_aux(A,Inc,Na,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_2380_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_2381_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(array(A),aa(list(A),assn,snga_assn(A,A2),Xs),array_upd(A,I,Xa,A2),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_er(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A2),Xa)) ) ) ).

% upd_rule
tff(fact_2382_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_em(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_2383_VEBTi_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,size_size(vEBT_VEBTi),X13)),aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size(3)
tff(fact_2384_vebt__succi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T2: vEBT_VEBT] : refine_Imp_refines(option(nat),vEBT_vebt_succi(Tia,Xa),vEBT_VEBT_vebt_succi(T2,Tia,Xa)) ).

% vebt_succi_refines
tff(fact_2385_vebt__predi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T2: vEBT_VEBT] : refine_Imp_refines(option(nat),vEBT_vebt_predi(Tia,Xa),vEBT_VEBT_vebt_predi(T2,Tia,Xa)) ).

% vebt_predi_refines
tff(fact_2386_vebt__memberi__refines,axiom,
    ! [Tia: vEBT_VEBTi,Xa: nat,T2: vEBT_VEBT] : refine_Imp_refines($o,vEBT_vebt_memberi(Tia,Xa),vEBT_V854960066525838166emberi(T2,Tia,Xa)) ).

% vebt_memberi_refines
tff(fact_2387_VEBTi_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H2: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),F22: fun($o,fun($o,B)),VEBTi: vEBT_VEBTi] : aa(B,A,H2,aa(vEBT_VEBTi,B,aa(fun($o,fun($o,B)),fun(vEBT_VEBTi,B),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBTi,B)),vEBT_case_VEBTi(B),F1),F22),VEBTi)) = aa(vEBT_VEBTi,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBTi,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A)),vEBT_case_VEBTi(A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_es(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),H2),F1)),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_et(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H2),F22)),VEBTi) ).

% VEBTi.case_distrib
tff(fact_2388_VEBTi_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F22: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBTi,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A)),vEBT_case_VEBTi(A),F1),F22),vEBT_Nodei(X11,X12,X13,X14)) = aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),F1,X11),X12),X13),X14) ).

% VEBTi.simps(5)
tff(fact_2389_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,vEBT_size_VEBTi,X13)),aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size_gen(1)
tff(fact_2390_TBOUND__minNull,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_VEBT_minNull(T2)
     => time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(T2,Tia,Xa),one_one(nat)) ) ).

% TBOUND_minNull
tff(fact_2391_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A5: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A5,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),I5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ))),
        finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_eu(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A5),Xs),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_2392_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ~ ! [L2: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L2))
               => ( ( aa(list(B),set(B),set2(B),L2) = A5 )
                 => ( aa(list(B),nat,size_size(list(B)),L2) != aa(set(B),nat,finite_card(B),A5) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_2393_card__lists__length__le,axiom,
    ! [A: $tType,A5: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_dn(set(A),fun(nat,fun(list(A),$o)),A5),Na))) = 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))),set_ord_atMost(nat,Na)) ) ) ).

% card_lists_length_le
tff(fact_2394_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A12: nat,A23: nat,A32: A,P2: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A12),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A23),A32))))
     => ( ! [F2: fun(nat,fun(A,A)),A4: nat,B4: nat,Acc: A] :
            ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B4),Acc))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B4),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P2,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B4),aa(A,A,aa(nat,fun(A,A),F2,A4),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P2,F2),A4),B4),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P2,A0),A12),A23),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_2395_TBOUND__vebt__buildupi,axiom,
    ! [Na: nat] : time_TBOUND(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Na),vEBT_V441764108873111860ildupi(Na)) ).

% TBOUND_vebt_buildupi
tff(fact_2396_TBOUND__replicate,axiom,
    ! [A: $tType,Xa: heap_Time_Heap(A),C3: nat,Na: nat] :
      ( time_TBOUND(A,Xa,C3)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,Na,Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C3)),Na))) ) ).

% TBOUND_replicate
tff(fact_2397_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atMost(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_2398_fold__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F3,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_2399_fold__infinite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,fun(B,B)),Z: B] :
      ( ~ aa(set(A),$o,finite_finite2(A),A5)
     => ( finite_fold(A,B,F3,Z,A5) = Z ) ) ).

% fold_infinite
tff(fact_2400_finite__atMost,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),set_ord_atMost(nat,K)) ).

% finite_atMost
tff(fact_2401_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,Xa)),set_ord_atMost(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ).

% atMost_subset_iff
tff(fact_2402_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),set_ord_atMost(nat,U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_2403_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_atMost(nat,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) ) ).

% sum.atMost_Suc
tff(fact_2404_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_2405_fold__closed__eq,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),Z: B] :
      ( ! [A4: A,B4: B] :
          ( member(A,A4,A5)
         => ( member(B,B4,B5)
           => ( aa(B,B,aa(A,fun(B,B),F3,A4),B4) = aa(B,B,aa(A,fun(B,B),G3,A4),B4) ) ) )
     => ( ! [A4: A,B4: B] :
            ( member(A,A4,A5)
           => ( member(B,B4,B5)
             => member(B,aa(B,B,aa(A,fun(B,B),G3,A4),B4),B5) ) )
       => ( member(B,Z,B5)
         => ( finite_fold(A,B,F3,Z,A5) = finite_fold(A,B,G3,Z,A5) ) ) ) ) ).

% fold_closed_eq
tff(fact_2406_TBOUND__mono,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),T2: nat,T8: nat] :
      ( time_TBOUND(A,C3,T2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),T2),T8)
       => time_TBOUND(A,C3,T8) ) ) ).

% TBOUND_mono
tff(fact_2407_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H2: A] : bot_bot(set(A)) != set_ord_atMost(A,H2) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_2408_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ aa(set(A),$o,finite_finite2(A),set_ord_atMost(A,A2)) ) ).

% infinite_Iic
tff(fact_2409_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ev(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_2410_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ).

% lessThan_Suc_atMost
tff(fact_2411_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_2412_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_atMost(nat,K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_2413_TBOUND__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),T2: nat] :
      ( time_TBOUND(A,M,T2)
    <=> ! [H4: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H4)),T2) ) ).

% TBOUND_def
tff(fact_2414_TBOUNDI,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),T2: nat] :
      ( ! [H: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H)),T2)
     => time_TBOUND(A,M,T2) ) ).

% TBOUNDI
tff(fact_2415_TBOUNDD,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),T2: nat,H2: heap_ext(product_unit)] :
      ( time_TBOUND(A,M,T2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H2)),T2) ) ).

% TBOUNDD
tff(fact_2416_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_2417_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_2418_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bx(fun(nat,A),fun(nat,A),F3)),set_ord_atMost(nat,I)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_2419_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_2420_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat,D2: fun(nat,A)] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),C3),X5)),set_ord_atMost(nat,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),D2),X5)),set_ord_atMost(nat,Na))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
             => ( aa(nat,A,C3,I3) = aa(nat,A,D2,I3) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_2421_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,A,ord_max(A),Xa,A5) ) ) ) ).

% Max.eq_fold
tff(fact_2422_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),C3),X5)),set_ord_atMost(nat,Na)) = zero_zero(A)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
             => ( aa(nat,A,C3,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_2423_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C3: fun(nat,A),Na: nat,K: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ex(fun(nat,A),fun(A,fun(nat,A)),C3),W)),set_ord_atMost(nat,Na)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(nat,A,C3,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_2424_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na))) ) ).

% sum.atMost_shift
tff(fact_2425_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))) ) ).

% sum_gp_basic
tff(fact_2426_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))
        <=> ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Na)
              & ( aa(nat,A,C3,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_2427_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),C3),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_2428_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))),Na) ) ) ) ).

% polyfun_roots_card
tff(fact_2429_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),A2: A,Na: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C3),A2)),set_ord_atMost(nat,Na)) = zero_zero(A) )
         => ~ ! [B4: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z4),A2)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),B4),Z4)),set_ord_lessThan(nat,Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_2430_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Na: nat,B2: fun(nat,A),Xa: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
             => ( aa(nat,A,A2,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),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),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),B2),Xa)),set_ord_atMost(nat,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_2431_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))),Na) ) ) ) ) ).

% polyfun_rootbound
tff(fact_2432_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Na: nat,B2: fun(nat,nat),Xa: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
         => ( aa(nat,nat,A2,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fc(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xa)),set_ord_atMost(nat,M))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_fc(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xa)),set_ord_atMost(nat,Na))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fe(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_2433_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G3: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H2)),set_ord_atMost(nat,P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fg(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H2)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_2434_aux,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,assn)),A2: A,As3: list(A),C3: B,Cs: list(B)] : finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_fh(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P2),A2),As3),C3),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As3)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P2,A2),C3)),finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_eu(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P2),As3),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As3)))) ).

% aux
tff(fact_2435_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na) = A2 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fi(nat,fun(A,fun(A,fun(nat,A))),Na),Z),A2)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_2436_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),set_ord_atMost(nat,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fk(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),Xa),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_2437_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc))))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) )
           => ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xb),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_2438_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),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)),A2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc2))))
     => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_2439_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L))
              & ( aa(list(B),set(B),set2(B),L) = A5 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A5) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_2440_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C3: fun(nat,A),Na: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ? [M10: real] :
            ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M10),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,A)),C3),Z4)),set_ord_atMost(nat,Na)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_2441_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A2: fun(nat,A),Xa: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),A2),Xa)),set_ord_atMost(nat,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fn(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A2),Xa),Y)),set_ord_lessThan(nat,Na))) ) ) ) ).

% polyfun_diff
tff(fact_2442_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xa: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),bot_bot(set(B))))) = remove1(B,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_2443_TBOUND__vebt__minti,axiom,
    ! [T2: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_minti(T2),one_one(nat)) ).

% TBOUND_vebt_minti
tff(fact_2444_TBOUND__vebt__maxti,axiom,
    ! [T2: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_maxti(T2),one_one(nat)) ).

% TBOUND_vebt_maxti
tff(fact_2445_TBOUND__minNulli,axiom,
    ! [T2: vEBT_VEBTi] : time_TBOUND($o,vEBT_VEBT_minNulli(T2),one_one(nat)) ).

% TBOUND_minNulli
tff(fact_2446_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_2447_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H2: A,L3: A,H3: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H2) = set_or1337092689740270186AtMost(A,L3,H3) )
        <=> ( ( ( L = L3 )
              & ( H2 = H3 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H2)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L3),H3) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_2448_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U)) ).

% finite_atLeastAtMost
tff(fact_2449_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_2450_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_2451_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_2452_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_2453_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_2454_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2455_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C3),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C3 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2456_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_2457_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H2: A,H3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H2)),set_ord_atMost(A,H3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H2),H3) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_2458_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_2459_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_2460_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na)))) ) ).

% sum.cl_ivl_Suc
tff(fact_2461_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_2462_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2463_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fo(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_2464_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H2: A,L3: A,H3: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,H2)),set_or1337092689740270186AtMost(A,L3,H3)) ) ).

% not_Iic_le_Icc
tff(fact_2465_all__nat__less,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Na)
         => aa(nat,$o,P2,M5) )
    <=> ! [X5: nat] :
          ( member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
         => aa(nat,$o,P2,X5) ) ) ).

% all_nat_less
tff(fact_2466_ex__nat__less,axiom,
    ! [Na: nat,P2: fun(nat,$o)] :
      ( ? [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),Na)
          & aa(nat,$o,P2,M5) )
    <=> ? [X5: nat] :
          ( member(nat,X5,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
          & aa(nat,$o,P2,X5) ) ) ).

% ex_nat_less
tff(fact_2467_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_2468_atMost__atLeast0,axiom,
    ! [Na: nat] : set_ord_atMost(nat,Na) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Na) ).

% atMost_atLeast0
tff(fact_2469_complex__mod__minus__le__complex__mod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Xa))),real_V7770717601297561774m_norm(complex,Xa)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2470_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2)) ).

% complex_mod_triangle_ineq2
tff(fact_2471_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_2472_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2473_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_2474_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2475_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2476_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = set_or1337092689740270186AtMost(nat,M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_2477_card_Oeq__fold,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),nat,finite_card(A),A5) = finite_fold(A,nat,aTP_Lamp_ap(A,fun(nat,nat)),zero_zero(nat),A5) ).

% card.eq_fold
tff(fact_2478_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N6),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N6) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2479_minus__fold__remove,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5) = finite_fold(A,set(A),remove(A),B5,A5) ) ) ).

% minus_fold_remove
tff(fact_2480_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_2481_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_2482_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_2483_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_2484_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_2485_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,aa(nat,nat,suc,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_2486_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_2487_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% sum.last_plus
tff(fact_2488_image__Suc__lessThan,axiom,
    ! [Na: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,Na)) = set_or1337092689740270186AtMost(nat,one_one(nat),Na) ).

% image_Suc_lessThan
tff(fact_2489_image__Suc__atMost,axiom,
    ! [Na: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,Na)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Na)) ).

% image_Suc_atMost
tff(fact_2490_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(B),image(A,B,F3),A5) = finite_fold(A,set(B),aTP_Lamp_fp(fun(A,B),fun(A,fun(set(B),set(B))),F3),bot_bot(set(B)),A5) ) ) ).

% image_fold_insert
tff(fact_2491_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_2492_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_2493_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bw(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,aa(nat,nat,suc,Na))),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_2494_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fq(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_2495_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_2496_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_ord_lessThan(nat,Na)) ) ).

% sum.nested_swap'
tff(fact_2497_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B),B5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),S2)
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = sorted8670434370408473282of_set(A,B,Less_eq,F3,B5) )
           => ( aa(set(B),$o,finite_finite2(B),A5)
             => ( aa(set(B),$o,finite_finite2(B),B5)
               => ( A5 = B5 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_2498_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2499_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2500_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_2501_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G3,Na))) ) ).

% sum.head_if
tff(fact_2502_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X4: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N5))),K5) )
        <=> ? [N8: nat] :
            ! [N5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def2
tff(fact_2503_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X4: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N5))),K5) )
        <=> ? [N8: nat] :
            ! [N5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% lemma_NBseq_def
tff(fact_2504_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_atMost(nat,M))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)))) ) ).

% sum_up_index_split
tff(fact_2505_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,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_fu(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Na),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2506_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xa)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_2507_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xa: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_fv(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,Xa,Y)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),C3))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_2508_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_fw(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_2509_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_fx(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C3))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_2510_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) = A5 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_2511_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) = aa(set(B),nat,finite_card(B),A5) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_2512_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_2513_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_2514_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => distinct(A,aa(list(B),list(A),map(B,A,F3),sorted8670434370408473282of_set(A,B,Less_eq,F3,A5))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_2515_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fy(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,M)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_2516_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fz(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,Na)),aa(nat,A,F3,M)) ) ) ) ).

% sum_telescope''
tff(fact_2517_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% sum_power_shift
tff(fact_2518_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),minus_minus(set(nat)),set_ord_atMost(nat,Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_2519_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na))) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na))) ) ) ) ).

% sum_gp_multiplied
tff(fact_2520_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat,K: A] :
          ( ! [X5: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),C3),X5)),set_ord_atMost(nat,Na)) = K
        <=> ( ( aa(nat,A,C3,zero_zero(nat)) = K )
            & ! [X5: nat] :
                ( member(nat,X5,set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C3,X5) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_2521_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S2)
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_2522_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real))
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2523_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa))
        <=> ( Xa != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2524_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2525_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( real_V7770717601297561774m_norm(A,Xa) = zero_zero(real) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2526_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W2: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W2)),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),W2)))) ) ) ) ).

% norm_power_diff
tff(fact_2527_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S2: set(A),F3: fun(A,B),K4: real] :
          ( ! [X: A] :
              ( member(A,X,S2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),K4) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),S2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),S2))),K4)) ) ) ).

% sum_norm_bound
tff(fact_2528_range__subset__card,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: word(A),D2: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),A2,B2)),set_or1337092689740270186AtMost(word(A),C3,D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),D2)
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),B2),A2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),D2),C3)) ) ) ) ) ).

% range_subset_card
tff(fact_2529_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2530_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_2531_aset_I7_J,axiom,
    ! [D5: int,A5: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ).

% aset(7)
tff(fact_2532_aset_I5_J,axiom,
    ! [D5: int,T2: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,T2,A5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),T2) ) ) ) ) ).

% aset(5)
tff(fact_2533_aset_I4_J,axiom,
    ! [D5: int,T2: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,T2,A5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_2534_aset_I3_J,axiom,
    ! [D5: int,T2: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),A5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_2535_bset_I7_J,axiom,
    ! [D5: int,T2: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,T2,B5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)) ) ) ) ) ).

% bset(7)
tff(fact_2536_bset_I5_J,axiom,
    ! [D5: int,B5: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)),T2) ) ) ) ).

% bset(5)
tff(fact_2537_bset_I4_J,axiom,
    ! [D5: int,T2: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,T2,B5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
           => ( ( X3 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_2538_bset_I3_J,axiom,
    ! [D5: int,T2: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
           => ( ( X3 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_2539_periodic__finite__ex,axiom,
    ! [D2: int,P2: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X: int,K2: int] :
            ( aa(int,$o,P2,X)
          <=> aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [X_13: int] : aa(int,$o,P2,X_13)
        <=> ? [X5: int] :
              ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P2,X5) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2540_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),R2: word(A),Y: word(A),Sa: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),Xa,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R2)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Sa)),one_one(word(A)))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),R2)),one_one(word(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Sa)),one_one(word(A))))
             => ( ( Sa != zero_zero(word(A)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),R2),Sa) ) ) ) ) ) ).

% word_subset_less
tff(fact_2541_aset_I8_J,axiom,
    ! [D5: int,A5: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ).

% aset(8)
tff(fact_2542_aset_I6_J,axiom,
    ! [D5: int,T2: int,A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int)),A5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb3),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),T2) ) ) ) ) ).

% aset(6)
tff(fact_2543_bset_I8_J,axiom,
    ! [D5: int,T2: int,B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),minus_minus(int),T2),one_one(int)),B5)
       => ! [X3: int] :
            ( ! [Xa5: int] :
                ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)) ) ) ) ) ).

% bset(8)
tff(fact_2544_bset_I6_J,axiom,
    ! [D5: int,B5: set(int),T2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa5: int] :
              ( member(int,Xa5,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa5) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),T2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),X3),D5)),T2) ) ) ) ).

% bset(6)
tff(fact_2545_cpmi,axiom,
    ! [D5: int,P2: fun(int,$o),P4: fun(int,$o),B5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X),Z4)
           => ( aa(int,$o,P2,X)
            <=> aa(int,$o,P4,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb4: int] :
                      ( member(int,Xb4,B5)
                     => ( X != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb4),Xa2) ) ) )
             => ( aa(int,$o,P2,X)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),D5)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P4,X)
              <=> aa(int,$o,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P2,X_13)
            <=> ( ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P4,X5) )
                | ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B5)
                        & aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2546_cppi,axiom,
    ! [D5: int,P2: fun(int,$o),P4: fun(int,$o),A5: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D5)
     => ( ? [Z4: int] :
          ! [X: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),X)
           => ( aa(int,$o,P2,X)
            <=> aa(int,$o,P4,X) ) )
       => ( ! [X: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb4: int] :
                      ( member(int,Xb4,A5)
                     => ( X != aa(int,int,aa(int,fun(int,int),minus_minus(int),Xb4),Xa2) ) ) )
             => ( aa(int,$o,P2,X)
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),X),D5)) ) )
         => ( ! [X: int,K2: int] :
                ( aa(int,$o,P4,X)
              <=> aa(int,$o,P4,aa(int,int,aa(int,fun(int,int),minus_minus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_13: int] : aa(int,$o,P2,X_13)
            <=> ( ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P4,X5) )
                | ? [X5: int] :
                    ( member(int,X5,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A5)
                        & aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa3),X5)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2547_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2548_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa)) ) ).

% norm_ge_zero
tff(fact_2549_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S2: set(A),F3: fun(A,B),G3: fun(A,real)] :
          ( ! [X: A] :
              ( member(A,X,S2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(A,real,G3,X)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),S2))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),G3),S2)) ) ) ).

% sum_norm_le
tff(fact_2550_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),aTP_Lamp_ga(fun(B,A),fun(B,real),F3)),A5)) ) ).

% norm_sum
tff(fact_2551_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),Na) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2552_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,R2: real,Y: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),Sa)) ) ) ) ).

% norm_mult_less
tff(fact_2553_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_2554_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),E3) ) ) ).

% norm_triangle_lt
tff(fact_2555_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,R2: real,Y: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Y)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),Sa)) ) ) ) ).

% norm_add_less
tff(fact_2556_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C3)) ) ) ).

% norm_add_leD
tff(fact_2557_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),E3) ) ) ).

% norm_triangle_le
tff(fact_2558_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2559_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R2: real,B2: A,Sa: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),Sa)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),Sa)) ) ) ) ).

% norm_triangle_mono
tff(fact_2560_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2561_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xa)),Na)) ) ).

% norm_power_ineq
tff(fact_2562_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)))) ) ).

% norm_triangle_sub
tff(fact_2563_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))) ) ).

% norm_triangle_ineq4
tff(fact_2564_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2565_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),E3) ) ) ).

% norm_triangle_le_diff
tff(fact_2566_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ).

% norm_diff_ineq
tff(fact_2567_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_2568_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W2),Na) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2569_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C3: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),C3))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2570_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_bv(fun(nat,A),fun(nat,A),F3)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_2571_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xa: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),A5)) = insort_key(A,B,Less_eq,F3,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_2572_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),
            zero_zero(A),
            $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ) ).

% sum_gp
tff(fact_2573_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ar(set(A),fun(nat,fun(list(A),$o)),A5),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_de(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_2574_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A5: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A5))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_gb(nat,fun(set(A),fun(list(A),$o)),K),A5))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_de(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_2575_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),Xa: B,A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),A5)),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ~ member(B,Xa,A5)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),A5)) = insort_key(A,B,Less_eq,F3,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F3,A5)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_2576_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_2577_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_2578_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_2579_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_2580_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_2581_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_2582_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)) ).

% nat_mult_div_cancel_disj
tff(fact_2583_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_2584_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_2585_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_2586_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_2587_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_2588_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_2589_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_2590_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_2591_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_2592_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != 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),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2593_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: 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),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2594_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2595_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2596_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2597_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2598_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_2599_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5) = zero_zero(B) )
          <=> ? [X5: A] :
                ( member(A,X5,A5)
                & ( aa(A,B,F3,X5) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_2600_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_2601_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_2602_prod__eq__1__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5) = one_one(nat) )
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => ( aa(A,nat,F3,X5) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_2603_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gc(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_2604_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gd(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_2605_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_2606_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_2607_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_2608_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_2609_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_2610_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_2611_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_2612_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_2613_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: 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),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2614_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2615_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xa: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod.insert
tff(fact_2616_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,Na))),aa(nat,A,G3,Na)) ) ).

% prod.lessThan_Suc
tff(fact_2617_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) ) ).

% prod.atMost_Suc
tff(fact_2618_prod__pos__nat__iff,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A5))
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X5)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_2619_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_2620_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_2621_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_2622_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_2623_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_ge(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_2624_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G3,Na))) ) ).

% prod.op_ivl_Suc
tff(fact_2625_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na)))) ) ).

% prod.cl_ivl_Suc
tff(fact_2626_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: fun(nat,A),D2: fun(nat,A),A5: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,A),fun(nat,A),aTP_Lamp_gf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D2)),A5) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A5)
            & member(nat,zero_zero(nat),A5) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C3,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_2627_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2628_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_2629_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))) ).

% real_of_nat_div4
tff(fact_2630_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F3: fun(B,A),A5: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A5))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),aTP_Lamp_gg(fun(B,A),fun(B,real),F3)),A5)) ) ).

% norm_prod_le
tff(fact_2631_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)),B2)),A2) ) ).

% word_div_mult_le
tff(fact_2632_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Z)),Y) ) ) ).

% div_to_mult_word_lt
tff(fact_2633_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),B5: set(B),G3: fun(A,fun(B,C)),R3: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),$o,finite_finite2(B),B5)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_gh(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G3),R3)),A5) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_gj(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A5),G3),R3)),B5) ) ) ) ) ).

% prod.swap_restrict
tff(fact_2634_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_nonneg
tff(fact_2635_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ).

% prod_mono
tff(fact_2636_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F3,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_pos
tff(fact_2637_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ).

% prod_ge_1
tff(fact_2638_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ? [X3: A] :
                ( member(A,X3,A5)
                & ( aa(A,B,F3,X3) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_2639_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)) ) ) ) ).

% divide_right_mono_neg
tff(fact_2640_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_2641_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_2642_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_2643_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_2644_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_2645_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ).

% divide_right_mono
tff(fact_2646_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_2647_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_2648_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_2649_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_2650_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_2651_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_2652_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_2653_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_2654_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ).

% divide_strict_right_mono
tff(fact_2655_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2656_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_2657_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2658_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2659_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) ) ) ) ) ).

% eq_divide_imp
tff(fact_2660_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_2661_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_2662_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) = A2 )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_2663_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_2664_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa)))) ).

% real_of_nat_div2
tff(fact_2665_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2666_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2667_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Xa)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_2668_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_2669_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_2670_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_2671_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),A5),P2))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_gk(fun(A,B),fun(fun(A,$o),fun(A,B)),G3),P2)),A5) ) ) ) ).

% prod.inter_filter
tff(fact_2672_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),H2: fun(A,B),G3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),S2) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_gl(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H2),G3)),aa(set(A),set(C),image(A,C,G3),S2)) ) ) ) ).

% prod.image_gen
tff(fact_2673_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C3: word(A),A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),C3)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),C3))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),C3)),B2) ) ) ) ).

% More_Word.word_div_mult
tff(fact_2674_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xa)),K) ) ) ) ).

% div_lt_mult
tff(fact_2675_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_2676_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% word_less_div
tff(fact_2677_word__div__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),V3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),V3)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W2),V3) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_2678_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A5: set(A),F3: fun(A,B),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( aa(A,B,F3,A2) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))))) = $ite(member(A,A2,A5),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ) ).

% prod_diff1
tff(fact_2679_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_2680_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_2681_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_2682_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R3: fun(A,fun(A,$o)),S2: set(B),H2: fun(B,A),G3: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R3,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R3,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( aa(set(B),$o,finite_finite2(B),S2)
             => ( ! [X: B] :
                    ( member(B,X,S2)
                   => aa(A,$o,aa(A,fun(A,$o),R3,aa(B,A,H2,X)),aa(B,A,G3,X)) )
               => aa(A,$o,aa(A,fun(A,$o),R3,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H2),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G3),S2)) ) ) ) ) ) ).

% prod.related
tff(fact_2683_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C3: A,B2: A,D2: A,G3: fun(A,B),H2: fun(A,B)] :
          ( ( A2 = C3 )
         => ( ( B2 = D2 )
           => ( ! [X: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),D2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),set_or7035219750837199246ssThan(A,C3,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_2684_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(member(A,Xa,A5),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5))) ) ) ) ).

% prod.insert_if
tff(fact_2685_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S5: set(A),T6: set(B),S2: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G3: fun(A,C),H2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S5)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6)) )
               => ( ! [B4: B] :
                      ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
                     => ( aa(A,B,J,aa(B,A,I,B4)) = B4 ) )
                 => ( ! [B4: B] :
                        ( member(B,B4,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),T3),T6))
                       => member(A,aa(B,A,I,B4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),S5)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S5)
                         => ( aa(A,C,G3,A4) = one_one(C) ) )
                     => ( ! [B4: B] :
                            ( member(B,B4,T6)
                           => ( aa(B,C,H2,B4) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H2,aa(A,B,J,A4)) = aa(A,C,G3,A4) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G3),S2) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H2),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_2686_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,P3: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_2687_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_le
tff(fact_2688_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_less
tff(fact_2689_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_2690_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_2691_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_2692_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_2693_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_2694_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_2695_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2696_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_2697_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2698_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_2699_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_2700_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_2701_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_2702_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_2703_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_less_divide_eq
tff(fact_2704_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_less_eq
tff(fact_2705_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_2706_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_2707_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_2708_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_2709_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2710_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: 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),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_2711_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),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),Xa),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_2712_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( ( Y != 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),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_2713_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_2714_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W2: A] :
          ( ( Y != 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),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),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),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_2715_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,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),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_2716_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_2717_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_2718_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_2719_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: 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),Xa),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_2720_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),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),Xa),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_2721_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xa: A,W2: A] :
          ( ( Y != 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),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),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),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_2722_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,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),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_2723_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2724_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2725_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C3 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2726_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) = A2 )
        <=> $ite(C3 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2727_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2728_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S2: set(A),T3: set(B),G3: fun(A,B),H2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G3),S2)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_gn(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G3),H2)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H2),S2) ) ) ) ) ) ).

% prod.group
tff(fact_2729_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_go(fun(A,B),fun(A,$o),G3)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_2730_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_2731_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_2732_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_2733_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_2734_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_2735_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_2736_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_2737_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_2738_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_2739_zdiv__mono2__neg,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_2740_zdiv__mono1__neg,axiom,
    ! [A2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_2741_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_2742_zdiv__mono2,axiom,
    ! [A2: int,B3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B3)) ) ) ) ).

% zdiv_mono2
tff(fact_2743_zdiv__mono1,axiom,
    ! [A2: int,A3: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B2)) ) ) ).

% zdiv_mono1
tff(fact_2744_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),K)),Xa) ) ) ).

% int_div_less_self
tff(fact_2745_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(nat,fun(nat,A)),G3),Na)),set_ord_lessThan(nat,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,Na)) ) ).

% prod.nat_diff_reindex
tff(fact_2746_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( ( Na != one_one(word(A)) )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Na)),Xa) ) ) ) ).

% div_less_dividend_word
tff(fact_2747_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xa)),K) ) ) ) ).

% div_le_mult
tff(fact_2748_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_2749_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F3,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_2750_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_2751_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H2,X) = one_one(B) ) )
             => ( ! [X: A] :
                    ( member(A,X,S2)
                   => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_2752_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_2753_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,G3,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_2754_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C4: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),B5))
                     => ( aa(A,B,H2,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),C4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),C4) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_2755_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C4: set(A),A5: set(A),B5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),A5))
                   => ( aa(A,B,G3,A4) = one_one(B) ) )
               => ( ! [B4: A] :
                      ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),C4),B5))
                     => ( aa(A,B,H2,B4) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),B5) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),C4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),C4) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_2756_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_2757_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G3,Na)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_2758_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_2759_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_2760_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G3,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_2761_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,aa(nat,nat,suc,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_2762_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_2763_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% prod.last_plus
tff(fact_2764_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_2765_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_2766_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_2767_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_2768_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_2769_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_2770_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_divide_le_eq
tff(fact_2771_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_le_divide_eq
tff(fact_2772_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_2773_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_2774_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_2775_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_2776_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_2777_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xa: A,W2: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W2),Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_2778_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2779_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2780_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2781_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2782_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2783_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2784_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2785_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: 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),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2786_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2787_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2788_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Y: 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),Xa),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2789_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Na: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ) ) ).

% power_diff
tff(fact_2790_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W2,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Z),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W2),I5)))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aa(fun(A,B),fun(A,real),aTP_Lamp_gq(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W2)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_2791_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_2792_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G3,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_2793_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_2794_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = 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),Na)),Na)) ) ) ) ).

% div_geq
tff(fact_2795_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,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_gr(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Na),M)),set_or7035219750837199246ssThan(nat,Na,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_2796_msrevs_I1_J,axiom,
    ! [Na: nat,K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na)),M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),K) ) ) ).

% msrevs(1)
tff(fact_2797_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gs(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_2798_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_2799_verit__less__mono__div__int2,axiom,
    ! [A5: int,B5: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),Na))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_2800_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2801_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gv(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Na)),set_ord_lessThan(nat,Na)) ) ).

% prod.nested_swap'
tff(fact_2802_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [I2: A] :
                ( member(A,I2,A5)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,I2)),aa(A,B,G3,I2)) ) )
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)) ) ) ) ) ).

% prod_mono_strict
tff(fact_2803_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),Xa: A,G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_2804_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G3,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_2805_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_2806_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G3,Na))) ) ).

% prod.head_if
tff(fact_2807_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_gw(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_2808_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V3: A,R2: A,Sa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),Sa)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,aa(A,fun(A,A),minus_minus(A),V3),U))),Sa))),V3) ) ) ) ) ).

% scaling_mono
tff(fact_2809_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2810_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2811_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2812_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2813_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2814_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2815_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E3)
         => ~ ! [N: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),E3) ) ) ).

% nat_approx_posE
tff(fact_2816_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( ( Na != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Na))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2817_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_ord_atMost(nat,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gm(fun(nat,A),fun(nat,A),G3)),set_ord_lessThan(nat,Na))) ) ).

% prod.atMost_shift
tff(fact_2818_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,Na,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_gx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G3),Na),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_2819_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_2820_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_2821_split__zdiv,axiom,
    ! [P2: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P2,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P2,I3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P2,I3) ) ) ) ) ).

% split_zdiv
tff(fact_2822_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A5: set(A),F3: fun(A,B),Na: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),Na) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Na)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Na),K)) ) ) ) ) ).

% prod_le_power
tff(fact_2823_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [B4: A] :
                  ( member(A,B4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F3,B4)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)) ) ) ) ) ) ).

% prod_mono2
tff(fact_2824_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_2825_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_gy(A,fun(fun(A,B),fun(B,fun(A,B))),A2),B2),C3)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),aa(nat,B,aa(B,fun(nat,B),power_power(B),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),S2)),one_one(nat)))),aa(nat,B,aa(B,fun(nat,B),power_power(B),C3),aa(set(A),nat,finite_card(A),S2))) ) ) ) ).

% prod_gen_delta
tff(fact_2826_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A5: set(A),F3: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),aa(B,B,aa(B,fun(B,B),divide_divide(B),K4),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A5)))) )
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ( A5 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),K4) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_2827_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),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_2828_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_2829_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_gz(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_2830_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_ha(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C3)),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)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_2831_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G3: fun(nat,A),H2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H2)),set_ord_atMost(nat,P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G3),H2)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_2832_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_ord_atMost(nat,Na)) = $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,suc,Na)))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ).

% sum_gp0
tff(fact_2833_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),Xa)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) = $ite(Xa = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),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),Xa),aa(nat,nat,suc,Na))))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Xa))) ) ).

% sum_gp_offset
tff(fact_2834_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_2835_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A2: 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),C3)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_2836_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C3: A,A2: 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),C3),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_2837_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_2838_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C3: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_2839_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_2840_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_2841_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          ( ( C3 != 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),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_2842_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) = $ite(C3 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_2843_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_2844_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) ) ) ).

% div_less
tff(fact_2845_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = M )
      <=> ( Na = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_2846_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_2847_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_2848_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_2849_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_2850_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),K)) ) ).

% div_le_mono
tff(fact_2851_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ).

% div_le_dividend
tff(fact_2852_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2853_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),Na)) ).

% Suc_div_le_mono
tff(fact_2854_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_2855_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))),M) ).

% times_div_less_eq_dividend
tff(fact_2856_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_2857_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_2858_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% div_greater_zero_iff
tff(fact_2859_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_2860_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),M) ) ) ).

% div_less_dividend
tff(fact_2861_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q3)),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ) ) ).

% div_less_iff_less_mult
tff(fact_2862_td__gal__lt,axiom,
    ! [C3: nat,A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C3))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C3)),B2) ) ) ).

% td_gal_lt
tff(fact_2863_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_2864_zdiv__zmult2__eq,axiom,
    ! [C3: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C3) ) ) ).

% zdiv_zmult2_eq
tff(fact_2865_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: 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),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_2866_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: 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),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_2867_div__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        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),Na)),Na)) ) ).

% div_if
tff(fact_2868_div__nat__eqI,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q3)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_2869_split__div,axiom,
    ! [P2: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P2,zero_zero(nat)) )
        & ( ( Na != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I3)),J3) )
               => aa(nat,$o,P2,I3) ) ) ) ) ) ).

% split_div
tff(fact_2870_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_2871_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)))) ) ).

% dividend_less_times_div
tff(fact_2872_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Q3))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_2873_td__gal,axiom,
    ! [C3: nat,B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C3)),A2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C3)) ) ) ).

% td_gal
tff(fact_2874_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = $ite(Na = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),M),Na)))) ).

% divide_nat_def
tff(fact_2875_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) = 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),Na)),Na)) ) ) ) ).

% le_div_geq
tff(fact_2876_split__div_H,axiom,
    ! [P2: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
          & aa(nat,$o,P2,zero_zero(nat)) )
        | ? [Q7: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q7)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q7)))
            & aa(nat,$o,P2,Q7) ) ) ) ).

% split_div'
tff(fact_2877_power__sub,axiom,
    ! [Na: nat,M: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),Na)) ) ) ) ).

% power_sub
tff(fact_2878_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),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),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_2879_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2880_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_2881_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_2882_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_he(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_2883_set__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_hf(nat,fun(list(A),fun(list(A),$o)),Na),Xs)) ).

% set_n_lists
tff(fact_2884_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),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)),Na) = 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),Na),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_2885_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_2886_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),A2) = zero_zero(A) ) ).

% mod_self
tff(fact_2887_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_2888_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_2889_nat__mod__eq_H,axiom,
    ! [A2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = A2 ) ) ).

% nat_mod_eq'
tff(fact_2890_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = M ) ) ).

% mod_less
tff(fact_2891_binomial__Suc__n,axiom,
    ! [Na: nat] : binomial(aa(nat,nat,suc,Na),Na) = aa(nat,nat,suc,Na) ).

% binomial_Suc_n
tff(fact_2892_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_2893_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_2894_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_2895_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_2896_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2897_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2898_mod__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_2899_binomial__1,axiom,
    ! [Na: nat] : binomial(Na,aa(nat,nat,suc,zero_zero(nat))) = Na ).

% binomial_1
tff(fact_2900_binomial__0__Suc,axiom,
    ! [K: nat] : binomial(zero_zero(nat),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2901_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( binomial(Na,K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ).

% binomial_eq_0_iff
tff(fact_2902_binomial__Suc__Suc,axiom,
    ! [Na: nat,K: nat] : binomial(aa(nat,nat,suc,Na),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(Na,K)),binomial(Na,aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2903_binomial__n__0,axiom,
    ! [Na: nat] : binomial(Na,zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2904_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2905_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,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),Na)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self1
tff(fact_2906_Suc__mod__mult__self2,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,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),Na),K)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self2
tff(fact_2907_Suc__mod__mult__self3,axiom,
    ! [K: nat,Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,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),Na)),M))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self3
tff(fact_2908_Suc__mod__mult__self4,axiom,
    ! [Na: nat,K: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,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),Na),K)),M))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% Suc_mod_mult_self4
tff(fact_2909_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(Na,K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% zero_less_binomial_iff
tff(fact_2910_mod__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) ).

% mod_Suc_eq
tff(fact_2911_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),Na) ).

% mod_Suc_Suc_eq
tff(fact_2912_nat__mod__eq,axiom,
    ! [B2: nat,Na: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),Na)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Na) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_2913_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),M) ).

% mod_less_eq_dividend
tff(fact_2914_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
     => ( binomial(Na,K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2915_Suc__times__binomial,axiom,
    ! [K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),binomial(aa(nat,nat,suc,Na),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),binomial(Na,K)) ).

% Suc_times_binomial
tff(fact_2916_Suc__times__binomial__eq,axiom,
    ! [Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),binomial(Na,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(aa(nat,nat,suc,Na),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_2917_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( binomial(Na,K) = binomial(Na,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)) ) ) ).

% binomial_symmetric
tff(fact_2918_binomial__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),R2)) ) ).

% binomial_le_pow
tff(fact_2919_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2920_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2921_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2922_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Na) = $ite(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) = Na,zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))) ).

% mod_Suc
tff(fact_2923_mod__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat,P3: nat,M: nat] :
      ( aa(nat,$o,P2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),P3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),P3)
               => ( aa(nat,$o,P2,N)
                 => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N)),P3)) ) )
           => aa(nat,$o,P2,M) ) ) ) ) ).

% mod_induct
tff(fact_2924_gcd__nat__induct,axiom,
    ! [P2: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [M3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P2,M3),zero_zero(nat))
     => ( ! [M3: nat,N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P2,N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M3),N))
             => aa(nat,$o,aa(nat,fun(nat,$o),P2,M3),N) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P2,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_2925_nat__mod__lem,axiom,
    ! [Na: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),Na)
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Na) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_2926_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_less_divisor
tff(fact_2927_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_2928_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),L),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),L)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),L)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D2),Na)),L) = Na ) ) ) ).

% word_rot_lem
tff(fact_2929_nat__minus__mod,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),M))),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_2930_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na) ) ) ).

% mod_geq
tff(fact_2931_mod__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),M,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na)) ).

% mod_if
tff(fact_2932_mod__nat__sub,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y)),Z) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),Y) ) ) ).

% mod_nat_sub
tff(fact_2933_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
     => ? [Q5: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q5) ) ).

% mod_eq_0D
tff(fact_2934_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na) ) ) ).

% le_mod_geq
tff(fact_2935_nat__mod__eq__iff,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_2936_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(Na,K)) ) ).

% zero_less_binomial
tff(fact_2937_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),A2)) ).

% Suc_times_binomial_add
tff(fact_2938_binomial__Suc__Suc__eq__times,axiom,
    ! [Na: nat,K: nat] : binomial(aa(nat,nat,suc,Na),aa(nat,nat,suc,K)) = 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,Na)),binomial(Na,K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_2939_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(Na,M)),binomial(M,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(Na,K)),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% choose_mult
tff(fact_2940_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2941_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2942_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)),Na) ) ).

% mod_le_divisor
tff(fact_2943_div__less__mono,axiom,
    ! [A5: nat,B5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A5),Na) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),Na) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),Na)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2944_mod__nat__add,axiom,
    ! [Xa: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Z)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_2945_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Na) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
       => ? [Q5: nat] : Xa = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2946_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ~ ! [S3: nat] : Na != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_2947_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ~ ! [S3: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_2948_n__subsets,axiom,
    ! [A: $tType,A5: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_hg(set(A),fun(nat,fun(set(A),$o)),A5),K))) = binomial(aa(set(A),nat,finite_card(A),A5),K) ) ) ).

% n_subsets
tff(fact_2949_sum__choose__upper,axiom,
    ! [M: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hh(nat,fun(nat,nat),M)),set_ord_atMost(nat,Na)) = binomial(aa(nat,nat,suc,Na),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_2950_binomial__absorption,axiom,
    ! [K: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),binomial(Na,aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),K)) ).

% binomial_absorption
tff(fact_2951_split__mod,axiom,
    ! [P2: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P2,M) )
        & ( ( Na != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I3)),J3) )
               => aa(nat,$o,P2,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2952_mod__lemma,axiom,
    ! [C3: nat,R2: nat,B2: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q3),C3))),R2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C3)) ) ) ).

% mod_lemma
tff(fact_2953_sum__choose__lower,axiom,
    ! [R2: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_hi(nat,fun(nat,nat),R2)),set_ord_atMost(nat,Na)) = binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),Na)),Na) ).

% sum_choose_lower
tff(fact_2954_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),binomial(Na,K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2955_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( binomial(Na,K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2956_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),binomial(Na,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2957_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C3))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2958_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2959_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_hj(nat,fun(nat,fun(nat,nat)),M),Na)),set_ord_atMost(nat,M)) = binomial(aa(nat,nat,suc,Na),M) ) ) ).

% sum_choose_diagonal
tff(fact_2960_nth__rotate1,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_2961_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( binomial(Na,aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),aa(nat,nat,suc,K))),binomial(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_2962_verit__le__mono__div,axiom,
    ! [A5: nat,B5: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A5),B5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A5),Na)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),Na) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),Na)) ) ) ).

% verit_le_mono_div
tff(fact_2963_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(Na,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hk(nat,fun(nat,fun(nat,A)),K),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_2964_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,Na),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_hm(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Na,Xs))) ).

% n_lists.simps(2)
tff(fact_2965_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( ( Na != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hn(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_2966_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_ho(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_2967_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summary: vEBT_VEBT,Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(Mmo,Deg,Tree_list,Summary)),vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi)) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          aa(assn,assn,
            aa(assn,fun(assn,assn),times_times(assn),
              pure_assn(( ( Mmoi = Mmo )
                & ( Degi = Deg ) ))),
            aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Summaryi))),
        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_hp(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list),Tree_array))) ).

% vebt_assn_raw.simps(2)
tff(fact_2968_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] :
          comm_s3205402744901411588hammer(A,A2,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_hq(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_2969_Set__filter__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( filter3(A,P2,A5) = finite_fold(A,set(A),aTP_Lamp_hr(fun(A,$o),fun(A,fun(set(A),set(A))),P2),bot_bot(set(A)),A5) ) ) ).

% Set_filter_fold
tff(fact_2970_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_2971_member__filter,axiom,
    ! [A: $tType,Xa: A,P2: fun(A,$o),A5: set(A)] :
      ( member(A,Xa,filter3(A,P2,A5))
    <=> ( member(A,Xa,A5)
        & aa(A,$o,P2,Xa) ) ) ).

% member_filter
tff(fact_2972_ex__assn__const,axiom,
    ! [A: $tType,C3: assn] : ex_assn(A,aTP_Lamp_hs(assn,fun(A,assn),C3)) = C3 ).

% ex_assn_const
tff(fact_2973_triv__exI,axiom,
    ! [A: $tType,Q2: fun(A,assn),Xa: A] : entails(aa(A,assn,Q2,Xa),ex_assn(A,Q2)) ).

% triv_exI
tff(fact_2974_mod__ex__dist,axiom,
    ! [A: $tType,P2: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P2)),H2)
    <=> ? [X5: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P2,X5)),H2) ) ).

% mod_ex_dist
tff(fact_2975_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2976_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2977_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_2978_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_2979_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V3: A] :
      ( ( aa(B,option(A),M,K) = none(A) )
     => ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_ht(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),V3),ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_2980_mod__h__bot__iff_I8_J,axiom,
    ! [A: $tType,R3: fun(A,assn),H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,R3)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> ? [X5: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,R3,X5)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ).

% mod_h_bot_iff(8)
tff(fact_2981_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),V3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),V3)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W2),V3) = W2 ) ) ) ).

% mod_word_less
tff(fact_2982_Set_Ofilter__def,axiom,
    ! [A: $tType,P2: fun(A,$o),A5: set(A)] : filter3(A,P2,A5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_hu(fun(A,$o),fun(set(A),fun(A,$o)),P2),A5)) ).

% Set.filter_def
tff(fact_2983_ex__distrib__star,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_hv(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_star
tff(fact_2984_ent__ex__preI,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] :
      ( ! [X: A] : entails(aa(A,assn,P2,X),Q2)
     => entails(ex_assn(A,P2),Q2) ) ).

% ent_ex_preI
tff(fact_2985_ent__ex__postI,axiom,
    ! [A: $tType,P2: assn,Q2: fun(A,assn),Xa: A] :
      ( entails(P2,aa(A,assn,Q2,Xa))
     => entails(P2,ex_assn(A,Q2)) ) ).

% ent_ex_postI
tff(fact_2986_ex__one__point__gen,axiom,
    ! [A: $tType,P2: fun(A,assn),V3: A] :
      ( ! [H: product_prod(heap_ext(product_unit),set(nat)),X: A] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P2,X)),H)
         => ( X = V3 ) )
     => ( ex_assn(A,P2) = aa(A,assn,P2,V3) ) ) ).

% ex_one_point_gen
tff(fact_2987_mod__exI,axiom,
    ! [A: $tType,P2: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X3: A] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P2,X3)),H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P2)),H2) ) ).

% mod_exI
tff(fact_2988_mod__exE,axiom,
    ! [A: $tType,P2: fun(A,assn),H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(ex_assn(A,P2)),H2)
     => ~ ! [X: A] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(A,assn,P2,X)),H2) ) ).

% mod_exE
tff(fact_2989_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_pos
tff(fact_2990_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2991_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% neg_mod_bound
tff(fact_2992_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2993_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2994_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2995_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
    <=> ? [Q7: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q7) ) ).

% zmod_eq_0_iff
tff(fact_2996_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
     => ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).

% zmod_eq_0D
tff(fact_2997_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Na)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),M),Na)),Na) ) ) ).

% word_mod_less_divisor
tff(fact_2998_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Na) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2999_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A2,Na) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3000_zmod__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zmod_int
tff(fact_3001_finite__filter,axiom,
    ! [A: $tType,S2: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => aa(set(A),$o,finite_finite2(A),filter3(A,P2,S2)) ) ).

% finite_filter
tff(fact_3002_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_3003_int__mod__eq,axiom,
    ! [B2: int,Na: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_3004_int__mod__lem,axiom,
    ! [Na: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),Na) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na) = B2 ) ) ) ).

% int_mod_lem
tff(fact_3005_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_3006_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3007_int__mod__ge,axiom,
    ! [A2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Na)) ) ) ).

% int_mod_ge
tff(fact_3008_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_3009_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_3010_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_3011_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_3012_int__mod__le_H,axiom,
    ! [B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),Na))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na)),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),Na)) ) ).

% int_mod_le'
tff(fact_3013_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_3014_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_3015_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_3016_zdiv__mono__strict,axiom,
    ! [A5: int,B5: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A5),Na) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),Na) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),Na)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_3017_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),Na)) ) ).

% pochhammer_rec
tff(fact_3018_int__mod__ge_H,axiom,
    ! [B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),Na)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Na)) ) ) ).

% int_mod_ge'
tff(fact_3019_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3020_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3021_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Na))) ) ).

% pochhammer_Suc
tff(fact_3022_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Na: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Na)) = 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),Na))),comm_s3205402744901411588hammer(A,Z,Na)) ) ).

% pochhammer_rec'
tff(fact_3023_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3024_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Na: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3025_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Na) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Na)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3026_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3027_mod__power__lem,axiom,
    ! [A2: int,Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Na)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),M)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Na)) ) ) ).

% mod_power_lem
tff(fact_3028_split__zmod,axiom,
    ! [P2: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P2,Na) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P2,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I3: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
             => aa(int,$o,P2,J3) ) ) ) ) ).

% split_zmod
tff(fact_3029_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),R2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3030_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3031_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),one_one(int))),L)) ) ) ).

% minus_mod_int_eq
tff(fact_3032_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_3033_mod__add__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_3034_mod__sub__if__z,axiom,
    ! [Xa: int,Z: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)),Z) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Xa),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_3035_zmod__zmult2__eq,axiom,
    ! [C3: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C3))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_3036_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_3037_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_3038_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( comm_s3205402744901411588hammer(A,Z,Na) = 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),Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_3039_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,Na) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_3040_verit__le__mono__div__int,axiom,
    ! [A5: int,B5: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A5),B5)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),Na)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),Na) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_3041_split__neg__lemma,axiom,
    ! [K: int,P2: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P2,I3),J3) ) ) ) ).

% split_neg_lemma
tff(fact_3042_split__pos__lemma,axiom,
    ! [K: int,P2: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),K))
      <=> ! [I3: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I3)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P2,I3),J3) ) ) ) ).

% split_pos_lemma
tff(fact_3043_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hw(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_3044_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hx(A,fun(nat,fun(nat,A)),A2),Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3045_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_hy(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_3046_vebt__assn__raw_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xa),Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A4) )
                      & ( (Bi) = (B4) ) )) ) ) )
       => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2) )
             => ! [Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] :
                  ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                 => ( Y != aa(assn,assn,
                        aa(assn,fun(assn,assn),times_times(assn),
                          aa(assn,assn,
                            aa(assn,fun(assn,assn),times_times(assn),
                              pure_assn(( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ))),
                            aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),Summaryi2))),
                        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_hp(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [V: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] : Xa = vEBT_Node(V,Va,Vb3,Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xa = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xaa = vEBT_Nodei(V,Va,Vb3,Vc3)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_3047_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( gbinomial(A,A2,K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3048_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),exp(real,aa(real,real,uminus_uminus(real),Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_3049_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),exp(real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_3050_TBOUND__VEBT__case,axiom,
    ! [A: $tType,Tia: vEBT_VEBTi,F3: fun($o,fun($o,heap_Time_Heap(A))),Bnd: fun($o,fun($o,nat)),F8: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),Bnd2: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat))))] :
      ( ! [A4: $o,B4: $o] :
          ( ( Tia = vEBT_Leafi((A4),(B4)) )
         => time_TBOUND(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F3,(A4)),(B4)),aa($o,nat,aa($o,fun($o,nat),Bnd,(A4)),(B4))) )
     => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeArray: array(vEBT_VEBTi),Summary2: vEBT_VEBTi] :
            ( ( Tia = vEBT_Nodei(Info2,Deg2,TreeArray,Summary2) )
           => time_TBOUND(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F8,Info2),Deg2),TreeArray),Summary2),aa(vEBT_VEBTi,nat,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat))),Bnd2,Info2),Deg2),TreeArray),Summary2)) )
       => time_TBOUND(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),fun(fun($o,fun($o,heap_Time_Heap(A))),fun(vEBT_VEBTi,heap_Time_Heap(A))),vEBT_case_VEBTi(heap_Time_Heap(A)),F8),F3),Tia),aa(vEBT_VEBTi,nat,aa(fun($o,fun($o,nat)),fun(vEBT_VEBTi,nat),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat)))),fun(fun($o,fun($o,nat)),fun(vEBT_VEBTi,nat)),vEBT_case_VEBTi(nat),Bnd2),Bnd),Tia)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3051_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leafi((X21),(X22)) = vEBT_Leafi((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBTi.inject(2)
tff(fact_3052_exp__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Y)) ) ).

% exp_less_mono
tff(fact_3053_exp__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% exp_less_cancel_iff
tff(fact_3054_exp__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% exp_le_cancel_iff
tff(fact_3055_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3056_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : gbinomial(A,zero_zero(A),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_3057_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : gbinomial(A,A2,zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3058_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : gbinomial(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_3059_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% one_less_exp_iff
tff(fact_3060_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_3061_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% one_le_exp_iff
tff(fact_3062_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3063_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3064_exp__less__cancel,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% exp_less_cancel
tff(fact_3065_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Xa))),exp(real,real_V7770717601297561774m_norm(A,Xa))) ) ).

% norm_exp
tff(fact_3066_VEBTi_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi,X21: $o,X22: $o] : vEBT_Nodei(X11,X12,X13,X14) != vEBT_Leafi((X21),(X22)) ).

% VEBTi.distinct(1)
tff(fact_3067_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: array(vEBT_VEBTi),X142: vEBT_VEBTi] : Y != vEBT_Nodei(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leafi((X212),(X222)) ) ).

% VEBTi.exhaust
tff(fact_3068_VEBTi_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F22: fun($o,fun($o,A)),X21: $o,X22: $o] : aa(vEBT_VEBTi,A,aa(fun($o,fun($o,A)),fun(vEBT_VEBTi,A),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A)),vEBT_case_VEBTi(A),F1),F22),vEBT_Leafi((X21),(X22))) = aa($o,A,aa($o,fun($o,A),F22,(X21)),(X22)) ).

% VEBTi.simps(6)
tff(fact_3069_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_3070_exp__gt__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),exp(real,Xa)) ).

% exp_gt_zero
tff(fact_3071_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] : exp(real,X) = Y ) ).

% exp_total
tff(fact_3072_exp__ge__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),exp(real,Xa)) ).

% exp_ge_zero
tff(fact_3073_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3074_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Leafi((X21),(X22))) = zero_zero(nat) ).

% VEBTi.size(4)
tff(fact_3075_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero(nat) ).

% VEBTi.size_gen(2)
tff(fact_3076_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa)) ) ).

% exp_gt_one
tff(fact_3077_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ).

% exp_ge_add_one_self
tff(fact_3078_vebt__assn__raw_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,vEBT_VEBTi)] :
      ( ! [A4: $o,B4: $o,Ai: $o,Bi: $o] : Xa != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((A4),(B4))),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] : Xa != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))
       => ( ! [V: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xa != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(V,Va,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xa != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V,Va,Vb3,Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_3079_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),gbinomial(A,A2,K)),gbinomial(A,A2,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3080_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xa: vEBT_VEBTi] :
      ( ( Xa != vEBT_Leafi($false,$false) )
     => ( ! [Uv: $o] : Xa != vEBT_Leafi($true,(Uv))
       => ( ! [Uu: $o] : Xa != vEBT_Leafi((Uu),$true)
         => ( ! [Uw: nat,Ux: array(vEBT_VEBTi),Uy: vEBT_VEBTi] : Xa != vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux,Uy)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] : Xa != vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_3081_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na),K) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3082_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Leaf((A2),(B2))),vEBT_Leafi((Ai2),(Bi2))) = pure_assn(( ( (Ai2) = (A2) )
        & ( (Bi2) = (B2) ) )) ).

% vebt_assn_raw.simps(1)
tff(fact_3083_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V3: option(product_prod(nat,nat)),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Vd2: $o,Ve2: $o] : aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(V3,Va2,Vb,Vc)),vEBT_Leafi((Vd2),(Ve2))) = bot_bot(assn) ).

% vebt_assn_raw.simps(3)
tff(fact_3084_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_hz(fun(A,B),fun(A,B),F3)),I5) ) ) ) ).

% exp_sum
tff(fact_3085_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3086_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),one_one(real)))
          & ( exp(real,X) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3087_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,A2,aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),aa(nat,nat,suc,K))),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),K)) ) ).

% gbinomial_addition_formula
tff(fact_3088_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,K)),A2) = 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)),gbinomial(A,A2,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))),gbinomial(A,A2,aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3089_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),gbinomial(A,A2,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)),gbinomial(A,A2,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))),gbinomial(A,A2,aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3090_vebt__minti_Ocases,axiom,
    ! [Xa: vEBT_VEBTi] :
      ( ! [A4: $o,B4: $o] : Xa != vEBT_Leafi((A4),(B4))
     => ( ! [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xa != vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xa != vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) ) ) ).

% vebt_minti.cases
tff(fact_3091_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),gbinomial(A,A2,K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3092_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: 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))),gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),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),A2),one_one(A))),gbinomial(A,A2,K)) ) ).

% Suc_times_gbinomial
tff(fact_3093_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: 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))),gbinomial(A,A2,aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),K)) ) ).

% gbinomial_absorption
tff(fact_3094_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,M)),gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,K)),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3095_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),aa(nat,A,semiring_1_of_nat(A),Na)))),Na) = exp(A,Xa) ) ) ) ).

% exp_divide_power_eq
tff(fact_3096_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3097_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),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),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),gbinomial(A,A2,K)) ) ).

% gbinomial_factors
tff(fact_3098_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( gbinomial(A,A2,K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),K),one_one(nat)))),gbinomial(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A)),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3099_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : gbinomial(A,A2,K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ia(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_3100_vebt__assn__def,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi] : aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia) = ex_assn(vEBT_VEBT,aa(vEBT_VEBTi,fun(vEBT_VEBT,assn),aa(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn)),aTP_Lamp_ib(nat,fun(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn))),Na),Sa),Tia)) ).

% vebt_assn_def
tff(fact_3101_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ic(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3102_vebt__assn__raw_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xa),Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel),aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A4) )
                          & ( (Bi) = (B4) ) )) )
                   => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel),aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((A4),(B4))),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2) )
               => ! [Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] :
                    ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                   => ( ( Y = aa(assn,assn,
                            aa(assn,fun(assn,assn),times_times(assn),
                              aa(assn,assn,
                                aa(assn,fun(assn,assn),times_times(assn),
                                  pure_assn(( ( Mmoi2 = Mmo2 )
                                    & ( Degi2 = Deg2 ) ))),
                                aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),Summaryi2))),
                            ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_hp(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) )
                     => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel),aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))) ) ) )
           => ( ! [V: option(product_prod(nat,nat)),Va: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(V,Va,Vb3,Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel),aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(V,Va,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xa = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V: option(product_prod(nat,nat)),Va: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] :
                        ( ( Xaa = vEBT_Nodei(V,Va,Vb3,Vc3) )
                       => ( ( Y = bot_bot(assn) )
                         => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel),aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V,Va,Vb3,Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_3103_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          gbinomial(A,A2,K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_id(A,fun(nat,fun(A,A)),A2),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_3104_vebt__minti_Oelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leafi((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),
                  $ite((B4),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),heap_Time_return(option(nat),none(nat))) ) ) )
       => ( ( ? [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xa = vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != heap_Time_return(option(nat),none(nat)) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xa = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_3105_vebt__maxti_Oelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leafi((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),
                  $ite((A4),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),heap_Time_return(option(nat),none(nat))) ) ) )
       => ( ( ? [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xa = vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != heap_Time_return(option(nat),none(nat)) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xa = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)
               => ( Y != heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_3106_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : gbinomial(A,A2,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_ie(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_3107_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3108_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_3109_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Na))),semiring_char_0_fact(A,Na)) ) ).

% fact_Suc
tff(fact_3110_fact__ge__self,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),semiring_char_0_fact(nat,Na)) ).

% fact_ge_self
tff(fact_3111_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ).

% fact_mono_nat
tff(fact_3112_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Na: nat] : semiring_char_0_fact(A,Na) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3113_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_3114_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_zero
tff(fact_3115_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Na)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3116_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_gt_zero
tff(fact_3117_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_1
tff(fact_3118_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ).

% fact_mono
tff(fact_3119_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_3120_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ) ).

% fact_less_mono
tff(fact_3121_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3122_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),Na))) ) ).

% fact_le_power
tff(fact_3123_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3124_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),binomial(Na,K)) = semiring_char_0_fact(nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_3125_prod__Suc__fact,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_fact
tff(fact_3126_prod__Suc__Suc__fact,axiom,
    ! [Na: 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)),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_3127_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : vEBT_vebt_minti(vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = heap_Time_return(option(nat),none(nat)) ).

% vebt_minti.simps(2)
tff(fact_3128_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : vEBT_vebt_maxti(vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) = heap_Time_return(option(nat),none(nat)) ).

% vebt_maxti.simps(2)
tff(fact_3129_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( binomial(Na,K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_3130_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz: vEBT_VEBTi] : vEBT_vebt_minti(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Mi)) ).

% vebt_minti.simps(3)
tff(fact_3131_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz: vEBT_VEBTi] : vEBT_vebt_maxti(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz)) = heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Ma)) ).

% vebt_maxti.simps(3)
tff(fact_3132_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Na)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_de(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3133_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = 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),Na))) ) ).

% fact_prod_Suc
tff(fact_3134_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = 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),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_rev
tff(fact_3135_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3136_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3137_vebt__maxti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxti(vEBT_Leafi((A2),(B2))) = $ite(
        (B2),
        heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),
        $ite((A2),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),heap_Time_return(option(nat),none(nat))) ) ).

% vebt_maxti.simps(1)
tff(fact_3138_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_minti(vEBT_Leafi((A2),(B2))) = $ite(
        (A2),
        heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),
        $ite((B2),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),heap_Time_return(option(nat),none(nat))) ) ).

% vebt_minti.simps(1)
tff(fact_3139_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),binomial(Na,K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))) ) ) ) ).

% fact_binomial
tff(fact_3140_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(Na,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Na)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))) ) ) ) ).

% binomial_fact
tff(fact_3141_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_de(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_3142_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K),Na)))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K))) ) ) ) ).

% fact_split
tff(fact_3143_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A2,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_if(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_3144_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),gbinomial(A,A2,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_if(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_3145_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : gbinomial(A,A2,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_ie(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_3146_vebt__maxti_Opelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xa) = Y )
     => ( aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leafi((A4),(B4)) )
             => ( ( Y = $ite(
                      (B4),
                      heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),
                      $ite((A4),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),heap_Time_return(option(nat),none(nat))) ) )
               => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),vEBT_Leafi((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] :
                ( ( Xa = vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = heap_Time_return(option(nat),none(nat)) )
                 => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xa = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Ma2)) )
                   => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_3147_vebt__minti_Opelims,axiom,
    ! [Xa: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xa) = Y )
     => ( aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leafi((A4),(B4)) )
             => ( ( Y = $ite(
                      (A4),
                      heap_Time_return(option(nat),aa(nat,option(nat),some(nat),zero_zero(nat))),
                      $ite((B4),heap_Time_return(option(nat),aa(nat,option(nat),some(nat),one_one(nat))),heap_Time_return(option(nat),none(nat))) ) )
               => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),vEBT_Leafi((A4),(B4))) ) )
         => ( ! [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] :
                ( ( Xa = vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = heap_Time_return(option(nat),none(nat)) )
                 => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xa = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2) )
                 => ( ( Y = heap_Time_return(option(nat),aa(nat,option(nat),some(nat),Mi2)) )
                   => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux,Uy,Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_3148_Maclaurin__lemma,axiom,
    ! [H2: real,F3: fun(real,real),J: fun(nat,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H2)
     => ? [B8: real] : aa(real,real,F3,H2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,real),fun(nat,real),aTP_Lamp_ig(real,fun(fun(nat,real),fun(nat,real)),H2),J)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),B8),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H2),Na)),semiring_char_0_fact(real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_3149_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: real,Na: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xa = zero_zero(real) )
         => ( ( Na != zero_zero(nat) )
           => ( aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ih(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xa),Diff)),set_ord_lessThan(nat,Na)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3150_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,suc,M)),Na)),semiring_char_0_fact(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_3151_sumr__cos__zero__one,axiom,
    ! [Na: nat] : aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ii(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Na))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3152_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T7))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
            & ( exp(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ij(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3153_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T7: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
      & ( exp(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ij(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_3154_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F4: set(A),I5: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ik(set(A),fun(fun(A,B),fun(A,$o)),I5),F3))),F4)
           => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I)),groups1027152243600224163dd_sum(A,B,F3,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3155_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S2))
           => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_il(set(A),fun(nat,fun(A,$o)),S2),Na)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_3156_Least__eq__0,axiom,
    ! [P2: fun(nat,$o)] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ord_Least(nat,P2) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_3157_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_idempotent
tff(fact_3158_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_3159_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_3160_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_3161_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_3162_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: 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),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_3163_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus_cancel
tff(fact_3164_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% abs_of_nat
tff(fact_3165_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_3166_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_3167_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_3168_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_3169_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P3,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_3170_sum_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),P3),I5) ) ) ) ).

% sum.eq_sum
tff(fact_3171_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3172_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F3),A5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_im(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs
tff(fact_3173_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_3174_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_3175_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_3176_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xa)) ) ) ).

% artanh_minus_real
tff(fact_3177_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F3: fun(B,A),A5: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_im(fun(B,A),fun(B,A),F3)),A5)) ) ).

% sum_abs_ge_zero
tff(fact_3178_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na))
        <=> ( ( A2 != zero_zero(A) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3179_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I5))) ) ) ) ).

% sum.insert'
tff(fact_3180_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2)) ) ).

% abs_minus_commute
tff(fact_3181_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_3182_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_3183_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_3184_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o)] :
          ( ? [X3: A] :
              ( aa(A,$o,P2,X3)
              & ! [Y3: A] :
                  ( aa(A,$o,P2,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
              & ! [Y3: A] :
                  ( ( aa(A,$o,P2,Y3)
                    & ! [Ya: A] :
                        ( aa(A,$o,P2,Ya)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Ya) ) )
                 => ( Y3 = X3 ) ) )
         => aa(A,$o,P2,ord_Least(A,P2)) ) ) ).

% Least1I
tff(fact_3185_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Z: A] :
          ( ? [X3: A] :
              ( aa(A,$o,P2,X3)
              & ! [Y3: A] :
                  ( aa(A,$o,P2,Y3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
              & ! [Y3: A] :
                  ( ( aa(A,$o,P2,Y3)
                    & ! [Ya: A] :
                        ( aa(A,$o,P2,Ya)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Ya) ) )
                 => ( Y3 = X3 ) ) )
         => ( aa(A,$o,P2,Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P2)),Z) ) ) ) ).

% Least1_le
tff(fact_3186_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xa: A,Q2: fun(A,$o)] :
          ( aa(A,$o,P2,Xa)
         => ( ! [Y3: A] :
                ( aa(A,$o,P2,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y3) )
           => ( ! [X: A] :
                  ( aa(A,$o,P2,X)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P2,Y4)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y4) )
                   => aa(A,$o,Q2,X) ) )
             => aa(A,$o,Q2,ord_Least(A,P2)) ) ) ) ) ).

% LeastI2_order
tff(fact_3187_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xa: A] :
          ( aa(A,$o,P2,Xa)
         => ( ! [Y3: A] :
                ( aa(A,$o,P2,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y3) )
           => ( ord_Least(A,P2) = Xa ) ) ) ) ).

% Least_equality
tff(fact_3188_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),A2: A,Q2: fun(A,$o)] :
          ( aa(A,$o,P2,A2)
         => ( ! [A4: A] :
                ( aa(A,$o,P2,A4)
               => ( ! [B10: A] :
                      ( aa(A,$o,P2,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B10) )
                 => aa(A,$o,Q2,A4) ) )
           => aa(A,$o,Q2,ord_Least(A,P2)) ) ) ) ).

% LeastI2_wellorder
tff(fact_3189_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P2,X_12)
         => ( ! [A4: A] :
                ( aa(A,$o,P2,A4)
               => ( ! [B10: A] :
                      ( aa(A,$o,P2,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B10) )
                 => aa(A,$o,Q2,A4) ) )
           => aa(A,$o,Q2,ord_Least(A,P2)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_3190_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G3,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_in(fun(B,A),fun(set(B),fun(B,$o)),G3),I5))) = groups1027152243600224163dd_sum(B,A,G3,I5) ) ).

% sum.non_neutral'
tff(fact_3191_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_3192_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_3193_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_3194_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq
tff(fact_3195_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ) ).

% abs_mult_less
tff(fact_3196_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),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),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_3197_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_3198_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_3199_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_3200_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_3201_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_3202_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_3203_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_3204_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_3205_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P2: fun(A,$o),K: A] :
          ( aa(A,$o,P2,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P2)),K) ) ) ).

% Least_le
tff(fact_3206_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P2))
         => ~ aa(A,$o,P2,K) ) ) ).

% not_less_Least
tff(fact_3207_finite__image__absD,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [S2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image(A,A,abs_abs(A)),S2))
         => aa(set(A),$o,finite_finite2(A),S2) ) ) ).

% finite_image_absD
tff(fact_3208_Least__Suc2,axiom,
    ! [P2: fun(nat,$o),Na: nat,Q2: fun(nat,$o),M: nat] :
      ( aa(nat,$o,P2,Na)
     => ( aa(nat,$o,Q2,M)
       => ( ~ aa(nat,$o,P2,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P2,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q2,K2) )
           => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,Q2)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_3209_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xa: A] :
          ( ! [E2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),E2) )
         => ( Xa = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_3210_Least__Suc,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,Na)
     => ( ~ aa(nat,$o,P2,zero_zero(nat))
       => ( ord_Least(nat,P2) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_io(fun(nat,$o),fun(nat,$o),P2))) ) ) ) ).

% Least_Suc
tff(fact_3211_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xa) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xa)) ) ) ) ).

% abs_mult_pos
tff(fact_3212_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_3213_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),Xa)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),Y)) ) ) ) ).

% abs_div_pos
tff(fact_3214_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Na)) ) ).

% zero_le_power_abs
tff(fact_3215_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_3216_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_3217_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_3218_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          aa(A,A,abs_abs(A),X3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),zero_zero(A)),aa(A,A,uminus_uminus(A),X3),X3) ) ).

% abs_if_raw
tff(fact_3219_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_3220_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)),aa(A,A,uminus_uminus(A),A2),A2) ) ).

% abs_if
tff(fact_3221_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C3: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D2)))),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),A2),C3))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_3222_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq4
tff(fact_3223_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_3224_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A2: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),A2))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_3225_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ip(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H2),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I5)),groups1027152243600224163dd_sum(A,B,H2,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3226_abs__real__def,axiom,
    ! [A2: real] :
      aa(real,real,abs_abs(real),A2) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real)),aa(real,real,uminus_uminus(real),A2),A2) ).

% abs_real_def
tff(fact_3227_sin__bound__lemma,axiom,
    ! [Xa: real,Y: real,U: real,V3: real] :
      ( ( Xa = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),U)),Y))),V3) ) ) ).

% sin_bound_lemma
tff(fact_3228_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_iq(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_3229_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
         => ( ? [X_12: A] : aa(A,$o,P2,X_12)
           => ( ord_Least(A,P2) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),P2)) ) ) ) ) ).

% Least_Min
tff(fact_3230_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] : aa(nat,A,infini527867602293511546merate(A,S2),zero_zero(nat)) = ord_Least(A,aTP_Lamp_ir(set(A),fun(A,$o),S2)) ) ).

% enumerate_0
tff(fact_3231_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,S2) = groups1027152243600224163dd_sum(A,B,G3,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3232_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G3,T3) = groups1027152243600224163dd_sum(A,B,G3,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3233_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,H2,I2) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S2)
                 => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,S2) = groups1027152243600224163dd_sum(A,B,H2,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3234_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = zero_zero(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S2)
                 => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
             => ( groups1027152243600224163dd_sum(A,B,G3,T3) = groups1027152243600224163dd_sum(A,B,H2,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3235_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),Xa))) ) ).

% abs_add_one_gt_zero
tff(fact_3236_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( ( Xa != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xa)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3237_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A))
           => ( Xa = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3238_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xa = Y )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3239_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),I5),G3)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),I5),H2)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ip(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H2),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G3,I5)),groups1027152243600224163dd_sum(A,B,H2,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3240_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_in(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_in(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_3241_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_3242_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xa: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xa,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xa),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,I2)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_is(fun(A,B),fun(fun(A,B),fun(A,B)),Xa),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3243_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_il(set(A),fun(nat,fun(A,$o)),S2),Na)) ) ) ) ).

% enumerate_Suc''
tff(fact_3244_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat] : aa(nat,A,infini527867602293511546merate(A,S2),aa(nat,nat,suc,Na)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),ord_Least(A,aTP_Lamp_ir(set(A),fun(A,$o),S2))),bot_bot(set(A))))),Na) ) ).

% enumerate_Suc
tff(fact_3245_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ik(set(A),fun(fun(A,B),fun(A,$o)),I5),F3)))
         => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,aa(B,fun(B,B),minus_minus(B),groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I)),groups1027152243600224163dd_sum(A,B,F3,I5)) ) ) ) ).

% sum_diff1'
tff(fact_3246_lemma__interval,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y4))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_3247_lemma__interval__lt,axiom,
    ! [A2: real,Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B2)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y4))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_3248_set__vebt__succ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_succ(T2,Xa) = $ite(
            ? [X5: nat] :
              ( member(nat,X5,vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),X5) ),
            aa(nat,option(nat),some(nat),ord_Least(nat,aa(nat,fun(nat,$o),aTP_Lamp_it(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),Xa))),
            none(nat) ) ) ) ).

% set_vebt_succ'
tff(fact_3249_arctan__add,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xa)),arctan(Y)) = arctan(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)))) ) ) ) ).

% arctan_add
tff(fact_3250_cos__coeff__Suc,axiom,
    ! [Na: nat] : cos_coeff(aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% cos_coeff_Suc
tff(fact_3251_bex__empty,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ~ ? [X3: A] :
          ( member(A,X3,bot_bot(set(A)))
          & aa(A,$o,P2,X3) ) ).

% bex_empty
tff(fact_3252_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A5: set(A),Q2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_iu(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),Q2)))
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_iv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q2),X5))) ) ) ) ).

% finite_Collect_bex
tff(fact_3253_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_3254_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% zero_less_arctan_iff
tff(fact_3255_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_3256_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_arctan_iff
tff(fact_3257_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_3258_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3259_set__vebt__pred_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_vebt_pred(T2,Xa) = $ite(
            ? [X5: nat] :
              ( member(nat,X5,vEBT_set_vebt(T2))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),Xa) ),
            aa(nat,option(nat),some(nat),order_Greatest(nat,aa(nat,fun(nat,$o),aTP_Lamp_iw(vEBT_VEBT,fun(nat,fun(nat,$o)),T2),Xa))),
            none(nat) ) ) ) ).

% set_vebt_pred'
tff(fact_3260_arctan__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),arctan(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% arctan_less_iff
tff(fact_3261_arctan__monotone,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),arctan(Y)) ) ).

% arctan_monotone
tff(fact_3262_arctan__monotone_H,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Y)) ) ).

% arctan_monotone'
tff(fact_3263_arctan__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% arctan_le_iff
tff(fact_3264_image__def,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : aa(set(B),set(A),image(B,A,F3),A5) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(A,$o)),F3),A5)) ).

% image_def
tff(fact_3265_Bex__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ? [X5: A] :
            ( member(A,X5,A5)
            & aa(A,$o,P2,X5) )
      <=> finite_fold(A,$o,aTP_Lamp_iy(fun(A,$o),fun(A,fun($o,$o)),P2),$false,A5) ) ) ).

% Bex_fold
tff(fact_3266_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),set_ord_atMost(int,K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_3267_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S2)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),set_ord_lessThan(int,K3)) ) ).

% finite_int_iff_bounded
tff(fact_3268_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M5: int] :
        ? [N5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M5),aa(int,int,abs_abs(int),N5))
          & member(int,N5,S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_3269_infinite__int__iff__unbounded,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ! [M5: int] :
        ? [N5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M5),aa(int,int,abs_abs(int),N5))
          & member(int,N5,S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_3270_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_3271_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_3272_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F3: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I2))),aa(nat,int,F3,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F3,Na))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
                & ( aa(nat,int,F3,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3273_incr__lemma,axiom,
    ! [D2: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_3274_decr__lemma,axiom,
    ! [D2: int,Xa: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_3275_nat__ivt__aux,axiom,
    ! [Na: nat,F3: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,suc,I2))),aa(nat,int,F3,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F3,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F3,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3276_sin__coeff__Suc,axiom,
    ! [Na: nat] : sin_coeff(aa(nat,nat,suc,Na)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na))) ).

% sin_coeff_Suc
tff(fact_3277_nat0__intermed__int__val,axiom,
    ! [Na: nat,F3: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F3,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F3,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F3,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3278_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_3279_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_3280_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_3281_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P2,X3) )
           => ( find(A,P2,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_iz(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P2)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_3282_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,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_ja(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_3283_ln__inj__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xa = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_3284_ln__less__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_3285_ln__le__cancel__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_3286_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = zero_zero(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_3287_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_3288_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_3289_exp__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( exp(real,aa(real,real,ln_ln(real),Xa)) = Xa ) ) ).

% exp_ln
tff(fact_3290_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),Xa)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% exp_ln_iff
tff(fact_3291_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_3292_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_3293_find__SomeD_I1_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),Xa) )
     => aa(A,$o,P2,Xa) ) ).

% find_SomeD(1)
tff(fact_3294_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_less_self
tff(fact_3295_GreatestI__ex__nat,axiom,
    ! [P2: fun(nat,$o),B2: nat] :
      ( ? [X_12: nat] : aa(nat,$o,P2,X_12)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P2,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P2,order_Greatest(nat,P2)) ) ) ).

% GreatestI_ex_nat
tff(fact_3296_Greatest__le__nat,axiom,
    ! [P2: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P2,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P2,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P2)) ) ) ).

% Greatest_le_nat
tff(fact_3297_GreatestI__nat,axiom,
    ! [P2: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P2,K)
     => ( ! [Y3: nat] :
            ( aa(nat,$o,P2,Y3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y3),B2) )
       => aa(nat,$o,P2,order_Greatest(nat,P2)) ) ) ).

% GreatestI_nat
tff(fact_3298_ln__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_bound
tff(fact_3299_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_gt_zero
tff(fact_3300_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_3301_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_3302_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_ge_zero
tff(fact_3303_size__list__estimation,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(A,nat,F3,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),size_list(A,F3,Xs)) ) ) ).

% size_list_estimation
tff(fact_3304_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G3: fun(A,nat)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,G3,X)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F3,Xs)),size_list(A,G3,Xs)) ) ).

% size_list_pointwise
tff(fact_3305_size__list__estimation_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F3,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F3,Xs)) ) ) ).

% size_list_estimation'
tff(fact_3306_find__SomeD_I2_J,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),Xa) )
     => member(A,Xa,aa(list(A),set(A),set2(A),Xs)) ) ).

% find_SomeD(2)
tff(fact_3307_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xa: A,Q2: fun(A,$o)] :
          ( aa(A,$o,P2,Xa)
         => ( ! [Y3: A] :
                ( aa(A,$o,P2,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( ! [X: A] :
                  ( aa(A,$o,P2,X)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P2,Y4)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X) )
                   => aa(A,$o,Q2,X) ) )
             => aa(A,$o,Q2,order_Greatest(A,P2)) ) ) ) ) ).

% GreatestI2_order
tff(fact_3308_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o),Xa: A] :
          ( aa(A,$o,P2,Xa)
         => ( ! [Y3: A] :
                ( aa(A,$o,P2,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( order_Greatest(A,P2) = Xa ) ) ) ) ).

% Greatest_equality
tff(fact_3309_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_3310_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_3311_ln__mult,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_3312_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),one_one(real)) )
       => ( Xa = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_3313_ln__div,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_3314_ln__ge__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Y)),Xa) ) ) ).

% ln_ge_iff
tff(fact_3315_ln__x__over__x__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xa)),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_3316_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_jb(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_3317_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_3318_ln__diff__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_3319_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_3320_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_realpow
tff(fact_3321_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xa))),aa(real,real,uminus_uminus(real),Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_3322_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F3: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F3,I2)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F3),I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_jc(fun(A,real),fun(A,real),F3)),I5) ) ) ) ).

% ln_prod
tff(fact_3323_find__Some__iff2,axiom,
    ! [A: $tType,Xa: A,P2: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xa) = find(A,P2,Xs) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3))
          & ( Xa = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_3324_find__Some__iff,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P2,Xs) = aa(A,option(A),some(A),Xa) )
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P2,aa(nat,A,nth(A,Xs),I3))
          & ( Xa = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I3)
             => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_3325_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X21: A,X22: list(A)] : size_list(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X21)),size_list(A,Xa,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_3326_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F3: fun(nat,A),Ns: list(nat)] :
          ( ! [X: nat,Y3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,X)),aa(nat,A,F3,Y3)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(A,aa(list(nat),list(A),map(nat,A,F3),Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_3327_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X4: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X4)
     => ( aa(set(A),$o,finite_finite2(A),X4)
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,nat),fun(A,nat),aTP_Lamp_jd(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X4) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_3328_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_je(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_3329_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_je(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_3330_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P2: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A12))
     => ( ! [I2: int,J2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P2,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P2,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P2,A0),A12) ) ) ).

% upto.pinduct
tff(fact_3331_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X5: A] :
              ( member(A,X5,aa(list(A),set(A),set2(A),Ns))
             => ( X5 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_3332_sum__list__0,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aTP_Lamp_jf(B,A)),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_3333_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_3334_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_3335_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Xs: list(A)] :
          ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),groups8242544230860333062m_list(A,Xs)) ) ) ).

% member_le_sum_list
tff(fact_3336_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_3337_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X5: A] :
                ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
               => ( X5 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_3338_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_3339_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs))) ) ).

% sum_list_abs
tff(fact_3340_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = foldr(A,A,plus_plus(A),Xs,zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_3341_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ).

% sum_list_mono
tff(fact_3342_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns)) ) ) ).

% elem_le_sum_list
tff(fact_3343_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_cz(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_3344_card__length__sum__list__rec,axiom,
    ! [M: nat,N6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_jg(nat,fun(nat,fun(list(nat),$o)),M),N6))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_jh(nat,fun(nat,fun(list(nat),$o)),M),N6)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_ji(nat,fun(nat,fun(list(nat),$o)),M),N6)))) ) ) ).

% card_length_sum_list_rec
tff(fact_3345_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_3346_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K,Xa)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),Xa)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_3347_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q3 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3348_bezw__0,axiom,
    ! [Xa: nat] : bezw(Xa,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_3349_max__extp_Omax__extI,axiom,
    ! [A: $tType,X4: set(A),Y6: set(A),R3: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(A),$o,finite_finite2(A),Y6)
       => ( ( Y6 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,Y6)
                    & aa(A,$o,aa(A,fun(A,$o),R3,X),Xa2) ) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),X4),Y6) ) ) ) ) ).

% max_extp.max_extI
tff(fact_3350_max__extp_Osimps,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),A12),A23)
    <=> ( aa(set(A),$o,finite_finite2(A),A12)
        & aa(set(A),$o,finite_finite2(A),A23)
        & ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
        & ! [X5: A] :
            ( member(A,X5,A12)
           => ? [Xa3: A] :
                ( member(A,Xa3,A23)
                & aa(A,$o,aa(A,fun(A,$o),R3,X5),Xa3) ) ) ) ) ).

% max_extp.simps
tff(fact_3351_unique__quotient,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
       => ( Q3 = Q6 ) ) ) ).

% unique_quotient
tff(fact_3352_unique__remainder,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int,Q6: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q6),R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_3353_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_3354_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_3355_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q3 ) ) ).

% div_int_unique
tff(fact_3356_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_3357_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)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ).

% eucl_rel_int
tff(fact_3358_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R2: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_3359_max__extp_Ocases,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,R3),A12),A23)
     => ~ ( aa(set(A),$o,finite_finite2(A),A12)
         => ( aa(set(A),$o,finite_finite2(A),A23)
           => ( ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
             => ~ ! [X3: A] :
                    ( member(A,X3,A12)
                   => ? [Xa5: A] :
                        ( member(A,Xa5,A23)
                        & aa(A,$o,aa(A,fun(A,$o),R3,X3),Xa5) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_3360_finite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S2)
     => ? [R: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R,set_ord_lessThan(nat,aa(set(nat),nat,finite_card(nat),S2)))
          & ! [N7: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N7),aa(set(nat),nat,finite_card(nat),S2))
             => member(nat,aa(nat,nat,R,N7),S2) ) ) ) ).

% finite_enumerate
tff(fact_3361_map__distinct__upd__conv,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F3: fun(A,B),Xa: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( distinct(A,L)
       => ( list_update(B,aa(list(A),list(B),map(A,B,F3),L),I,Xa) = aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,aa(nat,A,nth(A,L),I),Xa)),L) ) ) ) ).

% map_distinct_upd_conv
tff(fact_3362_in__lex__prod,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set(product_prod(A,A)),Sa: set(product_prod(B,B))] :
      ( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),lex_prod(A,B,R2,Sa))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A3),R2)
        | ( ( A2 = A3 )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3),Sa) ) ) ) ).

% in_lex_prod
tff(fact_3363_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( ? [K6: real] :
            ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K6))
         => real_V4916620083959148203axioms(A,B,F3) ) ) ).

% bounded_linear_axioms.intro
tff(fact_3364_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F3)
        <=> ? [K5: real] :
            ! [X5: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X5))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X5)),K5)) ) ) ).

% bounded_linear_axioms_def
tff(fact_3365_image__update,axiom,
    ! [B: $tType,A: $tType,Xa: A,A5: set(A),F3: fun(A,B),Na: B] :
      ( ~ member(A,Xa,A5)
     => ( aa(set(A),set(B),image(A,B,fun_upd(A,B,F3,Xa,Na)),A5) = aa(set(A),set(B),image(A,B,F3),A5) ) ) ).

% image_update
tff(fact_3366_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C3: B,P2: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_jj(fun(A,B),fun(B,fun(A,$o)),F3),C3)))
     => ( aa(fun(A,B),$o,P2,aTP_Lamp_cf(B,fun(A,B),C3))
       => ( ! [A4: A,B4: B,F2: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_jk(B,fun(fun(A,B),fun(A,$o)),C3),F2)))
             => ( ( aa(A,B,F2,A4) = C3 )
               => ( ( B4 != C3 )
                 => ( aa(fun(A,B),$o,P2,F2)
                   => aa(fun(A,B),$o,P2,fun_upd(A,B,F2,A4,B4)) ) ) ) )
         => aa(fun(A,B),$o,P2,F3) ) ) ) ).

% finite_update_induct
tff(fact_3367_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xa: B,Y: A,A5: set(B)] :
      aa(set(B),set(A),image(B,A,fun_upd(B,A,F3,Xa,Y)),A5) = $ite(member(B,Xa,A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A5),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),bot_bot(set(B)))))),aa(set(B),set(A),image(B,A,F3),A5)) ).

% fun_upd_image
tff(fact_3368_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A),R2: A,Sa: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,R2,A5)
           => ( member(A,Sa,A5)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),Sa)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R2)),aa(A,B,F3,Sa)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_3369_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [R: A,S3: A] :
              ( member(A,R,A5)
             => ( member(A,S3,A5)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R),S3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R)),aa(A,B,F3,S3)) ) ) )
         => strict_mono_on(A,B,F3,A5) ) ) ).

% strict_mono_onI
tff(fact_3370_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A5: set(A)] :
          ( strict_mono_on(A,B,F3,A5)
        <=> ! [R5: A,S6: A] :
              ( ( member(A,R5,A5)
                & member(A,S6,A5)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S6) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,R5)),aa(A,B,F3,S6)) ) ) ) ).

% strict_mono_on_def
tff(fact_3371_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A5: set(A),Xa: A,Y: A] :
          ( strict_mono_on(A,B,F3,A5)
         => ( member(A,Xa,A5)
           => ( member(A,Y,A5)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,F3,Y)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_3372_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_3373_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),Xa: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( finite_fold(A,B,F3,Z,A5) = aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_3374_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,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_jl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_3375_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: itself(A)] : semiri4206861660011772517g_char(A,Uu2) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_jm(nat,$o))) ) ).

% semiring_char_def
tff(fact_3376_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_aj(nat,fun(nat,$o)),aTP_Lamp_ai(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_3377_Suc__funpow,axiom,
    ! [Na: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),Na),suc) = aa(nat,fun(nat,nat),plus_plus(nat),Na) ).

% Suc_funpow
tff(fact_3378_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),Xa: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),Xa) = Xa ).

% funpow_0
tff(fact_3379_relpowp__Suc__I2,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xa: A,Y: A,Na: nat,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),P2,Xa),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Na)),P2),Xa),Z) ) ) ).

% relpowp_Suc_I2
tff(fact_3380_relpowp__Suc__E2,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Na)),P2),Xa),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P2,Xa),Y3)
           => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Y3),Z) ) ) ).

% relpowp_Suc_E2
tff(fact_3381_relpowp__Suc__D2,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Na)),P2),Xa),Z)
     => ? [Y3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),P2,Xa),Y3)
          & aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Y3),Z) ) ) ).

% relpowp_Suc_D2
tff(fact_3382_relpowp__Suc__I,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Y: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Xa),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),P2,Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Na)),P2),Xa),Z) ) ) ).

% relpowp_Suc_I
tff(fact_3383_relpowp__Suc__E,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),aa(nat,nat,suc,Na)),P2),Xa),Z)
     => ~ ! [Y3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Xa),Y3)
           => ~ aa(A,$o,aa(A,fun(A,$o),P2,Y3),Z) ) ) ).

% relpowp_Suc_E
tff(fact_3384_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),R3) = fequal(A) ).

% relpowp.simps(1)
tff(fact_3385_relpowp__0__E,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xa: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P2),Xa),Y)
     => ( Xa = Y ) ) ).

% relpowp_0_E
tff(fact_3386_relpowp__0__I,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o)),Xa: A] : aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P2),Xa),Xa) ).

% relpowp_0_I
tff(fact_3387_comp__fun__commute__on_Ofun__left__comm,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,Y: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xa,S2)
       => ( member(A,Y,S2)
         => ( aa(B,B,aa(A,fun(B,B),F3,Y),aa(B,B,aa(A,fun(B,B),F3,Xa),Z)) = aa(B,B,aa(A,fun(B,B),F3,Xa),aa(B,B,aa(A,fun(B,B),F3,Y),Z)) ) ) ) ) ).

% comp_fun_commute_on.fun_left_comm
tff(fact_3388_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),Na: nat,Xa: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3),Xa)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3),aa(A,A,F3,Xa)) ).

% funpow_swap1
tff(fact_3389_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => finite4664212375090638736ute_on(A,B,S2,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_jn(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_3390_funpow__mult,axiom,
    ! [A: $tType,Na: nat,M: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),F3) ).

% funpow_mult
tff(fact_3391_relpowp__E,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Xa),Z)
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( Na = aa(nat,nat,suc,M3) )
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M3),P2),Xa),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),P2,Y3),Z) ) ) ) ) ).

% relpowp_E
tff(fact_3392_relpowp__E2,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Xa),Z)
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y3: A,M3: nat] :
              ( ( Na = aa(nat,nat,suc,M3) )
             => ( aa(A,$o,aa(A,fun(A,$o),P2,Xa),Y3)
               => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M3),P2),Y3),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_3393_relpowp__fun__conv,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,fun(A,$o)),Xa: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P2),Xa),Y)
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = Xa )
          & ( aa(nat,A,F7,Na) = Y )
          & ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),Na)
             => aa(A,$o,aa(A,fun(A,$o),P2,aa(nat,A,F7,I3)),aa(nat,A,F7,aa(nat,nat,suc,I3))) ) ) ) ).

% relpowp_fun_conv
tff(fact_3394_relpowp__bot,axiom,
    ! [A: $tType,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_3395_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),Na) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_3396_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(A,fun(B,B)),A5: set(A),Sa: B,T2: B,B5: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( finite4664212375090638736ute_on(A,B,S2,G3)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( ! [X: A] :
                  ( member(A,X,A5)
                 => ( aa(A,fun(B,B),F3,X) = aa(A,fun(B,B),G3,X) ) )
             => ( ( Sa = T2 )
               => ( ( A5 = B5 )
                 => ( finite_fold(A,B,F3,Sa,A5) = finite_fold(A,B,G3,T2,B5) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_3397_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xa),Z),A5) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_3398_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xa),Z),A5) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_3399_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,A5)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_3400_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_3401_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q7: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q7: int] :
            ( ( A12 = K3 )
            & ( A23 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q7),R5) )
            & ( aa(int,int,sgn_sgn(int),R5) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L4))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q7),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_3402_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
     => ( ( ( A23 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A12) ) )
       => ( ! [Q5: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),zero_zero(int)) )
             => ( ( A23 != zero_zero(int) )
               => ( A12 != aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A23) ) ) )
         => ~ ! [R: int,Q5: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R) )
               => ( ( aa(int,int,sgn_sgn(int),R) = aa(int,int,sgn_sgn(int),A23) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R)),aa(int,int,abs_abs(int),A23))
                   => ( A12 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),A23)),R) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3403_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = linorder_insort_key(A,A,aTP_Lamp_jo(A,A),Xa,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_3404_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_3405_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_3406_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,sgn_sgn(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_3407_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,sgn_sgn(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_3408_length__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),linorder_insort_key(A,B,F3,Xa,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_insort
tff(fact_3409_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_3410_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_3411_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = linorder_insort_key(A,A,aTP_Lamp_jo(A,A),Xa,linord4507533701916653071of_set(A,A5)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_3412_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_3413_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_3414_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_3415_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( aa(A,A,sgn_sgn(A),Xa) = zero_zero(A) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_3416_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_3417_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_3418_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xa: A,Y: A,Ys: list(A)] :
          linorder_insort_key(A,B,F3,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,F3,Y)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),linorder_insort_key(A,B,F3,Xa,Ys))) ) ).

% insort_key.simps(2)
tff(fact_3419_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_jo(A,A),Xa,Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_3420_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_3421_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_3422_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B),A2: A] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,A2)),aa(A,B,F3,X)) )
         => ( linorder_insort_key(A,B,F3,A2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_3423_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xa: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),linorder_insort_key(B,A,F3,Xa,Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_3424_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          aa(A,A,sgn_sgn(A),Xa) = $ite(
            Xa = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_3425_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_3426_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_3427_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),Xa)) = $ite(Xa = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_3428_div__sgn__abs__cancel,axiom,
    ! [V3: int,K: int,L: int] :
      ( ( V3 != zero_zero(int) )
     => ( 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),V3)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V3)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_3429_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( linorder_insort_key(A,A,aTP_Lamp_jo(A,A),A2,remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_3430_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( linord4507533701916653071of_set(A,A5) = linorder_insort_key(A,A,aTP_Lamp_jo(A,A),Xa,linord4507533701916653071of_set(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_3431_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q3: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3432_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sinh(A,Xa) = zero_zero(A) )
        <=> member(A,exp(A,Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_3433_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1962203154675924110t_prod(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I),I5)) = $ite(member(A,I,I5),groups1962203154675924110t_prod(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,I)),groups1962203154675924110t_prod(A,B,P3,I5))) ) ) ) ).

% prod.insert'
tff(fact_3434_time__array__nth,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P3: array(A),I: nat,H2: heap_ext(product_unit)] :
          time_time(A,array_nth(A,P3,I),H2) = $ite(time_fails(A,array_nth(A,P3,I),H2),zero_zero(nat),one_one(nat)) ) ).

% time_array_nth
tff(fact_3435_precise__extr__pure_I2_J,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,assn)),P2: $o] :
      ( precise(A,B,aa($o,fun(A,fun(B,assn)),aTP_Lamp_jp(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),R3),(P2)))
    <=> ( (P2)
       => precise(A,B,R3) ) ) ).

% precise_extr_pure(2)
tff(fact_3436_precise__extr__pure_I1_J,axiom,
    ! [B: $tType,A: $tType,P2: $o,R3: fun(A,fun(B,assn))] :
      ( precise(A,B,aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_jq($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(P2)),R3))
    <=> ( (P2)
       => precise(A,B,R3) ) ) ).

% precise_extr_pure(1)
tff(fact_3437_sinh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% sinh_real_less_iff
tff(fact_3438_sinh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% sinh_real_le_iff
tff(fact_3439_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% sinh_real_pos_iff
tff(fact_3440_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_3441_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sgn_sgn(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_3442_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_sgn_iff
tff(fact_3443_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_3444_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_3445_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_3446_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : groups1962203154675924110t_prod(B,A,P3,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_3447_prod_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),P3),I5) ) ) ) ).

% prod.eq_sum
tff(fact_3448_snga__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(list(A),array(A),aTP_Lamp_jr(list(A),fun(array(A),assn))) ) ).

% snga_prec
tff(fact_3449_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_js(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H2),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I5)),groups1962203154675924110t_prod(A,B,H2,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_3450_sngr__prec,axiom,
    ! [A: $tType] :
      ( heap(A)
     => precise(A,ref(A),aTP_Lamp_jt(A,fun(ref(A),assn))) ) ).

% sngr_prec
tff(fact_3451_sgn__real__def,axiom,
    ! [A2: real] :
      aa(real,real,sgn_sgn(real),A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_3452_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S2)
                 => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
             => ( groups1962203154675924110t_prod(A,B,G3,T3) = groups1962203154675924110t_prod(A,B,H2,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_3453_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,H2,I2) = one_one(B) ) )
           => ( ! [X: A] :
                  ( member(A,X,S2)
                 => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
             => ( groups1962203154675924110t_prod(A,B,G3,S2) = groups1962203154675924110t_prod(A,B,H2,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_3454_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G3,T3) = groups1962203154675924110t_prod(A,B,G3,S2) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_3455_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),T3: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
         => ( ! [X: A] :
                ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
               => ( aa(A,B,G3,X) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G3,S2) = groups1962203154675924110t_prod(A,B,G3,T3) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_3456_preciseD_H,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,assn)),A2: A,P3: B,F4: assn,H2: product_prod(heap_ext(product_unit),set(nat)),A3: A,F6: assn] :
      ( precise(A,B,R3)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A2),P3)),F4)),H2)
       => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A3),P3)),F6)),H2)
         => ( A2 = A3 ) ) ) ) ).

% preciseD'
tff(fact_3457_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G3: fun(A,B),H2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),I5),G3)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),I5),H2)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_js(fun(A,B),fun(fun(A,B),fun(A,B)),G3),H2),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G3,I5)),groups1962203154675924110t_prod(A,B,H2,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_3458_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ju(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_ju(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_3459_time__array__upd,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xa: A,P3: array(A),H2: heap_ext(product_unit)] :
          time_time(array(A),array_upd(A,I,Xa,P3),H2) = $ite(time_fails(array(A),array_upd(A,I,Xa,P3),H2),zero_zero(nat),one_one(nat)) ) ).

% time_array_upd
tff(fact_3460_sgn__power__injE,axiom,
    ! [A2: real,Na: nat,Xa: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),Na)) = Xa )
     => ( ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Na)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_3461_time__refines,axiom,
    ! [A: $tType,C3: heap_Time_Heap(A),C8: heap_Time_Heap(A),H2: heap_ext(product_unit)] :
      ( refine_Imp_refines(A,C3,C8)
     => ( ~ time_fails(A,C8,H2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C3,H2)),time_time(A,C8,H2)) ) ) ).

% time_refines
tff(fact_3462_divmod__nat__def,axiom,
    ! [M: nat,Na: nat] : divmod_nat(M,Na) = 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),Na)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na)) ).

% divmod_nat_def
tff(fact_3463_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,Xa),Z),A5) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_3464_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,A5: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)),S2)
       => ( aa(set(A),$o,finite_finite2(A),A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,A5)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_3465_ln__root,axiom,
    ! [Na: nat,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_3466_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V3: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V3))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_3467_real__root__Suc__0,axiom,
    ! [Xa: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xa) = Xa ).

% real_root_Suc_0
tff(fact_3468_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = aa(real,real,root(Na),Y) )
      <=> ( Xa = Y ) ) ) ).

% real_root_eq_iff
tff(fact_3469_root__0,axiom,
    ! [Xa: real] : aa(real,real,root(zero_zero(nat)),Xa) = zero_zero(real) ).

% root_0
tff(fact_3470_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_3471_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ).

% real_root_less_iff
tff(fact_3472_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ).

% real_root_le_iff
tff(fact_3473_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = one_one(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_3474_real__root__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_3475_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_3476_real__root__gt__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_3477_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_3478_real__root__ge__0__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_3479_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_3480_real__root__gt__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_3481_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_3482_real__root__ge__1__iff,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Na),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_3483_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_3484_comp__fun__idem__on_Ofun__left__idem,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( member(A,Xa,S2)
       => ( aa(B,B,aa(A,fun(B,B),F3,Xa),aa(B,B,aa(A,fun(B,B),F3,Xa),Z)) = aa(B,B,aa(A,fun(B,B),F3,Xa),Z) ) ) ) ).

% comp_fun_idem_on.fun_left_idem
tff(fact_3485_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Na),Xa)) = aa(real,real,sgn_sgn(real),Xa) ) ) ).

% sgn_root
tff(fact_3486_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_3487_comp__fun__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => finite4664212375090638736ute_on(A,B,S2,F3) ) ).

% comp_fun_idem_on.axioms(1)
tff(fact_3488_in__graphI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V3: A] :
      ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V3) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V3),graph(B,A,M)) ) ).

% in_graphI
tff(fact_3489_in__graphD,axiom,
    ! [A: $tType,B: $tType,K: A,V3: B,M: fun(A,option(B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3),graph(A,B,M))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V3) ) ) ).

% in_graphD
tff(fact_3490_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_less_mono
tff(fact_3491_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Y)) ) ) ).

% real_root_le_mono
tff(fact_3492_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_3493_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,abs_abs(real),Xa)) = aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_abs
tff(fact_3494_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_3495_real__root__strict__decreasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N6),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_3496_root__abs__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Na))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_3497_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_3498_real__root__strict__increasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N6),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_3499_real__root__decreasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N6),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_3500_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_3501_real__root__pos__unique,axiom,
    ! [Na: nat,Y: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Na) = Xa )
         => ( aa(real,real,root(Na),Xa) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_3502_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_3503_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(Na),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_3504_root__sgn__power,axiom,
    ! [Na: nat,Y: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y)),Na))) = Y ) ) ).

% root_sgn_power
tff(fact_3505_real__root__increasing,axiom,
    ! [Na: nat,N6: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N6)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N6),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_3506_split__root,axiom,
    ! [P2: fun(real,$o),Na: nat,Xa: real] :
      ( aa(real,$o,P2,aa(real,real,root(Na),Xa))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(real,$o,P2,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y5)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Y5)),Na)) = Xa )
             => aa(real,$o,P2,Y5) ) ) ) ) ).

% split_root
tff(fact_3507_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),Xa) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ) ).

% root_powr_inverse
tff(fact_3508_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Y: B] : restrict_map(A,B,fun_upd(A,option(B),M,Xa,aa(B,option(B),some(B),Y)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_3509_log__base__root,axiom,
    ! [Na: nat,B2: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
       => ( aa(real,real,log(aa(real,real,root(Na),B2)),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),Xa)) ) ) ) ).

% log_base_root
tff(fact_3510_log__root,axiom,
    ! [Na: nat,A2: real,B2: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( aa(real,real,log(B2),aa(real,real,root(Na),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B2),A2)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_3511_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),D5: set(A),Xa: A] :
      fun_upd(A,option(B),restrict_map(A,B,M,D5),Xa,none(B)) = $ite(member(A,Xa,D5),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),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),restrict_map(A,B,M,D5)) ).

% fun_upd_None_restrict
tff(fact_3512_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W2: A,Z: A] :
          ( ( powr(A,W2,Z) = zero_zero(A) )
        <=> ( W2 = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_3513_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_3514_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] :
          powr(A,Xa,zero_zero(A)) = $ite(Xa = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_3515_powr__gt__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xa,A2))
    <=> ( Xa != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_3516_powr__nonneg__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,Xa)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_3517_powr__less__cancel__iff,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_3518_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X3: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X3) = none(B) ).

% restrict_map_to_empty
tff(fact_3519_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_3520_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_3521_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_3522_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_3523_log__less__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_3524_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_3525_powr__eq__one__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( ( powr(real,A2,Xa) = one_one(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3526_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( powr(real,Xa,one_one(real)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_3527_powr__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,one_one(real)) = Xa ) ) ).

% powr_one
tff(fact_3528_powr__le__cancel__iff,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_3529_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_3530_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_3531_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_3532_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_3533_log__le__cancel__iff,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_3534_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3535_powr__log__cancel,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( powr(real,A2,aa(real,real,log(A2),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_3536_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xa: A,D5: set(A),M: fun(A,option(B)),Y: option(B)] :
      ( member(A,Xa,D5)
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D5),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),Xa,Y) ) ) ).

% fun_upd_restrict_conv
tff(fact_3537_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Y: option(B),D5: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),M,Xa,Y),D5) = $ite(member(A,Xa,D5),fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),Xa,Y),restrict_map(A,B,M,D5)) ).

% restrict_fun_upd
tff(fact_3538_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_3539_restrict__map__subset__eq,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),R3: set(A),M6: fun(A,option(B)),R6: set(A)] :
      ( ( restrict_map(A,B,M,R3) = M6 )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R6),R3)
       => ( restrict_map(A,B,M,R6) = restrict_map(A,B,M6,R6) ) ) ) ).

% restrict_map_subset_eq
tff(fact_3540_le__map__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),X4: set(A)] : aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),restrict_map(A,B,M,X4)),M) ) ).

% le_map_restrict
tff(fact_3541_restrict__map__eq_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A5: set(B),K: B,V3: A] :
      ( ( aa(B,option(A),restrict_map(B,A,M,A5),K) = aa(A,option(A),some(A),V3) )
    <=> ( ( aa(B,option(A),M,K) = aa(A,option(A),some(A),V3) )
        & member(B,K,A5) ) ) ).

% restrict_map_eq(2)
tff(fact_3542_less__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa) ) ) ) ).

% less_log_iff
tff(fact_3543_log__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_3544_less__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),Xa)),Y) ) ) ) ).

% less_powr_iff
tff(fact_3545_powr__less__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,log(B2),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_3546_le__log__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa) ) ) ) ).

% le_log_iff
tff(fact_3547_log__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xa)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_3548_le__powr__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),Xa)),Y) ) ) ) ).

% le_powr_iff
tff(fact_3549_powr__le__iff,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,log(B2),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_3550_powr__less__mono2__neg,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3551_powr__non__neg,axiom,
    ! [A2: real,Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A2,Xa)),zero_zero(real)) ).

% powr_non_neg
tff(fact_3552_powr__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_3553_powr__ge__pzero,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xa,Y)) ).

% powr_ge_pzero
tff(fact_3554_powr__less__cancel,axiom,
    ! [Xa: real,A2: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2) ) ) ).

% powr_less_cancel
tff(fact_3555_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Xa,B2)) ) ) ).

% powr_less_mono
tff(fact_3556_powr__mono,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Xa,B2)) ) ) ).

% powr_mono
tff(fact_3557_add__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),Xa)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3558_log__add__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),Xa)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3559_minus__log__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),Y),aa(real,real,log(B2),Xa)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3560_graph__restrictD_I1_J,axiom,
    ! [B: $tType,A: $tType,K: A,V3: B,M: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3),graph(A,B,restrict_map(A,B,M,A5)))
     => member(A,K,A5) ) ).

% graph_restrictD(1)
tff(fact_3561_log__minus__eq__powr,axiom,
    ! [B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(B2),Xa)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3562_powr__less__mono2,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_3563_powr__mono2_H,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xa,A2)) ) ) ) ).

% powr_mono2'
tff(fact_3564_gr__one__powr,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xa,Y)) ) ) ).

% gr_one_powr
tff(fact_3565_powr__inj,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xa) = powr(real,A2,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% powr_inj
tff(fact_3566_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xa,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3567_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_3568_powr__le1,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_3569_powr__divide,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_3570_powr__mult,axiom,
    ! [Xa: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xa,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_3571_map__restrict__insert__none__simp,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),Xa: B,Sa: set(B)] :
      ( ( aa(B,option(A),M,Xa) = none(A) )
     => ( restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Xa),Sa))) = restrict_map(B,A,M,aa(set(B),set(B),uminus_uminus(set(B)),Sa)) ) ) ).

% map_restrict_insert_none_simp
tff(fact_3572_restrict__map__upd,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),S2: set(A),K: A,V3: B] : fun_upd(A,option(B),restrict_map(A,B,F3,S2),K,aa(B,option(B),some(B),V3)) = restrict_map(A,B,fun_upd(A,option(B),F3,K,aa(B,option(B),some(B),V3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),S2)) ).

% restrict_map_upd
tff(fact_3573_graph__restrictD_I2_J,axiom,
    ! [A: $tType,B: $tType,K: A,V3: B,M: fun(A,option(B)),A5: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3),graph(A,B,restrict_map(A,B,M,A5)))
     => ( aa(A,option(B),M,K) = aa(B,option(B),some(B),V3) ) ) ).

% graph_restrictD(2)
tff(fact_3574_log__base__change,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_3575_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na) ) ) ).

% powr_realpow
tff(fact_3576_log__of__power__eq,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_3577_less__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),M)) ) ) ).

% less_log_of_power
tff(fact_3578_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D5: set(A),Xa: A,Y: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D5),Xa,Y) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))),Xa,Y) ).

% fun_upd_restrict
tff(fact_3579_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa) ) ) ).

% powr_neg_one
tff(fact_3580_powr__mult__base,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,Xa,Y)) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_3581_log__mult,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_3582_le__log__of__power,axiom,
    ! [B2: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),M)) ) ) ).

% le_log_of_power
tff(fact_3583_log__base__pow,axiom,
    ! [A2: real,Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Na)),Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A2),Xa)),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_3584_log__nat__power,axiom,
    ! [Xa: real,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(B2),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B2),Xa)) ) ) ).

% log_nat_power
tff(fact_3585_log__divide,axiom,
    ! [A2: real,Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),Y)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,log(A2),Xa)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_3586_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),Xa: A] : restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) = fun_upd(A,option(B),F3,Xa,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_3587_map__upd__eq__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A] : fun_upd(A,option(B),M,Xa,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))) ).

% map_upd_eq_restrict
tff(fact_3588_log__of__power__less,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_3589_ln__powr__bound,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xa,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_3590_ln__powr__bound2,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xa),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_3591_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( aa(real,real,log(A2),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(real,real,ln_ln(real),A2))),aa(real,real,log(B2),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_3592_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A,A2: A] :
          powr(A,Xa,A2) = $ite(Xa = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xa)))) ) ).

% powr_def
tff(fact_3593_log__of__power__le,axiom,
    ! [M: nat,B2: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_3594_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),Xa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_3595_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D5: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D5)
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys),D5) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),D5),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_3596_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_je(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_3597_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_ej(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_3598_mod__type_Ointro,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Rep: fun(A,int),Abs: fun(int,A),Na: int] :
          ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Na))
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na)
           => ( ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
             => ( ( one_one(A) = aa(int,A,Abs,one_one(int)) )
               => ( ! [X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y3))),Na))
                 => ( ! [X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X),Y3) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y3))),Na))
                   => ( ! [X: A,Y3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y3) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(A,int,Rep,X)),aa(A,int,Rep,Y3))),Na))
                     => ( ! [X: A] : aa(A,A,uminus_uminus(A),X) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(A,int,Rep,X))),Na))
                       => numeral_mod_type(A,Na,Rep,Abs) ) ) ) ) ) ) ) ) ) ).

% mod_type.intro
tff(fact_3599_subset__mset_Omin__arg__le_I1_J,axiom,
    ! [A: $tType,Na: multiset(A),M: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),Na),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),Na))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),Na) = Na ) ) ).

% subset_mset.min_arg_le(1)
tff(fact_3600_subset__mset_Omin__arg__le_I2_J,axiom,
    ! [A: $tType,M: multiset(A),Na: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),Na))
    <=> ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),min(multiset(A),subseteq_mset(A)),M),Na) = M ) ) ).

% subset_mset.min_arg_le(2)
tff(fact_3601_Id__onI,axiom,
    ! [A: $tType,A2: A,A5: set(A)] :
      ( member(A,A2,A5)
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A2),id_on(A,A5)) ) ).

% Id_onI
tff(fact_3602_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_3603_Id__on__empty,axiom,
    ! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).

% Id_on_empty
tff(fact_3604_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,M: fun(A,option(B)),Ys: list(B),Y: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( map_upds(A,B,M,Xs,list_update(B,Ys,I,Y)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_3605_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_3606_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% zero_less_ceiling
tff(fact_3607_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_3608_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% one_le_ceiling
tff(fact_3609_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_3610_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ).

% one_less_ceiling
tff(fact_3611_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_3612_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) ) ) ).

% zero_le_ceiling
tff(fact_3613_mod__type_ORep__less__n,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),Xa: A] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(A,int,Rep,Xa)),Na) ) ) ).

% mod_type.Rep_less_n
tff(fact_3614_mod__type_ORep__le__n,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),Xa: A] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(A,int,Rep,Xa)),Na) ) ) ).

% mod_type.Rep_le_n
tff(fact_3615_mod__type_ORep__0,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( aa(A,int,Rep,zero_zero(A)) = zero_zero(int) ) ) ) ).

% mod_type.Rep_0
tff(fact_3616_mod__type_Ozero__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) ) ) ) ).

% mod_type.zero_def
tff(fact_3617_Id__on__iff,axiom,
    ! [A: $tType,Xa: A,Y: A,A5: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),id_on(A,A5))
    <=> ( ( Xa = Y )
        & member(A,Xa,A5) ) ) ).

% Id_on_iff
tff(fact_3618_Id__on__eqI,axiom,
    ! [A: $tType,A2: A,B2: A,A5: set(A)] :
      ( ( A2 = B2 )
     => ( member(A,A2,A5)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),id_on(A,A5)) ) ) ).

% Id_on_eqI
tff(fact_3619_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A5: set(A)] :
      ( member(product_prod(A,A),C3,id_on(A,A5))
     => ~ ! [X: A] :
            ( member(A,X,A5)
           => ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ) ).

% Id_onE
tff(fact_3620_mod__type_Osize0,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na) ) ) ).

% mod_type.size0
tff(fact_3621_mod__type_Osize1,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na) ) ) ).

% mod_type.size1
tff(fact_3622_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xa)) ) ) ).

% ceiling_mono
tff(fact_3623_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% ceiling_less_cancel
tff(fact_3624_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_3625_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_3626_Id__on__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( id_on(A,A5) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_jv(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_3627_mod__type__def,axiom,
    ! [A: $tType] :
      ( ( minus(A)
        & one(A)
        & plus(A)
        & times(A)
        & uminus(A)
        & zero(A) )
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A)] :
          ( numeral_mod_type(A,Na,Rep,Abs)
        <=> ( type_definition(A,int,Rep,Abs,set_or7035219750837199246ssThan(int,zero_zero(int),Na))
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Na)
            & ( zero_zero(A) = aa(int,A,Abs,zero_zero(int)) )
            & ( one_one(A) = aa(int,A,Abs,one_one(int)) )
            & ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),Y5) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(A,int,Rep,X5)),aa(A,int,Rep,Y5))),Na))
            & ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),times_times(A),X5),Y5) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(A,int,Rep,X5)),aa(A,int,Rep,Y5))),Na))
            & ! [X5: A,Y5: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),Y5) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(A,int,Rep,X5)),aa(A,int,Rep,Y5))),Na))
            & ! [X5: A] : aa(A,A,uminus_uminus(A),X5) = aa(int,A,Abs,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(A,int,Rep,X5))),Na)) ) ) ) ).

% mod_type_def
tff(fact_3628_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Na)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,Xa) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_3629_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_3630_comp__fun__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( finite4980608107308702382axioms(A,B,S2,F3)
       => finite673082921795544331dem_on(A,B,S2,F3) ) ) ).

% comp_fun_idem_on.intro
tff(fact_3631_comp__fun__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
    <=> ( finite4664212375090638736ute_on(A,B,S2,F3)
        & finite4980608107308702382axioms(A,B,S2,F3) ) ) ).

% comp_fun_idem_on_def
tff(fact_3632_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_3633_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_3634_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_3635_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% of_int_le_iff
tff(fact_3636_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% of_int_less_iff
tff(fact_3637_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_3638_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_3639_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_3640_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_3641_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_3642_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_3643_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_3644_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_3645_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_3646_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3647_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),Xa) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3648_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B2: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3649_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W2: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W2)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W2)),Xa) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3650_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_3651_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_3652_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z3)),Xa) ) ).

% ex_of_int_less
tff(fact_3653_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ).

% le_of_int_ceiling
tff(fact_3654_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_3655_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),A2) ) ) ).

% ceiling_le
tff(fact_3656_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% less_ceiling_iff
tff(fact_3657_real__of__int__div4,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))) ).

% real_of_int_div4
tff(fact_3658_comp__fun__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => finite4980608107308702382axioms(A,B,S2,F3) ) ).

% comp_fun_idem_on.axioms(2)
tff(fact_3659_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_3660_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_3661_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% of_int_leD
tff(fact_3662_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% of_int_lessD
tff(fact_3663_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_3664_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [X: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y4)),Xa)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int)))) )
             => ( Y4 = X ) ) ) ) ).

% floor_exists1
tff(fact_3665_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3666_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3667_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),Xa) ) ) ).

% of_nat_less_of_int_iff
tff(fact_3668_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real))) ) ).

% int_le_real_less
tff(fact_3669_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_3670_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))),one_one(A))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ) ).

% ceiling_correct
tff(fact_3671_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xa) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_3672_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archimedean_ceiling(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_3673_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,$o),T2: A] :
          ( aa(int,$o,P2,archimedean_ceiling(A,T2))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),T2),aa(int,A,ring_1_of_int(A),I3)) )
             => aa(int,$o,P2,I3) ) ) ) ).

% ceiling_split
tff(fact_3674_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_3675_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% le_ceiling_iff
tff(fact_3676_real__of__int__div2,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa)))) ).

% real_of_int_div2
tff(fact_3677_real__of__int__div3,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Na)),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),Xa)))),one_one(real)) ).

% real_of_int_div3
tff(fact_3678_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)) ) ) ).

% ceiling_divide_upper
tff(fact_3679_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3680_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_3681_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B2: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),Xa)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B2,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_3682_mod__ring_Ocases,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: int,Rep: fun(A,int),Abs: fun(int,A),Xa: A] :
          ( numeral_mod_ring(A,Na,Rep,Abs)
         => ~ ! [Z3: int] :
                ( ( Xa = aa(int,A,ring_1_of_int(A),Z3) )
               => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
                 => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),Na) ) ) ) ) ).

% mod_ring.cases
tff(fact_3683_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_3684_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_3685_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_3686_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_3687_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% zless_nat_conj
tff(fact_3688_nat__zminus__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_3689_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_3690_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ) ).

% zero_le_floor
tff(fact_3691_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_3692_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% zero_less_floor
tff(fact_3693_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_3694_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_3695_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% one_le_floor
tff(fact_3696_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_less_one
tff(fact_3697_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),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_3698_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xa))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_3699_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_3700_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_3701_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,A2))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B2)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_3702_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3703_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_3704_le__nat__floor,axiom,
    ! [Xa: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xa)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(int,nat,nat2,archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_3705_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_3706_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa) ) ).

% of_int_floor_le
tff(fact_3707_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_3708_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ).

% floor_less_cancel
tff(fact_3709_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_3710_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% floor_le_ceiling
tff(fact_3711_nat__mono,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ).

% nat_mono
tff(fact_3712_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z6) )
        <=> ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_3713_all__nat,axiom,
    ! [P2: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P2,X_13)
    <=> ! [X5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => aa(nat,$o,P2,aa(int,nat,nat2,X5)) ) ) ).

% all_nat
tff(fact_3714_ex__nat,axiom,
    ! [P2: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P2,X_13)
    <=> ? [X5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
          & aa(nat,$o,P2,aa(int,nat,nat2,X5)) ) ) ).

% ex_nat
tff(fact_3715_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% le_floor_iff
tff(fact_3716_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_3717_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% le_floor_add
tff(fact_3718_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3719_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_3720_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3721_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3722_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3723_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_mono_iff
tff(fact_3724_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_3725_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),Na)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_3726_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_3727_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_3728_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = aa(int,nat,nat2,Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_3729_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
         => ( aa(int,word(A),ring_1_of_int(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(int,nat,nat2,Xa)) ) ) ) ).

% word_of_int_nat
tff(fact_3730_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_3731_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_3732_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3733_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),archimedean_ceiling(A,Xa)),archim6421214686448440834_floor(A,Xa))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_3734_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_3735_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_3736_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_3737_split__nat,axiom,
    ! [P2: fun(nat,$o),I: int] :
      ( aa(nat,$o,P2,aa(int,nat,nat2,I))
    <=> ( ! [N5: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N5) )
           => aa(nat,$o,P2,N5) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P2,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_3738_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W2)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_3739_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) ) ) ) ).

% nat_add_distrib
tff(fact_3740_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ) ).

% le_nat_iff
tff(fact_3741_Suc__as__int,axiom,
    ! [X3: nat] : aa(nat,nat,suc,X3) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).

% Suc_as_int
tff(fact_3742_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) ) ) ).

% nat_mult_distrib
tff(fact_3743_nat__diff__distrib_H,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_3744_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z6)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z6),Z)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z6)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) ) ) ) ).

% nat_diff_distrib
tff(fact_3745_nat__div__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_3746_nat__div__distrib_H,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_3747_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),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_3748_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(int,nat,nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_3749_nat__mod__distrib,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_3750_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(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)))) ).

% div_abs_eq_div_nat
tff(fact_3751_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_3752_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P2: fun(int,$o),T2: A] :
          ( aa(int,$o,P2,archim6421214686448440834_floor(A,T2))
        <=> ! [I3: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I3)),T2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I3)),one_one(A))) )
             => aa(int,$o,P2,I3) ) ) ) ).

% floor_split
tff(fact_3753_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xa) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3754_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xa) = Z ) ) ) ) ).

% floor_unique
tff(fact_3755_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% less_floor_iff
tff(fact_3756_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_3757_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_3758_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_3759_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_3760_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),M)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_3761_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z6)) = 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),Z6))) ) ) ).

% nat_mult_distrib_neg
tff(fact_3762_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      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),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),B2),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),B2)) ).

% nat_abs_int_diff
tff(fact_3763_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_ord_lessThan(nat,aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_3764_diff__nat__eq__if,axiom,
    ! [Z: int,Z6: int] :
      aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z6)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z6),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,aa(int,fun(int,int),minus_minus(int),Z),Z6),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_3765_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),Q3)),P3) ) ) ).

% floor_divide_lower
tff(fact_3766_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_3767_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_3768_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_3769_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),P3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P3),Q3)))),one_one(A))),Q3)) ) ) ).

% floor_divide_upper
tff(fact_3770_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,Na)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_3771_card__UNION,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A5)
     => ( ! [X: set(A)] :
            ( member(set(A),X,A5)
           => aa(set(A),$o,finite_finite2(A),X) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_jw(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_jx(set(set(A)),fun(set(set(A)),$o),A5)))) ) ) ) ).

% card_UNION
tff(fact_3772_floor__real__def,axiom,
    ! [Xa: real] : archim6421214686448440834_floor(real,Xa) = the(int,aTP_Lamp_jy(real,fun(int,$o),Xa)) ).

% floor_real_def
tff(fact_3773_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_ka(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B5)) ) ).

% comp_fun_commute_product_fold
tff(fact_3774_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3775_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_3776_finite__Inter,axiom,
    ! [A: $tType,M7: set(set(A))] :
      ( ? [X3: set(A)] :
          ( member(set(A),X3,M7)
          & aa(set(A),$o,finite_finite2(A),X3) )
     => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),M7)) ) ).

% finite_Inter
tff(fact_3777_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3778_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3779_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3780_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3781_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_3782_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_3783_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% cSup_singleton
tff(fact_3784_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xa,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_3785_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,Xa)) = Xa ) ) ) ).

% cSup_atLeastAtMost
tff(fact_3786_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% cInf_singleton
tff(fact_3787_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,Xa)) = Xa ) ) ) ).

% cSup_atLeastLessThan
tff(fact_3788_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xa,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_3789_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Xa,Y)) = Xa ) ) ) ).

% Inf_atLeastAtMost
tff(fact_3790_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,Xa)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_3791_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,Xa)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_3792_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xa,Y)) = Xa ) ) ) ).

% Inf_atLeastLessThan
tff(fact_3793_finite__UN,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5)))
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,X5)) ) ) ) ).

% finite_UN
tff(fact_3794_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Inf_Inf(A),set_ord_atMost(A,Xa)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_3795_finite__Union,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A5)
     => ( ! [M10: set(A)] :
            ( member(set(A),M10,A5)
           => aa(set(A),$o,finite_finite2(A),M10) )
       => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) ) ) ).

% finite_Union
tff(fact_3796_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),C3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_kb(B,fun(A,B),C3)),A5)) = C3 ) ) ) ).

% cSUP_const
tff(fact_3797_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),C3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_kb(B,fun(A,B),C3)),A5)) = C3 ) ) ) ).

% cINF_const
tff(fact_3798_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [A4: A] :
            ( member(A,A4,A5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B5,A4)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))) ) ) ).

% finite_UN_I
tff(fact_3799_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( ? [X3: A] :
          ( member(A,X3,I5)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X3)) )
     => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5))) ) ).

% finite_INT
tff(fact_3800_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3801_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3802_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_3803_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_3804_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X4: set(A)] :
          ( member(A,Z,X4)
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_3805_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X4: set(A),A2: A] :
          ( ! [X: A] :
              ( member(A,X,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X) )
         => ( ! [Y3: A] :
                ( ! [X3: A] :
                    ( member(A,X3,X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_3806_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X4: set(A)] :
          ( member(A,Z,X4)
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_3807_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X4: set(A),A2: A] :
          ( ! [X: A] :
              ( member(A,X,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
         => ( ! [Y3: A] :
                ( ! [X3: A] :
                    ( member(A,X3,X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_3808_Some__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(A,option(A),some(A),aa(set(A),A,complete_Sup_Sup(A),A5)) = aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A5)) ) ) ) ).

% Some_Sup
tff(fact_3809_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_3810_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A2)) = A2 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_3811_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,inverse_inverse(A),B2) )
         => ( ( A2 != zero_zero(A) )
           => ( ( B2 != zero_zero(A) )
             => ( A2 = B2 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_3812_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( aa(A,A,inverse_inverse(A),A2) = zero_zero(A) )
         => ( A2 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_3813_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_3814_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A2)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A2)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_3815_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_kc(fun(B,B),fun(A,fun(B,B)),F3)) ).

% comp_fun_commute_const
tff(fact_3816_Some__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(B,option(B),some(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))) = aa(set(option(B)),option(B),complete_Sup_Sup(option(B)),aa(set(A),set(option(B)),image(A,option(B),aTP_Lamp_kd(fun(A,B),fun(A,option(B)),F3)),A5)) ) ) ) ).

% Some_SUP
tff(fact_3817_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xa))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3818_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_3819_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_3820_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3821_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3822_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3823_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3824_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_3825_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_3826_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_3827_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_3828_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A2)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_3829_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A2: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X) )
           => ( ! [Y3: A] :
                  ( ! [X3: A] :
                      ( member(A,X3,X4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),A2) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X4) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_3830_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X4)) ) ) ) ).

% cInf_greatest
tff(fact_3831_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Xa) ) ) ) ).

% cInf_le_finite
tff(fact_3832_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Z)
           => ? [X: A] :
                ( member(A,X,X4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_3833_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Xa: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => ( ! [X: A] :
                  ( member(A,X,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X4)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_3834_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A2: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
           => ( ! [Y3: A] :
                  ( ! [X3: A] :
                      ( member(A,X3,X4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y3) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Y3) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X4) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_3835_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X4)),Z) ) ) ) ).

% cSup_least
tff(fact_3836_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),X4)) ) ) ) ).

% le_cSup_finite
tff(fact_3837_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X4))
         => ( ( X4 != bot_bot(set(A)) )
           => ~ ! [X: A] :
                  ( member(A,X,X4)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X) ) ) ) ) ).

% less_cSupE
tff(fact_3838_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X4))
           => ? [X: A] :
                ( member(A,X,X4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X) ) ) ) ) ).

% less_cSupD
tff(fact_3839_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P2: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_hr(fun(A,$o),fun(A,fun(set(A),set(A))),P2)) ).

% comp_fun_commute_filter_fold
tff(fact_3840_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Xa: A,A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => ( ! [X: A] :
                  ( member(A,X,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A2) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X4)),A2) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_3841_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ).

% card_Union_le_sum_card
tff(fact_3842_finite__UNION__then__finite,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A5: set(B),A2: B] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),A5)))
     => ( member(B,A2,A5)
       => aa(set(A),$o,finite_finite2(A),aa(B,set(A),B5,A2)) ) ) ).

% finite_UNION_then_finite
tff(fact_3843_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),G3: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_jn(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F3),G3)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_3844_comp__fun__commute_Ofoldl__f__commute,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),A2: A,B2: B,Xs: list(A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(B,B,aa(A,fun(B,B),F3,A2),aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_ke(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),B2),Xs)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_ke(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),aa(B,B,aa(A,fun(B,B),F3,A2),B2)),Xs) ) ) ).

% comp_fun_commute.foldl_f_commute
tff(fact_3845_finite__UnionD,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5))
     => aa(set(set(A)),$o,finite_finite2(set(A)),A5) ) ).

% finite_UnionD
tff(fact_3846_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_3847_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_3848_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3849_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_3850_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% inverse_le_1_iff
tff(fact_3851_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_3852_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_3853_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),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),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_3854_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),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),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_3855_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_3856_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),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),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3857_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_3858_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),M: B,F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F3,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))) ) ) ) ).

% cINF_greatest
tff(fact_3859_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X4))
            <=> ! [X5: A] :
                  ( member(A,X5,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X5) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_3860_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A5: set(A),F3: fun(A,B),M7: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),M7) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))),M7) ) ) ) ).

% cSUP_least
tff(fact_3861_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X4)),A2)
            <=> ! [X5: A] :
                  ( member(A,X5,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),A2) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_3862_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S2))),A2) ) ) ) ).

% cInf_abs_ge
tff(fact_3863_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S2))),A2) ) ) ) ).

% cSup_abs_le
tff(fact_3864_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_3865_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Max_Sup
tff(fact_3866_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = aa(set(A),A,lattic643756798349783984er_Max(A),X4) ) ) ) ) ).

% cSup_eq_Max
tff(fact_3867_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X: set(A)] :
          ( member(set(A),X,U2)
         => aa(set(A),$o,finite_finite2(A),X) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ) ).

% card_Union_le_sum_card_weak
tff(fact_3868_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Min_Inf
tff(fact_3869_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = aa(set(A),A,lattic643756798350308766er_Min(A),X4) ) ) ) ) ).

% cInf_eq_Min
tff(fact_3870_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_3871_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% inverse_less_1_iff
tff(fact_3872_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_3873_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_3874_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_3875_comp__fun__commute_Ofoldr__conv__foldl,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),Xs: list(A),A2: B] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( foldr(A,B,F3,Xs,A2) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,aTP_Lamp_ke(fun(A,fun(B,B)),fun(B,fun(A,B)),F3)),A2),Xs) ) ) ).

% comp_fun_commute.foldr_conv_foldl
tff(fact_3876_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3877_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)))),Xa) ) ) ).

% reals_Archimedean
tff(fact_3878_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Inf_Inf(A),S2)),L))),E3) ) ) ) ).

% cInf_asclose
tff(fact_3879_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E3: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,S2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),L))),E3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(set(A),A,complete_Sup_Sup(A),S2)),L))),E3) ) ) ) ).

% cSup_asclose
tff(fact_3880_forall__pos__mono__1,axiom,
    ! [P2: fun(real,$o),E3: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P2,D3)
           => aa(real,$o,P2,E2) ) )
     => ( ! [N: nat] : aa(real,$o,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => aa(real,$o,P2,E3) ) ) ) ).

% forall_pos_mono_1
tff(fact_3881_forall__pos__mono,axiom,
    ! [P2: fun(real,$o),E3: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D3),E2)
         => ( aa(real,$o,P2,D3)
           => aa(real,$o,P2,E2) ) )
     => ( ! [N: nat] :
            ( ( N != zero_zero(nat) )
           => aa(real,$o,P2,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => aa(real,$o,P2,E3) ) ) ) ).

% forall_pos_mono
tff(fact_3882_real__arch__inverse,axiom,
    ! [E3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))),E3) ) ) ).

% real_arch_inverse
tff(fact_3883_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),S2)) = $ite(S2 = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,complete_Sup_Sup(A),S2))) ) ) ) ).

% Sup_insert_finite
tff(fact_3884_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_inverse
tff(fact_3885_comp__fun__commute__fold__insert,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),A5: set(A),Xa: A,Z: B] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(set(A),$o,finite_finite2(A),A5)
       => ( ~ member(A,Xa,A5)
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(B,B,aa(A,fun(B,B),F3,Xa),finite_fold(A,B,F3,Z,A5)) ) ) ) ) ).

% comp_fun_commute_fold_insert
tff(fact_3886_the__elem__def,axiom,
    ! [A: $tType,X4: set(A)] : the_elem(A,X4) = the(A,aTP_Lamp_kf(set(A),fun(A,$o),X4)) ).

% the_elem_def
tff(fact_3887_finite__subset__Union,axiom,
    ! [A: $tType,A5: set(A),B11: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11))
       => ~ ! [F9: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F9)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F9),B11)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F9)) ) ) ) ) ).

% finite_subset_Union
tff(fact_3888_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N))),Xa) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3889_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3890_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))),image(A,set(product_prod(A,A)),aTP_Lamp_kg(A,set(product_prod(A,A)))),A5)) ).

% Id_on_def
tff(fact_3891_log__inverse,axiom,
    ! [A2: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_3892_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_kh(fun(A,set(B)),fun(A,nat),A5)),I5)) ) ).

% card_UN_le
tff(fact_3893_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)),image(A,set(A),aTP_Lamp_ki(A,set(A))),A5)) = A5 ).

% UN_singleton
tff(fact_3894_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B5: fun(B,set(A)),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kj(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B5)),C4)) = $ite(C4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4)))) ).

% UN_simps(1)
tff(fact_3895_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_3896_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A5)) = bot_bot(A) )
        <=> ! [X5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X5)
             => ? [Xa3: B] :
                  ( member(B,Xa3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,Xa3)),X5) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_3897_subset__mset_OcINF__const,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C3: multiset(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),aTP_Lamp_kk(multiset(B),fun(A,multiset(B)),C3)),A5)) = C3 ) ) ).

% subset_mset.cINF_const
tff(fact_3898_subset__mset_OcSUP__const,axiom,
    ! [A: $tType,B: $tType,A5: set(A),C3: multiset(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),aTP_Lamp_kk(multiset(B),fun(A,multiset(B)),C3)),A5)) = C3 ) ) ).

% subset_mset.cSUP_const
tff(fact_3899_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A5) )
        <=> ! [X5: A] :
              ( member(A,X5,A5)
             => ( X5 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_3900_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = bot_bot(A) )
        <=> ! [X5: A] :
              ( member(A,X5,A5)
             => ( X5 = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_3901_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_3902_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A5) = bot_bot(A) )
        <=> ! [X5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X5)
             => ? [Xa3: A] :
                  ( member(A,Xa3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X5) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_3903_Sup__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% Sup_empty
tff(fact_3904_ccSup__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% ccSup_empty
tff(fact_3905_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_kl(B,A)),A5)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_3906_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),image(B,A,aTP_Lamp_km(B,A)),A5)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_3907_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B5),A5)) = bot_bot(A) )
        <=> ! [X5: B] :
              ( member(B,X5,A5)
             => ( aa(B,A,B5,X5) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_3908_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: fun(B,A),A5: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,B5),A5)) )
        <=> ! [X5: B] :
              ( member(B,X5,A5)
             => ( aa(B,A,B5,X5) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_3909_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_kn(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% ccSUP_const
tff(fact_3910_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,aTP_Lamp_ko(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% SUP_const
tff(fact_3911_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_kn(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% ccINF_const
tff(fact_3912_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: B] :
          ( ( A5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aTP_Lamp_ko(B,fun(A,B),F3)),A5)) = F3 ) ) ) ).

% INF_const
tff(fact_3913_UN__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A5: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_kp(set(A),fun(B,set(A)),C3)),A5)) = $ite(A5 = bot_bot(set(B)),bot_bot(set(A)),C3) ).

% UN_constant
tff(fact_3914_Inf__nat__def1,axiom,
    ! [K4: set(nat)] :
      ( ( K4 != bot_bot(set(nat)) )
     => member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K4),K4) ) ).

% Inf_nat_def1
tff(fact_3915_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X3),Xa2)
    <=> 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),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_3916_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(set(product_prod(A,B))),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o)))),S2)),X3),Xa2)
    <=> 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),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_3917_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_kq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S2)),X3),Xa2)
    <=> 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),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))),image(C,set(product_prod(A,B)),R2),S2))) ) ).

% INF_INT_eq2
tff(fact_3918_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S2: set(C),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image(C,fun(A,fun(B,$o)),aTP_Lamp_kq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S2)),X3),Xa2)
    <=> 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),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))),image(C,set(product_prod(A,B)),R2),S2))) ) ).

% SUP_UN_eq2
tff(fact_3919_subset__mset_OcINF__greatest,axiom,
    ! [B: $tType,A: $tType,A5: set(A),M: multiset(B),F3: fun(A,multiset(B))] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( member(A,X,A5)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(A,multiset(B),F3,X)) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),M),aa(set(multiset(B)),multiset(B),complete_Inf_Inf(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),F3),A5))) ) ) ).

% subset_mset.cINF_greatest
tff(fact_3920_subset__mset_OcSUP__least,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(A,multiset(B)),M7: multiset(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ! [X: A] :
            ( member(A,X,A5)
           => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(A,multiset(B),F3,X)),M7) )
       => aa(multiset(B),$o,aa(multiset(B),fun(multiset(B),$o),subseteq_mset(B),aa(set(multiset(B)),multiset(B),complete_Sup_Sup(multiset(B)),aa(set(A),set(multiset(B)),image(A,multiset(B),F3),A5))),M7) ) ) ).

% subset_mset.cSUP_least
tff(fact_3921_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F4: fun(A,filter(B))] :
      ( ! [X7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5)
         => ( aa(set(A),$o,finite_finite2(A),X7)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),X7)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),B5)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_3922_Sup__nat__def,axiom,
    ! [X4: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X4) = $ite(X4 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X4)) ).

% Sup_nat_def
tff(fact_3923_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xa) = the(real,aTP_Lamp_kr(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_3924_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V3: A] :
          ( member(A,U,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% Sup_upper2
tff(fact_3925_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B2)
        <=> ! [X5: A] :
              ( member(A,X5,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),B2) ) ) ) ).

% Sup_le_iff
tff(fact_3926_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A5: set(A)] :
          ( member(A,Xa,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Sup_upper
tff(fact_3927_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),Z) ) ) ).

% Sup_least
tff(fact_3928_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( ! [A4: A] :
              ( member(A,A4,A5)
             => ? [X3: A] :
                  ( member(A,X3,B5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),X3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_mono
tff(fact_3929_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xa: A] :
          ( ! [Y3: A] :
              ( member(A,Y3,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
         => ( ! [Y3: A] :
                ( ! [Z4: A] :
                    ( member(A,Z4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z4),Y3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y3) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A5) = Xa ) ) ) ) ).

% Sup_eqI
tff(fact_3930_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2))
        <=> ? [X5: A] :
              ( member(A,X5,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),X5) ) ) ) ).

% less_Sup_iff
tff(fact_3931_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Z: A] :
          ( ! [X: A] :
              ( member(A,X,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A5)) ) ) ).

% Inf_greatest
tff(fact_3932_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A5))
        <=> ! [X5: A] :
              ( member(A,X5,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),X5) ) ) ) ).

% le_Inf_iff
tff(fact_3933_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A5: set(A),V3: A] :
          ( member(A,U,A5)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),V3) ) ) ) ).

% Inf_lower2
tff(fact_3934_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A5: set(A)] :
          ( member(A,Xa,A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xa) ) ) ).

% Inf_lower
tff(fact_3935_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( ! [B4: A] :
              ( member(A,B4,B5)
             => ? [X3: A] :
                  ( member(A,X3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B4) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_mono
tff(fact_3936_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),Xa: A] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),I2) )
         => ( ! [Y3: A] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),I4) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Xa) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A5) = Xa ) ) ) ) ).

% Inf_eqI
tff(fact_3937_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2)
        <=> ? [X5: A] :
              ( member(A,X5,S2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),A2) ) ) ) ).

% Inf_less_iff
tff(fact_3938_Union__empty__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) = bot_bot(set(A)) )
    <=> ! [X5: set(A)] :
          ( member(set(A),X5,A5)
         => ( X5 = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_3939_empty__Union__conv,axiom,
    ! [A: $tType,A5: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5) )
    <=> ! [X5: set(A)] :
          ( member(set(A),X5,A5)
         => ( X5 = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_3940_Union__subsetI,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(set(A))] :
      ( ! [X: set(A)] :
          ( member(set(A),X,A5)
         => ? [Y4: set(A)] :
              ( member(set(A),Y4,B5)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Y4) ) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) ) ).

% Union_subsetI
tff(fact_3941_Union__upper,axiom,
    ! [A: $tType,B5: set(A),A5: set(set(A))] :
      ( member(set(A),B5,A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)) ) ).

% Union_upper
tff(fact_3942_Union__least,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(A)] :
      ( ! [X7: set(A)] :
          ( member(set(A),X7,A5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),C4) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),C4) ) ).

% Union_least
tff(fact_3943_Inter__greatest,axiom,
    ! [A: $tType,A5: set(set(A)),C4: set(A)] :
      ( ! [X7: set(A)] :
          ( member(set(A),X7,A5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),X7) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)) ) ).

% Inter_greatest
tff(fact_3944_Inter__lower,axiom,
    ! [A: $tType,B5: set(A),A5: set(set(A))] :
      ( member(set(A),B5,A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),B5) ) ).

% Inter_lower
tff(fact_3945_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,A5: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A5))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
             => ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X5) ) ) ) ) ).

% le_Sup_iff
tff(fact_3946_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),Xa)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
             => ? [X5: A] :
                  ( member(A,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Y5) ) ) ) ) ).

% Inf_le_iff
tff(fact_3947_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ? [X3: B] :
                  ( member(B,X3,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,I2)),aa(B,C,G3,X3)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X3: A] :
                    ( member(A,X3,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,J2)),aa(A,C,F3,X3)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F3),A5)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G3),B5)) ) ) ) ) ).

% SUP_eq
tff(fact_3948_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V: A] :
              ( member(A,V,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ) ).

% less_eq_Sup
tff(fact_3949_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),G3: fun(B,C),F3: fun(A,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => ? [X3: B] :
                  ( member(B,X3,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G3,X3)),aa(A,C,F3,I2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B5)
               => ? [X3: A] :
                    ( member(A,X3,A5)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,X3)),aa(B,C,G3,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image(A,C,F3),A5)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,G3),B5)) ) ) ) ) ).

% INF_eq
tff(fact_3950_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5)) ) ) ).

% Sup_subset_mono
tff(fact_3951_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),U: A] :
          ( ! [V: A] :
              ( member(A,V,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V),U) )
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),U) ) ) ) ).

% Inf_less_eq
tff(fact_3952_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B5: set(A),A5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5)) ) ) ).

% Inf_superset_mono
tff(fact_3953_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F3: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F3,I2) = Xa ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),I5)) = Xa ) ) ) ) ).

% SUP_eq_const
tff(fact_3954_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F3: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F3,I2) = Xa ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),I5)) = Xa ) ) ) ) ).

% INF_eq_const
tff(fact_3955_Union__mono,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A5),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) ) ).

% Union_mono
tff(fact_3956_Union__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).

% Union_empty
tff(fact_3957_Inter__anti__mono,axiom,
    ! [A: $tType,B5: set(set(A)),A5: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B5),A5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5)) ) ).

% Inter_anti_mono
tff(fact_3958_Inter__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(A)] :
      ( ! [X7: set(A)] :
          ( member(set(A),X7,A5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X7),B5) )
     => ( ( A5 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),B5) ) ) ).

% Inter_subset
tff(fact_3959_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),U: B,F3: fun(A,B)] :
          ( member(A,I,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))) ) ) ) ).

% SUP_upper2
tff(fact_3960_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A5))),U)
        <=> ! [X5: B] :
              ( member(B,X5,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F3,X5)),U) ) ) ) ).

% SUP_le_iff
tff(fact_3961_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))) ) ) ).

% SUP_upper
tff(fact_3962_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G3),A5))) ) ) ).

% SUP_mono'
tff(fact_3963_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))),U) ) ) ).

% SUP_least
tff(fact_3964_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A5: set(A),B5: set(B),F3: fun(A,C),G3: fun(B,C)] :
          ( ! [N: A] :
              ( member(A,N,A5)
             => ? [X3: B] :
                  ( member(B,X3,B5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F3,N)),aa(B,C,G3,X3)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image(A,C,F3),A5))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image(B,C,G3),B5))) ) ) ).

% SUP_mono
tff(fact_3965_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B),Xa: B] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),Xa) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I4)),Y3) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),Y3) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5)) = Xa ) ) ) ) ).

% SUP_eqI
tff(fact_3966_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A5: set(B),Y: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A5))),Y)
         => ( member(B,I,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,I)),Y) ) ) ) ).

% SUP_lessD
tff(fact_3967_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A5)))
        <=> ? [X5: B] :
              ( member(B,X5,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(B,A,F3,X5)) ) ) ) ).

% less_SUP_iff
tff(fact_3968_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),U: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F3,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))) ) ) ).

% INF_greatest
tff(fact_3969_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A5)))
        <=> ! [X5: B] :
              ( member(B,X5,A5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F3,X5)) ) ) ) ).

% le_INF_iff
tff(fact_3970_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B),U: B] :
          ( member(A,I,A5)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))),U) ) ) ) ).

% INF_lower2
tff(fact_3971_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F3: fun(A,B),G3: fun(A,B),A5: set(A)] :
          ( ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G3),A5))) ) ) ).

% INF_mono'
tff(fact_3972_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A5: set(A),F3: fun(A,B)] :
          ( member(A,I,A5)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(A,B,F3,I)) ) ) ).

% INF_lower
tff(fact_3973_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B5: set(A),A5: set(B),F3: fun(B,C),G3: fun(A,C)] :
          ( ! [M3: A] :
              ( member(A,M3,B5)
             => ? [X3: B] :
                  ( member(B,X3,A5)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F3,X3)),aa(A,C,G3,M3)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image(B,C,F3),A5))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image(A,C,G3),B5))) ) ) ).

% INF_mono
tff(fact_3974_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),Xa: B,F3: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),aa(A,B,F3,I2)) )
         => ( ! [Y3: B] :
                ( ! [I4: A] :
                    ( member(A,I4,A5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),aa(A,B,F3,I4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y3),Xa) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5)) = Xa ) ) ) ) ).

% INF_eqI
tff(fact_3975_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F3: fun(B,A),A5: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A5)))
         => ( member(B,I,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),aa(B,A,F3,I)) ) ) ) ).

% less_INF_D
tff(fact_3976_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A5))),A2)
        <=> ? [X5: B] :
              ( member(B,X5,A5)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X5)),A2) ) ) ) ).

% INF_less_iff
tff(fact_3977_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)),image(B,set(A),aTP_Lamp_ks(B,set(A))),A5)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_3978_UN__empty,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_3979_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B5: 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)),image(B,set(A),B5),A5)) )
    <=> ! [X5: B] :
          ( member(B,X5,A5)
         => ( aa(B,set(A),B5,X5) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_3980_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A)),A5: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),A5)) = bot_bot(set(A)) )
    <=> ! [X5: B] :
          ( member(B,X5,A5)
         => ( aa(B,set(A),B5,X5) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_3981_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I5: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),I5))),B5)
    <=> ! [X5: B] :
          ( member(B,X5,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A5,X5)),B5) ) ) ).

% UN_subset_iff
tff(fact_3982_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A5: set(A),B5: fun(A,set(B))] :
      ( member(A,A2,A5)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B5,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))) ) ).

% UN_upper
tff(fact_3983_UN__least,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(A,set(B)),C4: set(B)] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B5,X)),C4) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))),C4) ) ).

% UN_least
tff(fact_3984_UN__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,X)),aa(A,set(B),G3,X)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F3),A5))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G3),B5))) ) ) ).

% UN_mono
tff(fact_3985_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A5: fun(B,set(A)),I5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),I5)))
    <=> ! [X5: B] :
          ( member(B,X5,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(B,set(A),A5,X5)) ) ) ).

% INT_subset_iff
tff(fact_3986_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(A),F3: fun(A,set(B)),G3: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,X)),aa(A,set(B),G3,X)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),F3),B5))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),G3),A5))) ) ) ).

% INT_anti_mono
tff(fact_3987_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A5: set(A),C4: set(B),B5: fun(A,set(B))] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C4),aa(A,set(B),B5,X)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C4),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))) ) ).

% INT_greatest
tff(fact_3988_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A5: set(A),B5: fun(A,set(B))] :
      ( member(A,A2,A5)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))),aa(A,set(B),B5,A2)) ) ).

% INT_lower
tff(fact_3989_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,F3: fun(B,A),A5: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A5)))
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
             => ? [X5: B] :
                  ( member(B,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),aa(B,A,F3,X5)) ) ) ) ) ).

% le_SUP_iff
tff(fact_3990_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A5))),Xa)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y5)
             => ? [X5: B] :
                  ( member(B,X5,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F3,X5)),Y5) ) ) ) ) ).

% INF_le_iff
tff(fact_3991_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C3: B,F3: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),aa(A,B,F3,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),I5)) = C3 )
            <=> ! [X5: A] :
                  ( member(A,X5,I5)
                 => ( aa(A,B,F3,X5) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_3992_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F3: fun(A,B),C3: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,I2)),C3) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),I5)) = C3 )
            <=> ! [X5: A] :
                  ( member(A,X5,I5)
                 => ( aa(A,B,F3,X5) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_3993_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Sup_Sup(A),A5)) ) ) ).

% Inf_le_Sup
tff(fact_3994_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,G3),B5))) ) ) ) ).

% SUP_subset_mono
tff(fact_3995_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( ! [X: A] :
                ( member(A,X,B5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,G3),B5))) ) ) ) ).

% INF_superset_mono
tff(fact_3996_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A5: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,aTP_Lamp_kt(A,fun(B,A),C3)),A5)) = $ite(A5 = bot_bot(set(B)),bot_bot(A),C3) ) ).

% SUP_constant
tff(fact_3997_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_3998_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A2: A,B5: fun(B,set(A)),C4: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4))) = $ite(C4 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kj(A,fun(fun(B,set(A)),fun(B,set(A))),A2),B5)),C4))) ).

% UN_extend_simps(1)
tff(fact_3999_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ( A5 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))) ) ) ).

% INF_le_SUP
tff(fact_4000_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_ku(fun(B,A),fun(B,set(A)),F3)),A5)) = aa(set(B),set(A),image(B,A,F3),A5) ).

% UNION_singleton_eq_range
tff(fact_4001_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4))) = $ite(C4 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4))) ).

% INT_extend_simps(4)
tff(fact_4002_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% ccpo_Sup_singleton
tff(fact_4003_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa2: product_prod(A,B)] : product_rec_prod(A,B,C,X3,Xa2) = the(C,product_rec_set_prod(A,B,C,X3,Xa2)) ).

% old.rec_prod_def
tff(fact_4004_floor__rat__def,axiom,
    ! [Xa: rat] : archim6421214686448440834_floor(rat,Xa) = the(int,aTP_Lamp_kw(rat,fun(int,$o),Xa)) ).

% floor_rat_def
tff(fact_4005_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),G3: fun(C,set(B)),Xa: C,X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),aa(C,set(B),G3,Xa)))),X4)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G3,Xa)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_kx(fun(B,set(A)),fun(set(A),fun(B,$o)),F3),X4))) ) ).

% UN_image_subset
tff(fact_4006_Inf__filter__not__bot,axiom,
    ! [A: $tType,B5: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X7),B5)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X7)
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X7) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4007_less__eq__rat__def,axiom,
    ! [Xa: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xa),Y)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xa),Y)
        | ( Xa = Y ) ) ) ).

% less_eq_rat_def
tff(fact_4008_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,sgn_sgn(rat),A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_4009_abs__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,abs_abs(rat),A2) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A2),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A2),A2) ).

% abs_rat_def
tff(fact_4010_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
     => ~ ! [S3: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S3)
           => ! [T7: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T7)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T7) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4011_conj__subset__def,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o),Q2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(fun(A,$o),set(A),collect(A),Q2)) ) ) ).

% conj_subset_def
tff(fact_4012_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),sin(real,Xa)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ky(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_4013_normalize__negative,axiom,
    ! [Q3: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q3),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_4014_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B5: set(A),Xa: B,A5: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( 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))),image(A,set(product_prod(B,A)),aTP_Lamp_kz(B,fun(A,set(product_prod(B,A))),Xa)),B5))),A5) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_jz(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xa),A5,B5) ) ) ).

% fold_union_pair
tff(fact_4015_Cauchy__iff2,axiom,
    ! [X4: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X4)
    <=> ! [J3: nat] :
        ? [M11: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
         => ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,X4,M5)),aa(nat,real,X4,N5)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_4016_Un__iff,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
    <=> ( member(A,C3,A5)
        | member(A,C3,B5) ) ) ).

% Un_iff
tff(fact_4017_UnCI,axiom,
    ! [A: $tType,C3: A,B5: set(A),A5: set(A)] :
      ( ( ~ member(A,C3,B5)
       => member(A,C3,A5) )
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ) ).

% UnCI
tff(fact_4018_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.bounded_iff
tff(fact_4019_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% le_sup_iff
tff(fact_4020_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xa) = Xa ) ).

% sup_bot_left
tff(fact_4021_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% sup_bot_right
tff(fact_4022_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_4023_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = bot_bot(A) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_4024_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = bot_bot(A) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_4025_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A2) = A2 ) ).

% sup_bot.left_neutral
tff(fact_4026_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
        <=> ( ( A2 = bot_bot(A) )
            & ( B2 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_4027_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),bot_bot(A)) = A2 ) ).

% sup_bot.right_neutral
tff(fact_4028_Un__empty,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ( ( A5 = bot_bot(set(A)) )
        & ( B5 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_4029_finite__Un,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5))
    <=> ( aa(set(A),$o,finite_finite2(A),F4)
        & aa(set(A),$o,finite_finite2(A),G5) ) ) ).

% finite_Un
tff(fact_4030_Un__subset__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),C4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4) ) ) ).

% Un_subset_iff
tff(fact_4031_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_4032_Un__insert__left,axiom,
    ! [A: $tType,A2: A,B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% Un_insert_left
tff(fact_4033_Un__insert__right,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ).

% Un_insert_right
tff(fact_4034_Un__Diff__cancel,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) ).

% Un_Diff_cancel
tff(fact_4035_Un__Diff__cancel2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A5) ).

% Un_Diff_cancel2
tff(fact_4036_Compl__Diff__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),B5) ).

% Compl_Diff_eq
tff(fact_4037_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_la(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4)))) ).

% UN_simps(3)
tff(fact_4038_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lb(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5)) ).

% UN_simps(2)
tff(fact_4039_image__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B5: set(B)] : aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),aa(set(B),set(A),image(B,A,F3),B5)) ).

% image_Un
tff(fact_4040_Un__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B5: set(A),D5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),D5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),D5)) ) ) ).

% Un_mono
tff(fact_4041_Un__least,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),C4) ) ) ).

% Un_least
tff(fact_4042_Un__upper1,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ).

% Un_upper1
tff(fact_4043_Un__upper2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ).

% Un_upper2
tff(fact_4044_Un__absorb1,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = B5 ) ) ).

% Un_absorb1
tff(fact_4045_Un__absorb2,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = A5 ) ) ).

% Un_absorb2
tff(fact_4046_subset__UnE,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
     => ~ ! [A11: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A11),A5)
           => ! [B12: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B12),B5)
               => ( C4 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A11),B12) ) ) ) ) ).

% subset_UnE
tff(fact_4047_subset__Un__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = B5 ) ) ).

% subset_Un_eq
tff(fact_4048_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI2
tff(fact_4049_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.coboundedI1
tff(fact_4050_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb_iff2
tff(fact_4051_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb_iff1
tff(fact_4052_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded2
tff(fact_4053_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ).

% sup.cobounded1
tff(fact_4054_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.order_iff
tff(fact_4055_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2) ) ) ) ).

% sup.boundedI
tff(fact_4056_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2) ) ) ) ).

% sup.boundedE
tff(fact_4057_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_4058_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Xa ) ) ) ).

% sup_absorb1
tff(fact_4059_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_4060_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_4061_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F3,X),Y3))
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),aa(A,A,aa(A,fun(A,A),F3,X),Y3))
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),X)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)),X) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F3,Xa),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_4062_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% sup.orderI
tff(fact_4063_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_4064_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_4065_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),Xa) ) ) ) ).

% sup_least
tff(fact_4066_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D2)) ) ) ) ).

% sup_mono
tff(fact_4067_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ) ).

% sup.mono
tff(fact_4068_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI2
tff(fact_4069_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% le_supI1
tff(fact_4070_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge2
tff(fact_4071_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% sup_ge1
tff(fact_4072_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa) ) ) ) ).

% le_supI
tff(fact_4073_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),Xa)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa) ) ) ) ).

% le_supE
tff(fact_4074_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(3)
tff(fact_4075_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)) ) ).

% inf_sup_ord(4)
tff(fact_4076_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% boolean_algebra.disj_zero_right
tff(fact_4077_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI1
tff(fact_4078_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% less_supI2
tff(fact_4079_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_4080_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_4081_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C3)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2) ) ) ) ).

% sup.strict_boundedE
tff(fact_4082_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_4083_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI1
tff(fact_4084_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)) ) ) ).

% sup.strict_coboundedI2
tff(fact_4085_foldl__un__empty__eq,axiom,
    ! [A: $tType,I: set(A),Ww: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),I),Ww) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I),aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),Ww)) ).

% foldl_un_empty_eq
tff(fact_4086_Un__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B5) = B5 ).

% Un_empty_left
tff(fact_4087_Un__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),bot_bot(set(A))) = A5 ).

% Un_empty_right
tff(fact_4088_Un__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4)) ).

% Un_Diff
tff(fact_4089_set__diff__diff__left,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% set_diff_diff_left
tff(fact_4090_Un__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_lc(set(A),fun(set(A),fun(A,$o)),A5),B5)) ).

% Un_def
tff(fact_4091_Collect__disj__eq,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ac(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P2)),aa(fun(A,$o),set(A),collect(A),Q2)) ).

% Collect_disj_eq
tff(fact_4092_Un__left__commute,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4)) ).

% Un_left_commute
tff(fact_4093_Un__left__absorb,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) ).

% Un_left_absorb
tff(fact_4094_Un__commute,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A5) ).

% Un_commute
tff(fact_4095_Un__absorb,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),A5) = A5 ).

% Un_absorb
tff(fact_4096_Un__assoc,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ).

% Un_assoc
tff(fact_4097_ball__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
         => aa(A,$o,P2,X5) )
    <=> ( ! [X5: A] :
            ( member(A,X5,A5)
           => aa(A,$o,P2,X5) )
        & ! [X5: A] :
            ( member(A,X5,B5)
           => aa(A,$o,P2,X5) ) ) ) ).

% ball_Un
tff(fact_4098_bex__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),P2: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
          & aa(A,$o,P2,X5) )
    <=> ( ? [X5: A] :
            ( member(A,X5,A5)
            & aa(A,$o,P2,X5) )
        | ? [X5: A] :
            ( member(A,X5,B5)
            & aa(A,$o,P2,X5) ) ) ) ).

% bex_Un
tff(fact_4099_UnI2,axiom,
    ! [A: $tType,C3: A,B5: set(A),A5: set(A)] :
      ( member(A,C3,B5)
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ) ).

% UnI2
tff(fact_4100_UnI1,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,A5)
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ) ).

% UnI1
tff(fact_4101_UnE,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
     => ( ~ member(A,C3,A5)
       => member(A,C3,B5) ) ) ).

% UnE
tff(fact_4102_infinite__Un,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3))
    <=> ( ~ aa(set(A),$o,finite_finite2(A),S2)
        | ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_Un
tff(fact_4103_Un__infinite,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) ) ).

% Un_infinite
tff(fact_4104_finite__UnI,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),F4)
     => ( aa(set(A),$o,finite_finite2(A),G5)
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F4),G5)) ) ) ).

% finite_UnI
tff(fact_4105_insert__def,axiom,
    ! [A: $tType,A2: A,B5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ah(A,fun(A,$o),A2))),B5) ).

% insert_def
tff(fact_4106_Collect__imp__eq,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ld(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2))),aa(fun(A,$o),set(A),collect(A),Q2)) ).

% Collect_imp_eq
tff(fact_4107_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_4108_sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),one_one(real)) ).

% sin_le_one
tff(fact_4109_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),aa(real,real,abs_abs(real),Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_4110_singleton__Un__iff,axiom,
    ! [A: $tType,Xa: A,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_4111_Un__singleton__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),Xa: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
    <=> ( ( ( A5 = bot_bot(set(A)) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B5 = bot_bot(set(A)) ) )
        | ( ( A5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) )
          & ( B5 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_4112_insert__is__Un,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))),A5) ).

% insert_is_Un
tff(fact_4113_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_4114_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_4115_Diff__partition,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = B5 ) ) ).

% Diff_partition
tff(fact_4116_Diff__subset__conv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C4)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) ) ).

% Diff_subset_conv
tff(fact_4117_foldl__set,axiom,
    ! [A: $tType,L: list(set(A))] : aa(list(set(A)),set(A),aa(set(A),fun(list(set(A)),set(A)),foldl(set(A),set(A),sup_sup(set(A))),bot_bot(set(A))),L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_le(list(set(A)),fun(set(A),$o),L))) ).

% foldl_set
tff(fact_4118_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] :
                  ( member(A,X,A5)
                 => ( member(A,Y3,A5)
                   => member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y3),A5) ) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ) ).

% finite_Sup_in
tff(fact_4119_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),sin(real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_4120_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xa)) ).

% sin_ge_minus_one
tff(fact_4121_abs__sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),one_one(real)) ).

% abs_sin_le_one
tff(fact_4122_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A5)),B5) = finite_fold(A,A,sup_sup(A),B5,A5) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_4123_card__Un__le,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),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),B5))) ).

% card_Un_le
tff(fact_4124_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_4125_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_4126_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),bot_bot(set(B))))) = A5 ).

% Union_image_empty
tff(fact_4127_union__fold__insert,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = finite_fold(A,set(A),insert(A),B5,A5) ) ) ).

% union_fold_insert
tff(fact_4128_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4))) = $ite(C4 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_la(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4))) ).

% UN_extend_simps(3)
tff(fact_4129_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C4: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5) = $ite(C4 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lb(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4))) ).

% UN_extend_simps(2)
tff(fact_4130_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,complete_Sup_Sup(A),A5) = finite_fold(A,A,sup_sup(A),bot_bot(A),A5) ) ) ) ).

% Sup_fold_sup
tff(fact_4131_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_4132_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_ord_atMost(A,U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_4133_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_4134_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A5)),aa(set(A),A,lattic643756798349783984er_Max(A),B5)) ) ) ) ) ) ) ).

% Max.union
tff(fact_4135_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_lf(A,fun(A,$o)),aTP_Lamp_lg(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_4136_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q3: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P3),Q3) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q3) ) ).

% normalize_denom_pos
tff(fact_4137_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_4138_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),I: B,B5: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),fun_upd(B,set(A),A5,I,B5)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I),bot_bot(set(B))))))),
        $ite(member(B,I,J4),B5,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_4139_UN__le__eq__Un0,axiom,
    ! [A: $tType,M7: fun(nat,set(A)),Na: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M7),set_ord_atMost(nat,Na))) = 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)),image(nat,set(A),M7),set_or1337092689740270186AtMost(nat,one_one(nat),Na)))),aa(nat,set(A),M7,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_4140_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M10: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M2)
               => ! [N7: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N7)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,M2)),aa(nat,A,X4,N7)))),E3) ) ) ) ) ) ).

% CauchyD
tff(fact_4141_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M12: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M3)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,M3)),aa(nat,A,X4,N)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% CauchyI
tff(fact_4142_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M11: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
                 => ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N5)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,M5)),aa(nat,A,X4,N5)))),E4) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_4143_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_lh(A,fun(nat,A),B5)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A5),B5) ) ).

% SUP_nat_binary
tff(fact_4144_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S2: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F3,I2)),S2)
     => ( aa(set(A),$o,finite_finite2(A),S2)
       => ( ? [N9: nat] :
              ( ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N9)
                 => ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N9)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F3,M3)),aa(nat,set(A),F3,N)) ) ) )
              & ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
                 => ( aa(nat,set(A),F3,N9) = aa(nat,set(A),F3,N) ) ) )
         => ( aa(nat,set(A),F3,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)),image(nat,set(A),F3),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_4145_card__partition,axiom,
    ! [A: $tType,C4: set(set(A)),K: nat] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),C4)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4))
       => ( ! [C2: set(A)] :
              ( member(set(A),C2,C4)
             => ( aa(set(A),nat,finite_card(A),C2) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( member(set(A),C1,C4)
               => ( member(set(A),C22,C4)
                 => ( ( C1 != C22 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C1),C22) = bot_bot(set(A)) ) ) ) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(set(A)),nat,finite_card(set(A)),C4)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) ) ) ) ) ) ).

% card_partition
tff(fact_4146_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xa,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_4147_UNIV__I,axiom,
    ! [A: $tType,Xa: A] : member(A,Xa,top_top(set(A))) ).

% UNIV_I
tff(fact_4148_IntI,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,A5)
     => ( member(A,C3,B5)
       => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) ) ) ).

% IntI
tff(fact_4149_Int__iff,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
    <=> ( member(A,C3,A5)
        & member(A,C3,B5) ) ) ).

% Int_iff
tff(fact_4150_mod__or__dist,axiom,
    ! [P2: assn,Q2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2)),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
        | aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H2) ) ) ).

% mod_or_dist
tff(fact_4151_merge__pure__or,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        | (B2) )) ).

% merge_pure_or
tff(fact_4152_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% inf.bounded_iff
tff(fact_4153_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% le_inf_iff
tff(fact_4154_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_4155_finite__Plus__UNIV__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),top_top(set(sum_sum(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_4156_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_4157_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_4158_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_4159_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_4160_Int__UNIV,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = top_top(set(A)) )
    <=> ( ( A5 = top_top(set(A)) )
        & ( B5 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_4161_finite__Int,axiom,
    ! [A: $tType,F4: set(A),G5: set(A)] :
      ( ( aa(set(A),$o,finite_finite2(A),F4)
        | aa(set(A),$o,finite_finite2(A),G5) )
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F4),G5)) ) ).

% finite_Int
tff(fact_4162_card__eq__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S2: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
        <=> ( S2 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV
tff(fact_4163_card__eq__UNIV2,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S2: set(A)] :
          ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) = aa(set(A),nat,finite_card(A),S2) )
        <=> ( S2 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV2
tff(fact_4164_Int__subset__iff,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),A5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),B5) ) ) ).

% Int_subset_iff
tff(fact_4165_Int__insert__right__if1,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( member(A,A2,A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) ) ) ).

% Int_insert_right_if1
tff(fact_4166_Int__insert__right__if0,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( ~ member(A,A2,A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ) ) ).

% Int_insert_right_if0
tff(fact_4167_insert__inter__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) ).

% insert_inter_insert
tff(fact_4168_Int__insert__left__if1,axiom,
    ! [A: $tType,A2: A,C4: set(A),B5: set(A)] :
      ( member(A,A2,C4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)),C4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ) ) ).

% Int_insert_left_if1
tff(fact_4169_Int__insert__left__if0,axiom,
    ! [A: $tType,A2: A,C4: set(A),B5: set(A)] :
      ( ~ member(A,A2,C4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4) ) ) ).

% Int_insert_left_if0
tff(fact_4170_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)),S2) = S2 ).

% Un_Int_eq(1)
tff(fact_4171_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_4172_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) = S2 ).

% Un_Int_eq(3)
tff(fact_4173_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_4174_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)),S2) = S2 ).

% Int_Un_eq(1)
tff(fact_4175_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_4176_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S2: set(A),T3: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = S2 ).

% Int_Un_eq(3)
tff(fact_4177_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_4178_restrict__map__UNIV,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B))] : restrict_map(A,B,F3,top_top(set(A))) = F3 ).

% restrict_map_UNIV
tff(fact_4179_Collect__const,axiom,
    ! [A: $tType,P2: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_li($o,fun(A,$o),(P2))) = $ite((P2),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_4180_finite__Collect__not,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ba(fun(A,$o),fun(A,$o),P2)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_4181_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_4182_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_4183_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A5: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A5) = top_top(A) )
        <=> ! [X5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),top_top(A))
             => ? [Xa3: A] :
                  ( member(A,Xa3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Xa3) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_4184_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_4185_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_4186_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),aa(A,A,uminus_uminus(A),Xa))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_4187_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_4188_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_4189_Inf__UNIV,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) = bot_bot(A) ) ) ).

% Inf_UNIV
tff(fact_4190_ccInf__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).

% ccInf_empty
tff(fact_4191_Inf__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).

% Inf_empty
tff(fact_4192_Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_4193_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)),B5) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B5)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_4194_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A2: A,A5: set(A),B5: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)),B5) )
    <=> ( ~ member(A,A2,B5)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ) ) ) ).

% insert_disjoint(2)
tff(fact_4195_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B5: set(A),A2: A,A5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A2,B5)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A5) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_4196_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A5: set(A),B2: A,B5: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B5)) )
    <=> ( ~ member(A,B2,A5)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ) ) ) ).

% disjoint_insert(2)
tff(fact_4197_card__ge__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [S2: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(A),nat,finite_card(A),S2))
        <=> ( S2 = top_top(set(A)) ) ) ) ).

% card_ge_UNIV
tff(fact_4198_Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_4199_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Na: nat] :
      ( ( aa(set(A),set(A),image(A,A,F3),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4200_finite__compl,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A5))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_4201_Compl__disjoint,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_4202_Compl__disjoint2,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),A5) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_4203_inter__compl__diff__conv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) ).

% inter_compl_diff_conv
tff(fact_4204_Diff__Compl,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).

% Diff_Compl
tff(fact_4205_mod__h__bot__iff_I7_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
        | aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_iff(7)
tff(fact_4206_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P2: fun(B,$o),F3: fun(B,A),G3: fun(B,A),S2: set(B)] : aa(set(B),set(A),image(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,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),F3),G3)),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,$o),set(B),collect(B),P2)))),aa(set(B),set(A),image(B,A,G3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_lk(fun(B,$o),fun(B,$o),P2))))) ).

% if_image_distrib
tff(fact_4207_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A5: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A5)) = top_top(A) )
        <=> ! [X5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),top_top(A))
             => ? [Xa3: B] :
                  ( member(B,Xa3,A5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),aa(B,A,F3,Xa3)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_4208_range__constant,axiom,
    ! [B: $tType,A: $tType,Xa: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_cg(A,fun(B,A)),Xa)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ).

% range_constant
tff(fact_4209_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_4210_INT__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A5: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_kp(set(A),fun(B,set(A)),C3)),A5)) = $ite(A5 = bot_bot(set(B)),top_top(set(A)),C3) ).

% INT_constant
tff(fact_4211_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_ord_lessThan(A,Xa)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_4212_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B2: B,A2: B] :
          ( nO_MATCH(A,B,Xa,B2)
         => ( ( B2 != zero_zero(B) )
           => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_4213_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ll(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4)))) ).

% INT_simps(2)
tff(fact_4214_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lm(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5)) ).

% INT_simps(1)
tff(fact_4215_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A5: fun(B,set(A)),B5: set(A),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5)) ).

% INT_simps(3)
tff(fact_4216_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kv(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4)) = $ite(C4 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),C4)))) ).

% INT_simps(4)
tff(fact_4217_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A5)),aa(set(A),A,complete_Sup_Sup(A),B5))) ) ).

% Sup_inter_less_eq
tff(fact_4218_Union__Int__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A5),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5))) ).

% Union_Int_subset
tff(fact_4219_subset__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),top_top(set(A))) ).

% subset_UNIV
tff(fact_4220_inter__eq__subsetI,axiom,
    ! [A: $tType,S2: set(A),S5: set(A),A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),S5)
     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S5) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),S2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),S2) ) ) ) ).

% inter_eq_subsetI
tff(fact_4221_Int__mono,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B5: set(A),D5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),D5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),D5)) ) ) ).

% Int_mono
tff(fact_4222_Int__lower1,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),A5) ).

% Int_lower1
tff(fact_4223_Int__lower2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),B5) ).

% Int_lower2
tff(fact_4224_Int__absorb1,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = B5 ) ) ).

% Int_absorb1
tff(fact_4225_Int__absorb2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = A5 ) ) ).

% Int_absorb2
tff(fact_4226_Int__greatest,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),B5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) ) ) ).

% Int_greatest
tff(fact_4227_Int__Collect__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),P2: fun(A,$o),Q2: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => ( aa(A,$o,P2,X)
             => aa(A,$o,Q2,X) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(fun(A,$o),set(A),collect(A),Q2))) ) ) ).

% Int_Collect_mono
tff(fact_4228_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_4229_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_4230_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_4231_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI2
tff(fact_4232_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.coboundedI1
tff(fact_4233_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb_iff2
tff(fact_4234_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb_iff1
tff(fact_4235_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2) ) ).

% inf.cobounded2
tff(fact_4236_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2) ) ).

% inf.cobounded1
tff(fact_4237_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.order_iff
tff(fact_4238_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)) ) ) ) ).

% inf_greatest
tff(fact_4239_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3)) ) ) ) ).

% inf.boundedI
tff(fact_4240_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% inf.boundedE
tff(fact_4241_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_4242_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% inf_absorb1
tff(fact_4243_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_4244_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_4245_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = Xa ) ) ) ).

% le_iff_inf
tff(fact_4246_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),Xa: A,Y: A] :
          ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X),Y3)),X)
         => ( ! [X: A,Y3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X),Y3)),Y3)
           => ( ! [X: A,Y3: A,Z3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),F3,Y3),Z3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = aa(A,A,aa(A,fun(A,A),F3,Xa),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_4247_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% inf.orderI
tff(fact_4248_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_4249_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI2
tff(fact_4250_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% le_infI1
tff(fact_4251_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C3),D2)) ) ) ) ).

% inf_mono
tff(fact_4252_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)) ) ) ) ).

% le_infI
tff(fact_4253_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B2) ) ) ) ).

% le_infE
tff(fact_4254_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_le2
tff(fact_4255_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_le1
tff(fact_4256_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Xa) ) ).

% inf_sup_ord(1)
tff(fact_4257_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Y) ) ).

% inf_sup_ord(2)
tff(fact_4258_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xa) = Y ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_4259_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Xa) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Xa) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),Y) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),Y) = top_top(A) )
               => ( Xa = Y ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4260_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,complete_Inf_Inf(A),A5) = finite_fold(A,A,inf_inf(A),top_top(A),A5) ) ) ) ).

% Inf_fold_inf
tff(fact_4261_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_4262_infinite__UNIV__char__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% infinite_UNIV_char_0
tff(fact_4263_ex__new__if__finite,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),A5)
       => ? [A4: A] : ~ member(A,A4,A5) ) ) ).

% ex_new_if_finite
tff(fact_4264_finite__class_Ofinite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_class.finite_UNIV
tff(fact_4265_ent__disjE,axiom,
    ! [A5: assn,C4: assn,B5: assn] :
      ( entails(A5,C4)
     => ( entails(B5,C4)
       => entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B5),C4) ) ) ).

% ent_disjE
tff(fact_4266_ent__disjI1,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2),R3)
     => entails(P2,R3) ) ).

% ent_disjI1
tff(fact_4267_ent__disjI2,axiom,
    ! [P2: assn,Q2: assn,R3: assn] :
      ( entails(aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),P2),Q2),R3)
     => entails(Q2,R3) ) ).

% ent_disjI2
tff(fact_4268_ent__disjI1_H,axiom,
    ! [A5: assn,B5: assn,C4: assn] :
      ( entails(A5,B5)
     => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B5),C4)) ) ).

% ent_disjI1'
tff(fact_4269_ent__disjI2_H,axiom,
    ! [A5: assn,C4: assn,B5: assn] :
      ( entails(A5,C4)
     => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),B5),C4)) ) ).

% ent_disjI2'
tff(fact_4270_ent__disjI1__direct,axiom,
    ! [A5: assn,B5: assn] : entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B5)) ).

% ent_disjI1_direct
tff(fact_4271_ent__disjI2__direct,axiom,
    ! [B5: assn,A5: assn] : entails(B5,aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B5)) ).

% ent_disjI2_direct
tff(fact_4272_star__or__dist1,axiom,
    ! [A5: assn,B5: assn,C4: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B5)),C4) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),C4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),C4)) ).

% star_or_dist1
tff(fact_4273_star__or__dist2,axiom,
    ! [C4: assn,A5: assn,B5: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),A5),B5)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),A5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C4),B5)) ).

% star_or_dist2
tff(fact_4274_Finite__Set_Ofinite__set,axiom,
    ! [A: $tType] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),top_top(set(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% Finite_Set.finite_set
tff(fact_4275_finite__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_prod
tff(fact_4276_finite__Prod__UNIV,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(B),$o,finite_finite2(B),top_top(set(B)))
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B)))) ) ) ).

% finite_Prod_UNIV
tff(fact_4277_finite__fun__UNIVD2,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ).

% finite_fun_UNIVD2
tff(fact_4278_sup__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5))) ).

% sup_set_def
tff(fact_4279_Un__Int__distrib2,axiom,
    ! [A: $tType,B5: set(A),C4: set(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A5)) ).

% Un_Int_distrib2
tff(fact_4280_Int__Un__distrib2,axiom,
    ! [A: $tType,B5: set(A),C4: set(A),A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A5)) ).

% Int_Un_distrib2
tff(fact_4281_Un__Int__distrib,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),C4)) ).

% Un_Int_distrib
tff(fact_4282_Int__Un__distrib,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)) ).

% Int_Un_distrib
tff(fact_4283_Un__UNIV__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_4284_Un__UNIV__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B5) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_4285_Un__Int__crazy,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C4),A5)) ).

% Un_Int_crazy
tff(fact_4286_infinite__UNIV__nat,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% infinite_UNIV_nat
tff(fact_4287_nat__not__finite,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% nat_not_finite
tff(fact_4288_insert__UNIV,axiom,
    ! [A: $tType,Xa: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_4289_IntE,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => ~ ( member(A,C3,A5)
         => ~ member(A,C3,B5) ) ) ).

% IntE
tff(fact_4290_IntD1,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => member(A,C3,A5) ) ).

% IntD1
tff(fact_4291_IntD2,axiom,
    ! [A: $tType,C3: A,A5: set(A),B5: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => member(A,C3,B5) ) ).

% IntD2
tff(fact_4292_Int__assoc,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ).

% Int_assoc
tff(fact_4293_UNIV__eq__I,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ! [X: A] : member(A,X,A5)
     => ( top_top(set(A)) = A5 ) ) ).

% UNIV_eq_I
tff(fact_4294_Int__absorb,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),A5) = A5 ).

% Int_absorb
tff(fact_4295_Int__commute,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),A5) ).

% Int_commute
tff(fact_4296_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X: A] : member(A,X,top_top(set(A))) ).

% UNIV_witness
tff(fact_4297_Int__UNIV__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B5) = B5 ).

% Int_UNIV_left
tff(fact_4298_Int__UNIV__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),top_top(set(A))) = A5 ).

% Int_UNIV_right
tff(fact_4299_Int__left__absorb,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).

% Int_left_absorb
tff(fact_4300_Int__left__commute,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)) ).

% Int_left_commute
tff(fact_4301_Collect__conj__eq,axiom,
    ! [A: $tType,P2: fun(A,$o),Q2: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P2),Q2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P2)),aa(fun(A,$o),set(A),collect(A),Q2)) ).

% Collect_conj_eq
tff(fact_4302_Int__Collect,axiom,
    ! [A: $tType,Xa: A,A5: set(A),P2: fun(A,$o)] :
      ( member(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))
    <=> ( member(A,Xa,A5)
        & aa(A,$o,P2,Xa) ) ) ).

% Int_Collect
tff(fact_4303_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_lo(A,$o)) ).

% UNIV_def
tff(fact_4304_Int__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_lp(set(A),fun(set(A),fun(A,$o)),A5),B5)) ).

% Int_def
tff(fact_4305_Int__insert__right,axiom,
    ! [A: $tType,A5: set(A),A2: A,B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)) = $ite(member(A,A2,A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) ).

% Int_insert_right
tff(fact_4306_Int__insert__left,axiom,
    ! [A: $tType,A2: A,B5: set(A),C4: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B5)),C4) = $ite(member(A,A2,C4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ).

% Int_insert_left
tff(fact_4307_Diff__Int__distrib2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) ).

% Diff_Int_distrib2
tff(fact_4308_Diff__Int__distrib,axiom,
    ! [A: $tType,C4: set(A),A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),A5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C4),B5)) ).

% Diff_Int_distrib
tff(fact_4309_Diff__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) ).

% Diff_Diff_Int
tff(fact_4310_Diff__Int2,axiom,
    ! [A: $tType,A5: set(A),C4: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),C4)),B5) ).

% Diff_Int2
tff(fact_4311_Int__Diff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),C4)) ).

% Int_Diff
tff(fact_4312_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xa: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,Xa,Y) = top_top(set(A)) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4313_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI2
tff(fact_4314_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C3) ) ) ).

% inf.strict_coboundedI1
tff(fact_4315_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_4316_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% inf.strict_boundedE
tff(fact_4317_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_4318_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_4319_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,Xa: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI2
tff(fact_4320_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),Xa) ) ) ).

% less_infI1
tff(fact_4321_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_4322_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_4323_disjointI,axiom,
    ! [A: $tType,A2: set(A),B2: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A2)
         => ~ member(A,X,B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ).

% disjointI
tff(fact_4324_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ! [X5: A] :
          ( member(A,X5,A5)
         => ! [Xa3: A] :
              ( member(A,Xa3,B5)
             => ( X5 != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_4325_Int__empty__right,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_4326_Int__empty__left,axiom,
    ! [A: $tType,B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B5) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_4327_disjoint__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> ! [X5: A] :
          ( member(A,X5,A5)
         => ~ member(A,X5,B5) ) ) ).

% disjoint_iff
tff(fact_4328_Int__emptyI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ! [X: A] :
          ( member(A,X,A5)
         => ~ member(A,X,B5) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_4329_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_4330_rangeI,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xa: B] : member(A,aa(B,A,F3,Xa),aa(set(B),set(A),image(B,A,F3),top_top(set(B)))) ).

% rangeI
tff(fact_4331_range__eqI,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A),Xa: B] :
      ( ( B2 = aa(B,A,F3,Xa) )
     => member(A,B2,aa(set(B),set(A),image(B,A,F3),top_top(set(B)))) ) ).

% range_eqI
tff(fact_4332_inf__top_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => semila1105856199041335345_order(A,inf_inf(A),top_top(A),ord_less_eq(A),ord_less(A)) ) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_4333_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F3: fun(B,A)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F3),top_top(set(B))))
     => ~ ! [X: B] : B2 != aa(B,A,F3,X) ) ).

% rangeE
tff(fact_4334_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),G3: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lq(fun(C,A),fun(fun(B,C),fun(B,A)),F3),G3)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F3),aa(set(B),set(C),image(B,C,G3),top_top(set(B)))) ).

% range_composition
tff(fact_4335_ex__distrib__or,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_lr(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_or
tff(fact_4336_ex__join__or,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: fun(A,assn)] : ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ls(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P2),Q2)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_lt(fun(A,assn),fun(fun(A,assn),fun(A,assn)),P2),Q2)) ).

% ex_join_or
tff(fact_4337_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z))) ) ).

% distrib_sup_le
tff(fact_4338_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ).

% distrib_inf_le
tff(fact_4339_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),B2)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_4340_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),A2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),B2)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_4341_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_4342_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B5: set(A),A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B5)),A2) = bot_bot(A) )
        <=> ! [X5: A] :
              ( member(A,X5,B5)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X5),A2) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_4343_mod__star__conv,axiom,
    ! [A5: assn,B5: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A5),B5)),H2)
    <=> ? [Hr: heap_ext(product_unit),As1: set(nat),As22: set(nat)] :
          ( ( H2 = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22)) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(A5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As1))
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(B5),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),Hr),As22)) ) ) ).

% mod_star_conv
tff(fact_4344_star__assnI,axiom,
    ! [P2: assn,H2: heap_ext(product_unit),As3: set(nat),Q2: assn,As4: set(nat)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3))
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As4))
       => ( ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As3),As4) = bot_bot(set(nat)) )
         => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As3),As4))) ) ) ) ).

% star_assnI
tff(fact_4345_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A5),B5))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),aa(set(B),set(A),image(B,A,F3),B5))) ).

% image_Int_subset
tff(fact_4346_disjoint__mono,axiom,
    ! [A: $tType,A2: set(A),A3: set(A),B2: set(A),B3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A2),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),B3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = bot_bot(set(A)) )
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A2),B2) = bot_bot(set(A)) ) ) ) ) ).

% disjoint_mono
tff(fact_4347_disjoint__alt__simp1,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = A5 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) ) ) ).

% disjoint_alt_simp1
tff(fact_4348_disjoint__alt__simp2,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) != A5 )
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp2
tff(fact_4349_Int__Diff__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_4350_Diff__triv,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5) = A5 ) ) ).

% Diff_triv
tff(fact_4351_insert__partition,axiom,
    ! [A: $tType,Xa: set(A),F4: set(set(A))] :
      ( ~ member(set(A),Xa,F4)
     => ( ! [X: set(A)] :
            ( member(set(A),X,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Xa),F4))
           => ! [Xa5: set(A)] :
                ( member(set(A),Xa5,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Xa),F4))
               => ( ( X != Xa5 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Xa),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F4)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_4352_Union__disjoint,axiom,
    ! [A: $tType,C4: set(set(A)),A5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)),A5) = bot_bot(set(A)) )
    <=> ! [X5: set(A)] :
          ( member(set(A),X5,C4)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X5),A5) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_4353_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),C4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),A5) ) ).

% Un_Int_assoc_eq
tff(fact_4354_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_4355_Un__Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = A5 ).

% Un_Diff_Int
tff(fact_4356_Int__Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = A5 ).

% Int_Diff_Un
tff(fact_4357_Diff__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C4)) ).

% Diff_Int
tff(fact_4358_Diff__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),C4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),C4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),C4)) ).

% Diff_Un
tff(fact_4359_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),B5: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),top_top(set(B)))),B5)
     => member(A,aa(B,A,F3,I),B5) ) ).

% range_subsetD
tff(fact_4360_Compl__Un,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Un
tff(fact_4361_Compl__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ).

% Compl_Int
tff(fact_4362_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),nat,finite_card(A),A5) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A5 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_4363_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H2)) ) ).

% not_UNIV_le_Icc
tff(fact_4364_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_ord_atMost(A,H2)) ) ).

% not_UNIV_le_Iic
tff(fact_4365_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B)),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X3),Xa2)
    <=> 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),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))),R3),S2)) ) ).

% sup_Un_eq2
tff(fact_4366_Compl__UNIV__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),top_top(set(A))) = bot_bot(set(A)) ).

% Compl_UNIV_eq
tff(fact_4367_Compl__empty__eq,axiom,
    ! [A: $tType] : aa(set(A),set(A),uminus_uminus(set(A)),bot_bot(set(A))) = top_top(set(A)) ).

% Compl_empty_eq
tff(fact_4368_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),A5) ).

% Compl_eq_Diff_UNIV
tff(fact_4369_Compl__partition,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),A5)) = top_top(set(A)) ).

% Compl_partition
tff(fact_4370_Compl__partition2,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A5)),A5) = top_top(set(A)) ).

% Compl_partition2
tff(fact_4371_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),F3: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,G3),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_lu(fun(B,A),fun(fun(A,C),fun(B,C)),G3),F3)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_4372_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_lv(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_4373_comp__fun__commute__def_H,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F3)
    <=> finite4664212375090638736ute_on(A,B,top_top(set(A)),F3) ) ).

% comp_fun_commute_def'
tff(fact_4374_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% inf_shunt
tff(fact_4375_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Y) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Y) ) ) ).

% sup_shunt
tff(fact_4376_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q3))),R2) ) ) ).

% sup_neg_inf
tff(fact_4377_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Y))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)) ) ) ).

% shunt2
tff(fact_4378_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Y)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z)) ) ) ).

% shunt1
tff(fact_4379_inter__Set__filter,axiom,
    ! [A: $tType,B5: set(A),A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B5)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = filter3(A,aTP_Lamp_a(set(A),fun(A,$o),A5),B5) ) ) ).

% inter_Set_filter
tff(fact_4380_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] :
                  ( member(A,X,A5)
                 => ( member(A,Y3,A5)
                   => member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y3),A5) ) )
             => member(A,aa(set(A),A,complete_Inf_Inf(A),A5),A5) ) ) ) ) ).

% finite_Inf_in
tff(fact_4381_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A5)),aa(set(A),A,complete_Inf_Inf(A),B5))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ).

% less_eq_Inf_inter
tff(fact_4382_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A),B5: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A5)),B5) = finite_fold(A,A,inf_inf(A),B5,A5) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_4383_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A5: A,B5: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_lh(A,fun(nat,A),B5)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A5),B5) ) ).

% INF_nat_binary
tff(fact_4384_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_4385_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_4386_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_4387_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_4388_disjoint__alt__simp3,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),A5)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) != bot_bot(set(A)) ) ) ).

% disjoint_alt_simp3
tff(fact_4389_top_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ordering_top(A,ord_less_eq(A),ord_less(A),top_top(A)) ) ).

% top.ordering_top_axioms
tff(fact_4390_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),B5)) ) ).

% disjoint_eq_subset_Compl
tff(fact_4391_Inter__Un__subset,axiom,
    ! [A: $tType,A5: set(set(A)),B5: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A5)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A5),B5))) ).

% Inter_Un_subset
tff(fact_4392_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A2: A,Xa: B] :
      ( ( aa(set(B),set(A),image(B,A,F3),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) )
     => ( aa(B,A,F3,Xa) = A2 ) ) ).

% range_eq_singletonD
tff(fact_4393_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_4394_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A5: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,aTP_Lamp_kt(A,fun(B,A),C3)),A5)) = $ite(A5 = bot_bot(set(B)),top_top(A),C3) ) ).

% INF_constant
tff(fact_4395_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),Xa: B,F3: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lw(B,fun(fun(A,B),fun(A,B)),Xa),F3)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xa),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_4396_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F3: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_lx(fun(A,B),fun(B,fun(A,B)),F3),Xa)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),I5))),Xa) ) ) ) ).

% INF_inf_const2
tff(fact_4397_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image(B,A,F3),A5))),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),uminus_uminus(set(B)),A5))) ) ).

% surj_Compl_image_subset
tff(fact_4398_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_ly(fun(A,B),fun(set(A),fun(A,B)),G3),B5)),A5) ) ) ) ).

% sum.inter_restrict
tff(fact_4399_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_lz(fun(A,B),fun(set(A),fun(A,B)),G3),B5)),A5) ) ) ) ).

% prod.inter_restrict
tff(fact_4400_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: fun(B,set(A)),C4: set(B)] :
      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)),aa(set(B),set(set(A)),image(B,set(A),B5),C4))) = $ite(C4 = bot_bot(set(B)),A5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ll(set(A),fun(fun(B,set(A)),fun(B,set(A))),A5),B5)),C4))) ).

% INT_extend_simps(2)
tff(fact_4401_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C4: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5) = $ite(C4 = bot_bot(set(B)),B5,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lm(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4))) ).

% INT_extend_simps(1)
tff(fact_4402_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_4403_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)),image(nat,set(A),aTP_Lamp_ma(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)),image(nat,set(A),A5),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_4404_INT__empty,axiom,
    ! [B: $tType,A: $tType,B5: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_4405_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_jo(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_4406_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S2: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H2,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3))
                   => ( aa(A,B,G3,I2) = zero_zero(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3))
                     => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),T3) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_4407_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),A5) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% sum.Int_Diff
tff(fact_4408_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ).

% sum.union_inter
tff(fact_4409_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xa: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_lessThan(A,K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_4410_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S2: set(A),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S2)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),T3),S2))
                 => ( aa(A,B,H2,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S2),T3))
                   => ( aa(A,B,G3,I2) = one_one(B) ) )
               => ( ! [X: A] :
                      ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T3))
                     => ( aa(A,B,G3,X) = aa(A,B,H2,X) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),S2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),T3) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_4411_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),G3: fun(A,B),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))) ) ) ) ).

% prod.Int_Diff
tff(fact_4412_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod.union_inter
tff(fact_4413_card__Un__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( 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),B5)) = 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),B5))),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),B5))) ) ) ) ).

% card_Un_Int
tff(fact_4414_card__Diff__subset__Int,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
     => ( 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),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ).

% card_Diff_subset_Int
tff(fact_4415_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_4416_range__enumerate,axiom,
    ! [S2: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S2)
     => ( aa(set(nat),set(nat),image(nat,nat,infini527867602293511546merate(nat,S2)),top_top(set(nat))) = S2 ) ) ).

% range_enumerate
tff(fact_4417_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),P2: fun(A,$o),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_mb(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P2),H2),G3)),A5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2))))) ) ) ) ).

% sum.If_cases
tff(fact_4418_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),P2: fun(A,$o),H2: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_mc(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P2),H2),G3)),A5) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P2))))) ) ) ) ).

% prod.If_cases
tff(fact_4419_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_4420_UN__finite__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),C4: set(A)] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),C4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A5),top_top(set(nat))))),C4) ) ).

% UN_finite_subset
tff(fact_4421_UN__finite2__eq,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: 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)),image(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(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)),image(nat,set(A),B5),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_4422_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A5: fun(B,set(A)),C4: set(B),B5: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A5),C4))),B5) = $ite(C4 = bot_bot(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),top_top(set(A))),B5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(set(A),fun(B,set(A))),A5),B5)),C4))) ).

% INT_extend_simps(3)
tff(fact_4423_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_4424_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum_Un
tff(fact_4425_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),B5)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_4426_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,G3,X) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_4427_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_4428_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = 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),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% sum.union_diff2
tff(fact_4429_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un2
tff(fact_4430_card__Un__disjoint,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = 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),B5)) = 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),B5)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_4431_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A5: set(A),B5: set(A),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ).

% prod.union_diff2
tff(fact_4432_sum__Un__nat,axiom,
    ! [A: $tType,A5: set(A),B5: set(A),F3: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,finite_finite2(A),B5)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),A5)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),B5))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ).

% sum_Un_nat
tff(fact_4433_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A),Na: nat,Sa: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),set_ord_lessThan(A,Sa))))
           => ( aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),set_ord_lessThan(A,Sa))),Na) = aa(nat,A,infini527867602293511546merate(A,S2),Na) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_4434_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,F3),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F3),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_4435_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys5: list(A)] :
            ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4436_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(A),A5: fun(A,set(B)),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X: A] :
                ( member(A,X,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X)) )
           => ( ! [X: A] :
                  ( member(A,X,I5)
                 => ! [Xa5: A] :
                      ( member(A,Xa5,I5)
                     => ( ( X != Xa5 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X)),aa(A,set(B),A5,Xa5)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5))) = 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_md(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A5),G3)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4437_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(A),A5: fun(A,set(B)),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X: A] :
                ( member(A,X,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X)) )
           => ( ! [X: A] :
                  ( member(A,X,I5)
                 => ! [Xa5: A] :
                      ( member(A,Xa5,I5)
                     => ( ( X != Xa5 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X)),aa(A,set(B),A5,Xa5)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_me(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A5),G3)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4438_word__atLeastLessThan__Suc__atLeastAtMost__union,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A),L: word(A),U: word(A)] :
          ( ( M != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),L),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),U)
             => ( aa(set(word(A)),set(word(A)),aa(set(word(A)),fun(set(word(A)),set(word(A))),sup_sup(set(word(A))),set_or1337092689740270186AtMost(word(A),L,M)),set_or1337092689740270186AtMost(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),M),one_one(word(A))),U)) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost_union
tff(fact_4439_UN__finite2__subset,axiom,
    ! [A: $tType,A5: fun(nat,set(A)),B5: fun(nat,set(A)),K: nat] :
      ( ! [N: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A5),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(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)),image(nat,set(A),B5),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_4440_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A5: set(A),B5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),B5)
           => ( ! [X: A] :
                  ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))
                 => ( aa(A,B,F3,X) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5))) ) ) ) ) ) ).

% prod_Un
tff(fact_4441_range__mod,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_mf(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_4442_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A5: set(A),B5: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),S2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),S2)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = finite_fold(A,B,F3,finite_fold(A,B,F3,Z,A5),B5) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_4443_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [X: A] :
            ( member(A,X,I5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A5,X)) )
       => ( ! [X: A] :
              ( member(A,X,I5)
             => ! [Xa5: A] :
                  ( member(A,Xa5,I5)
                 => ( ( X != Xa5 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A5,X)),aa(A,set(B),A5,Xa5)) = 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)),image(A,set(B),A5),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_kh(fun(A,set(B)),fun(A,nat),A5)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4444_card__nat,axiom,
    aa(set(nat),nat,finite_card(nat),top_top(set(nat))) = zero_zero(nat) ).

% card_nat
tff(fact_4445_card__option,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(set(option(A)),nat,finite_card(option(A)),top_top(set(option(A)))) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ) ).

% card_option
tff(fact_4446_one__less__int__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% one_less_int_card
tff(fact_4447_card__fun,axiom,
    ! [A: $tType,B: $tType] :
      aa(set(fun(A,B)),nat,finite_card(fun(A,B)),top_top(set(fun(A,B)))) = $ite(
        ( ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) != zero_zero(nat) )
          & ( aa(set(B),nat,finite_card(B),top_top(set(B))) != zero_zero(nat) ) )
        | ( aa(set(B),nat,finite_card(B),top_top(set(B))) = one_one(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(set(B),nat,finite_card(B),top_top(set(B)))),aa(set(A),nat,finite_card(A),top_top(set(A)))),
        zero_zero(nat) ) ).

% card_fun
tff(fact_4448_card__num0,axiom,
    aa(set(numeral_num0),nat,finite_card(numeral_num0),top_top(set(numeral_num0))) = zero_zero(nat) ).

% card_num0
tff(fact_4449_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_4450_merge__true__star,axiom,
    aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),top_top(assn)) = top_top(assn) ).

% merge_true_star
tff(fact_4451_merge__pure__and,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        & (B2) )) ).

% merge_pure_and
tff(fact_4452_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_4453_assn__basic__inequalities_I5_J,axiom,
    top_top(assn) != bot_bot(assn) ).

% assn_basic_inequalities(5)
tff(fact_4454_mod__h__bot__iff_I6_J,axiom,
    ! [P2: assn,Q2: assn,H2: heap_ext(product_unit)] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P2),Q2)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_iff(6)
tff(fact_4455_card__literal,axiom,
    aa(set(literal),nat,finite_card(literal),top_top(set(literal))) = zero_zero(nat) ).

% card_literal
tff(fact_4456_ent__conjE2,axiom,
    ! [B5: assn,C4: assn,A5: assn] :
      ( entails(B5,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A5),B5),C4) ) ).

% ent_conjE2
tff(fact_4457_ent__conjE1,axiom,
    ! [A5: assn,C4: assn,B5: assn] :
      ( entails(A5,C4)
     => entails(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A5),B5),C4) ) ).

% ent_conjE1
tff(fact_4458_ent__conjI,axiom,
    ! [A5: assn,B5: assn,C4: assn] :
      ( entails(A5,B5)
     => ( entails(A5,C4)
       => entails(A5,aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),B5),C4)) ) ) ).

% ent_conjI
tff(fact_4459_mod__and__dist,axiom,
    ! [P2: assn,Q2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),P2),Q2)),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
        & aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(Q2),H2) ) ) ).

% mod_and_dist
tff(fact_4460_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),top_top(fun(A,$o))) ).

% top_set_def
tff(fact_4461_inf__set__def,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aTP_Lamp_a(set(A),fun(A,$o),A5)),aTP_Lamp_a(set(A),fun(A,$o),B5))) ).

% inf_set_def
tff(fact_4462_ent__true,axiom,
    ! [P2: assn] : entails(P2,top_top(assn)) ).

% ent_true
tff(fact_4463_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(A,B)),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),S2)),X3),Xa2)
    <=> 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),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))),R3),S2)) ) ).

% inf_Int_eq2
tff(fact_4464_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X3),Xa2)
    <=> 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),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_4465_ex__distrib__and,axiom,
    ! [A: $tType,P2: fun(A,assn),Q2: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_mg(fun(A,assn),fun(assn,fun(A,assn)),P2),Q2)) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),ex_assn(A,P2)),Q2) ).

% ex_distrib_and
tff(fact_4466_mod__star__trueE,axiom,
    ! [P2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H2)
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] : ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H5) ) ).

% mod_star_trueE
tff(fact_4467_mod__star__trueI,axiom,
    ! [P2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H2)
     => aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H2) ) ).

% mod_star_trueI
tff(fact_4468_Inter__UNIV,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),top_top(set(set(A)))) = bot_bot(set(A)) ).

% Inter_UNIV
tff(fact_4469_prec__frame_H,axiom,
    ! [A: $tType,P2: fun(A,assn),Xa: A,F13: assn,Y: A,F24: assn,H2: product_prod(heap_ext(product_unit),set(nat)),R12: assn,R23: assn] :
      ( ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Xa)),F13)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Y)),F24))),H2)
       => ( Xa = Y ) )
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R23)),H2)
       => ( entails(R12,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Xa)),F13))
         => ( entails(R23,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Y)),F24))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame'
tff(fact_4470_prec__frame__expl,axiom,
    ! [A: $tType,P2: fun(A,assn),F13: assn,F24: assn,H2: product_prod(heap_ext(product_unit),set(nat)),R12: assn,R23: assn,Xa: A,Y: A] :
      ( ! [X: A,Y3: A] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,X)),F13)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Y3)),F24))),H2)
         => ( X = Y3 ) )
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R23)),H2)
       => ( entails(R12,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Xa)),F13))
         => ( entails(R23,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,P2,Y)),F24))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame_expl
tff(fact_4471_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_ext(product_unit)] : aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(top_top(assn)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat)))) ).

% mod_h_bot_iff(2)
tff(fact_4472_bit0_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size0
tff(fact_4473_bit1_Osize0,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size0
tff(fact_4474_bit1_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ).

% bit1.size1
tff(fact_4475_bit0_Osize1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ).

% bit0.size1
tff(fact_4476_preciseD,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,assn)),A2: A,P3: B,F4: assn,A3: A,F6: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( precise(A,B,R3)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A2),P3)),F4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A3),P3)),F6))),H2)
       => ( A2 = A3 ) ) ) ).

% preciseD
tff(fact_4477_preciseI,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,assn))] :
      ( ! [A4: A,A13: A,H: product_prod(heap_ext(product_unit),set(nat)),P5: B,F5: assn,F10: assn] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A4),P5)),F5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A13),P5)),F10))),H)
         => ( A4 = A13 ) )
     => precise(A,B,R3) ) ).

% preciseI
tff(fact_4478_precise__def,axiom,
    ! [B: $tType,A: $tType,R3: fun(A,fun(B,assn))] :
      ( precise(A,B,R3)
    <=> ! [A7: A,A14: A,H4: product_prod(heap_ext(product_unit),set(nat)),P6: B,F11: assn,F14: assn] :
          ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A7),P6)),F11)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),R3,A14),P6)),F14))),H4)
         => ( A7 = A14 ) ) ) ).

% precise_def
tff(fact_4479_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F4: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( member(A,I2,I5)
         => ! [J2: A] :
              ( member(A,J2,I5)
             => ? [X3: A] :
                  ( member(A,X3,I5)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,I2)),aa(A,filter(B),F4,J2))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),I5)) = bot_bot(filter(B)) )
      <=> ? [X5: A] :
            ( member(A,X5,I5)
            & ( aa(A,filter(B),F4,X5) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4480_bit1__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P2: fun(numeral_bit1(A),$o),Xa: numeral_bit1(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A))))))
               => aa(numeral_bit1(A),$o,P2,aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3)) ) )
         => aa(numeral_bit1(A),$o,P2,Xa) ) ) ).

% bit1_induct
tff(fact_4481_bit0__induct,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [P2: fun(numeral_bit0(A),$o),Xa: numeral_bit0(A)] :
          ( ! [Z3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A))))))
               => aa(numeral_bit0(A),$o,P2,aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3)) ) )
         => aa(numeral_bit0(A),$o,P2,Xa) ) ) ).

% bit0_induct
tff(fact_4482_bit1__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit1(A)] :
          ~ ! [Z3: int] :
              ( ( Xa = aa(int,numeral_bit1(A),ring_1_of_int(numeral_bit1(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))))) ) ) ) ).

% bit1_cases
tff(fact_4483_bit0__cases,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [Xa: numeral_bit0(A)] :
          ~ ! [Z3: int] :
              ( ( Xa = aa(int,numeral_bit0(A),ring_1_of_int(numeral_bit0(A)),Z3) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z3)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),aa(nat,int,semiring_1_of_nat(int),aa(set(numeral_bit0(A)),nat,finite_card(numeral_bit0(A)),top_top(set(numeral_bit0(A)))))) ) ) ) ).

% bit0_cases
tff(fact_4484_card__map__elide2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(word(A)),nat,finite_card(word(A)),top_top(set(word(A)))))
         => ( aa(set(word(A)),nat,finite_card(word(A)),aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) = Na ) ) ) ).

% card_map_elide2
tff(fact_4485_prec__frame,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,assn)),R12: assn,R23: assn,H2: product_prod(heap_ext(product_unit),set(nat)),Xa: A,P3: B,F13: assn,Y: A,F24: assn] :
      ( precise(A,B,P2)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),R12),R23)),H2)
       => ( entails(R12,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P2,Xa),P3)),F13))
         => ( entails(R23,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P2,Y),P3)),F24))
           => ( Xa = Y ) ) ) ) ) ).

% prec_frame
tff(fact_4486_card__map__elide,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(word(A)),nat,finite_card(word(A)),top_top(set(word(A)))))
         => ( aa(set(word(A)),nat,finite_card(word(A)),aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) = aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ) ) ).

% card_map_elide
tff(fact_4487_zero__less__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% zero_less_card_finite
tff(fact_4488_card__UNIV__sum,axiom,
    ! [B: $tType,A: $tType] :
      aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),top_top(set(sum_sum(A,B)))) = $ite(
        ( ( aa(set(A),nat,finite_card(A),top_top(set(A))) != zero_zero(nat) )
        & ( aa(set(B),nat,finite_card(B),top_top(set(B))) != zero_zero(nat) ) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),aa(set(B),nat,finite_card(B),top_top(set(B)))),
        zero_zero(nat) ) ).

% card_UNIV_sum
tff(fact_4489_finite__UNIV__fun,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
    <=> ( ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
          & aa(set(B),$o,finite_finite2(B),top_top(set(B))) )
        | ( aa(set(B),nat,finite_card(B),top_top(set(B))) = one_one(nat) ) ) ) ).

% finite_UNIV_fun
tff(fact_4490_one__less__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% one_less_card
tff(fact_4491_one__le__card__finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% one_le_card_finite
tff(fact_4492_card__UNIV__option,axiom,
    ! [A: $tType] :
      aa(set(option(A)),nat,finite_card(option(A)),top_top(set(option(A)))) = $ite(aa(set(A),nat,finite_card(A),top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),top_top(set(A)))),one_one(nat))) ).

% card_UNIV_option
tff(fact_4493_Inf__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).

% Inf_finite_empty
tff(fact_4494_Sup__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% Sup_finite_empty
tff(fact_4495_cclfp__def,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F3: fun(A,A)] : order_532582986084564980_cclfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_mh(fun(A,A),fun(nat,A),F3)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_4496_root__def,axiom,
    ! [Na: nat,Xa: real] :
      aa(real,real,root(Na),Xa) = $ite(Na = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_mi(nat,fun(real,real),Na),Xa)) ).

% root_def
tff(fact_4497_less__eq__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A2),B2)
    <=> ( A2 = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A2),B2) ) ) ).

% less_eq_assn_def
tff(fact_4498_minus__assn__def,axiom,
    ! [A2: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),minus_minus(assn),A2),B2) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),A2),aa(assn,assn,uminus_uminus(assn),B2)) ).

% minus_assn_def
tff(fact_4499_less__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F4),F6)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F6),F4) ) ) ).

% less_filter_def
tff(fact_4500_bot__finite__def,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( bot_bot(A) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% bot_finite_def
tff(fact_4501_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_4502_Pow__fold,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( pow(A,A5) = finite_fold(A,set(set(A)),aTP_Lamp_lv(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A5) ) ) ).

% Pow_fold
tff(fact_4503_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
        & ! [Ys3: list(A)] :
            ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
        & ! [Ys3: list(A),Zs3: list(A)] :
            ( ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & member(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys3 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4504_times__assn__raw_Oelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ times_assn_raw(Xa,Xaa,Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ? [As12: set(nat),As23: set(nat)] :
                ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
                & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_4505_PowI,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => member(set(A),A5,pow(A,B5)) ) ).

% PowI
tff(fact_4506_Pow__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( member(set(A),A5,pow(A,B5))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% Pow_iff
tff(fact_4507_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : slice(A,I,I,Xs) = nil(A) ).

% slice_eq_bounds_empty
tff(fact_4508_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).

% slice_Nil
tff(fact_4509_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_4510_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_4511_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_4512_take0,axiom,
    ! [A: $tType,X3: list(A)] : take(A,zero_zero(nat),X3) = nil(A) ).

% take0
tff(fact_4513_take__eq__Nil,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( take(A,Na,Xs) = nil(A) )
    <=> ( ( Na = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_4514_take__eq__Nil2,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Na,Xs) )
    <=> ( ( Na = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_4515_sum__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ( groups8242544230860333062m_list(A,nil(A)) = zero_zero(A) ) ) ).

% sum_list.Nil
tff(fact_4516_empty__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( nil(A) = replicate(A,Na,Xa) )
    <=> ( Na = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4517_replicate__empty,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( replicate(A,Na,Xa) = nil(A) )
    <=> ( Na = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4518_Misc_Olast__in__set,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => member(A,last(A,L),aa(list(A),set(A),set2(A),L)) ) ).

% Misc.last_in_set
tff(fact_4519_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_4520_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A] : groups4207007520872428315er_sum(B,A,F3,A2,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_4521_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( linord4507533701916653071of_set(A,A5) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_4522_finite__Pow__iff,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),pow(A,A5))
    <=> aa(set(A),$o,finite_finite2(A),A5) ) ).

% finite_Pow_iff
tff(fact_4523_n__lists__Nil,axiom,
    ! [A: $tType,Na: nat] :
      n_lists(A,Na,nil(A)) = $ite(Na = zero_zero(nat),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_4524_Pow__UNIV,axiom,
    ! [A: $tType] : pow(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_4525_Pow__Int__eq,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A5)),pow(A,B5)) ).

% Pow_Int_eq
tff(fact_4526_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4527_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( linord4507533701916653071of_set(A,A5) = nil(A) )
          <=> ( A5 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4528_Pow__empty,axiom,
    ! [A: $tType] : pow(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_4529_Pow__singleton__iff,axiom,
    ! [A: $tType,X4: set(A),Y6: set(A)] :
      ( ( pow(A,X4) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),Y6),bot_bot(set(set(A)))) )
    <=> ( ( X4 = bot_bot(set(A)) )
        & ( Y6 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_4530_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_4531_Pow__def,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,A5) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_an(set(A),fun(set(A),$o),A5)) ).

% Pow_def
tff(fact_4532_PowD,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( member(set(A),A5,pow(A,B5))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% PowD
tff(fact_4533_less__assn__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less(assn),A2),B2)
    <=> ( aa(assn,$o,aa(assn,fun(assn,$o),ord_less_eq(assn),A2),B2)
        & ( A2 != B2 ) ) ) ).

% less_assn_def
tff(fact_4534_Pow__bottom,axiom,
    ! [A: $tType,B5: set(A)] : member(set(A),bot_bot(set(A)),pow(A,B5)) ).

% Pow_bottom
tff(fact_4535_Pow__not__empty,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,A5) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_4536_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs: list(A)] : foldr(A,list(A),cons(A),Xs,nil(A)) = Xs ).

% Misc.foldr_Cons
tff(fact_4537_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) ) ).

% neq_NilE
tff(fact_4538_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),$o)),W1: list(A),W22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),nil(B))
     => ( ! [E2: A,W12: list(A),W23: list(B)] :
            ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,W12),W23)
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),W12)),W23) )
       => ( ! [E2: B,W13: list(A),W24: list(B)] :
              ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,W13),W24)
             => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24)) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,W1),W22) ) ) ) ).

% list_2pre_induct
tff(fact_4539_list__induct__first2,axiom,
    ! [A: $tType,P2: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P2,nil(A))
     => ( ! [X: A] : aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))
       => ( ! [X15: A,X23: A,Xs2: list(A)] :
              ( aa(list(A),$o,P2,Xs2)
             => aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) )
         => aa(list(A),$o,P2,Xs) ) ) ) ).

% list_induct_first2
tff(fact_4540_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),$o)),R3: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( ! [Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),nil(B))
     => ( ! [Ys5: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),Ys5)
       => ( ! [X: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R3,X),Y3)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5))
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)) ) )
         => ( ! [X: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                ( ~ aa(B,$o,aa(A,fun(B,$o),R3,X),Y3)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),Ys5)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),Ys5)) ) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_4541_Pow__top,axiom,
    ! [A: $tType,A5: set(A)] : member(set(A),A5,pow(A,A5)) ).

% Pow_top
tff(fact_4542_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_4543_length__nth__simps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% length_nth_simps(1)
tff(fact_4544_len__greater__imp__nonempty,axiom,
    ! [A: $tType,Xa: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_4545_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4546_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_4547_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_4548_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,Xa: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,fun(B,C))] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [F2: fun(A,fun(B,C)),A4: A,As: list(A),B4: B,Bs: list(B)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs)))
       => ( ! [A4: fun(A,fun(B,C)),V: A,Va: list(A)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)),nil(B)))
         => ~ ! [A4: fun(A,fun(B,C)),V: B,Va: list(B)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va))) ) ) ) ).

% zipf.cases
tff(fact_4549_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: product_prod(list(A),list(A))] :
          ( ! [L22: list(A)] : Xa != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),L22)
         => ( ! [V: A,Va: list(A)] : Xa != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)),nil(A))
           => ~ ! [X15: A,L1: list(A),X23: A,L22: list(A)] : Xa != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),L22)) ) ) ) ).

% merge.cases
tff(fact_4550_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
      ( ! [P: fun(A,fun(B,$o))] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P: fun(A,fun(B,$o)),A4: A,As: list(A),B4: B,Bs: list(B)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B4),Bs)))
       => ( ! [P: fun(A,fun(B,$o)),V: A,Va: list(A)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)),nil(B)))
         => ~ ! [P: fun(A,fun(B,$o)),V: B,Va: list(B)] : Xa != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_4551_partition__rev_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P: fun(A,$o),Yes: list(A),No: list(A)] : Xa != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P: fun(A,$o),Yes: list(A),No: list(A),X: A,Xs2: list(A)] : Xa != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2))) ) ).

% partition_rev.cases
tff(fact_4552_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,$o)),Sl: list(A)] : Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
     => ~ ! [R7: fun(A,fun(A,$o)),Sl: list(A),X: A,Xs2: list(A)] : Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_4553_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,$o)),X: A,Xs2: list(A),Y3: A,Ys5: list(A)] : Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))
     => ( ! [R7: fun(A,fun(A,$o)),Xs2: list(A)] : Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs2),nil(A)))
       => ~ ! [R7: fun(A,fun(A,$o)),V: A,Va: list(A)] : Xa != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_4554_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,Xa: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs1: list(A),Xs22: list(A)] : Xa != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),nil(A))
     => ( ! [Xs1: list(A),Xs22: list(A),X: A] : Xa != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))
       => ~ ! [Xs1: list(A),Xs22: list(A),X15: A,X23: A,Xs2: list(A)] : Xa != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_4555_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4556_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_4557_replicate__0,axiom,
    ! [A: $tType,Xa: A] : replicate(A,zero_zero(nat),Xa) = nil(A) ).

% replicate_0
tff(fact_4558_Pow__mono,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow(A,A5)),pow(A,B5)) ) ).

% Pow_mono
tff(fact_4559_subset__Pow__Union,axiom,
    ! [A: $tType,A5: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A5),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A5))) ).

% subset_Pow_Union
tff(fact_4560_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B5: set(A)] :
      ( ( aa(set(B),set(A),image(B,A,F3),A5) = B5 )
     => ( aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F3)),pow(B,A5)) = pow(A,B5) ) ) ).

% image_Pow_surj
tff(fact_4561_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_list(A,Xa,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_4562_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_4563_Fpow__subset__Pow,axiom,
    ! [A: $tType,A5: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A5)),pow(A,A5)) ).

% Fpow_subset_Pow
tff(fact_4564_length__compl__induct,axiom,
    ! [A: $tType,P2: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P2,nil(A))
     => ( ! [E2: A,L2: list(A)] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P2,Ll) )
           => aa(list(A),$o,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L2)) )
       => aa(list(A),$o,P2,L) ) ) ).

% length_compl_induct
tff(fact_4565_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A4: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) ) ).

% list_decomp_1
tff(fact_4566_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ).

% sorted1
tff(fact_4567_butlast__subset,axiom,
    ! [A: $tType,Xs: list(A),A5: set(A)] :
      ( ( Xs != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),butlast(A,Xs))),A5) ) ) ).

% butlast_subset
tff(fact_4568_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),butlast(A,Xs)) ) ) ) ).

% sorted_butlast
tff(fact_4569_Un__Pow__subset,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A5)),pow(A,B5))),pow(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))) ).

% Un_Pow_subset
tff(fact_4570_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( sorted8670434370408473282of_set(A,B,Less_eq,F3,bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_4571_Pow__insert,axiom,
    ! [A: $tType,A2: A,A5: set(A)] : pow(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A5)),aa(set(set(A)),set(set(A)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2)),pow(A,A5))) ).

% Pow_insert
tff(fact_4572_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B5: fun(B,set(A)),A5: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_mj(fun(B,set(A)),fun(B,set(set(A))),B5)),A5))),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B5),A5)))) ).

% UN_Pow_subset
tff(fact_4573_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X)),aa(A,B,G3,X)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G3),Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_4574_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_4575_Fpow__Pow__finite,axiom,
    ! [A: $tType,A5: set(A)] : finite_Fpow(A,A5) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow(A,A5)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_4576_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A5: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A5)),B5)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F3)),pow(B,A5))),pow(A,B5)) ) ).

% image_Pow_mono
tff(fact_4577_binomial__def,axiom,
    ! [Na: nat,K: nat] : binomial(Na,K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_mk(nat,fun(nat,fun(set(nat),$o)),Na),K))) ).

% binomial_def
tff(fact_4578_take__Cons_H,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      take(A,Na,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = $ite(Na = zero_zero(nat),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_4579_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S2: set(B),F3: fun(B,A),A5: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A5),S2)
       => ( aa(set(B),$o,finite_finite2(B),A5)
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F3,A5) = nil(B) )
          <=> ( A5 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4580_times__assn__raw_Osimps,axiom,
    ! [P2: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q2: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As3: set(nat)] :
      ( times_assn_raw(P2,Q2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As3))
    <=> ? [As1: set(nat),As22: set(nat)] :
          ( ( As3 = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
          & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,P2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As1))
          & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Q2,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As22)) ) ) ).

% times_assn_raw.simps
tff(fact_4581_times__assn__raw_Oelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( times_assn_raw(Xa,Xaa,Xb)
      <=> (Y) )
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ( (Y)
            <=> ~ ? [As1: set(nat),As22: set(nat)] :
                    ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_4582_times__assn__raw_Oelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( times_assn_raw(Xa,Xaa,Xb)
     => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
            ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
           => ~ ? [As13: set(nat),As24: set(nat)] :
                  ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                  & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
                  & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As24)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_4583_times__assn__raw_Opelims_I1_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat)),Y: $o] :
      ( ( times_assn_raw(Xa,Xaa,Xb)
      <=> (Y) )
     => ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
             => ( ( (Y)
                <=> ? [As1: set(nat),As22: set(nat)] :
                      ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As1),As22) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As1),As22) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As1))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As22)) ) )
               => ~ aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_4584_times__assn__raw_Opelims_I2_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( times_assn_raw(Xa,Xaa,Xb)
     => ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
             => ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ~ ? [As13: set(nat),As24: set(nat)] :
                      ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As13),As24) )
                      & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As13),As24) = bot_bot(set(nat)) )
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As13))
                      & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_4585_times__assn__raw_Opelims_I3_J,axiom,
    ! [Xa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xaa: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ( ~ times_assn_raw(Xa,Xaa,Xb)
     => ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),Xb)))
       => ~ ! [H: heap_ext(product_unit),As: set(nat)] :
              ( ( Xb = aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As) )
             => ( aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),$o,accp(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),times_assn_raw_rel),aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),Xa),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Xaa),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As))))
               => ? [As12: set(nat),As23: set(nat)] :
                    ( ( As = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),As12),As23) )
                    & ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),As12),As23) = bot_bot(set(nat)) )
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As12))
                    & aa(product_prod(heap_ext(product_unit),set(nat)),$o,Xaa,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_4586_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xa ).

% concat_inth
tff(fact_4587_op__conc__empty__img__id,axiom,
    ! [A: $tType,L5: set(list(A))] : aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A))),L5) = L5 ).

% op_conc_empty_img_id
tff(fact_4588_empty__append__eq__id,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),X3) = X3 ).

% empty_append_eq_id
tff(fact_4589_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_4590_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),linord4507533701916653071of_set(nat,set_ord_atMost(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_4591_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E5: A,L23: list(A),E3: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_4592_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L12: list(A),E5: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_4593_list__se__match_I4_J,axiom,
    ! [A: $tType,L23: list(A),A2: A,L12: list(A)] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) ) ) ) ) ).

% list_se_match(4)
tff(fact_4594_list__se__match_I3_J,axiom,
    ! [A: $tType,L12: list(A),A2: A,L23: list(A)] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(3)
tff(fact_4595_list__se__match_I2_J,axiom,
    ! [A: $tType,L23: list(A),L12: list(A),A2: A] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) ) ) ) ) ).

% list_se_match(2)
tff(fact_4596_list__se__match_I1_J,axiom,
    ! [A: $tType,L12: list(A),L23: list(A),A2: A] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(1)
tff(fact_4597_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E12: A,E23: A,L23: list(A),E1: A,E22: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_4598_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E1: A,E22: A,L12: list(A),E12: A,E23: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_4599_nth__append__first,axiom,
    ! [A: $tType,I: nat,L: list(A),L3: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_append_first
tff(fact_4600_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_4601_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),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),Xa),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),Xa,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_4602_foldl__conc__empty__eq,axiom,
    ! [A: $tType,I: list(A),Ww: list(list(A))] : aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),I),Ww) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),I),aa(list(list(A)),list(A),aa(list(A),fun(list(list(A)),list(A)),foldl(list(A),list(A),append(A)),nil(A)),Ww)) ).

% foldl_conc_empty_eq
tff(fact_4603_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: product_prod(list(list(A)),list(list(A)))] :
          ( ( Xa != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L2: list(A)] : Xa != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
           => ( ! [La: list(A),Acc22: list(list(A))] : Xa != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))
             => ( ! [La: list(A),Acc22: list(list(A)),L2: list(A)] : Xa != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
               => ~ ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] : Xa != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_4604_list__match__lel__lel,axiom,
    ! [A: $tType,C12: list(A),Qs: A,C23: list(A),C13: list(A),Qs2: A,C24: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C23)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C24)) )
     => ( ! [C21: list(A)] :
            ( ( C12 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C13),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
           => ( C24 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C23)) ) )
       => ( ( ( C13 = C12 )
           => ( ( Qs2 = Qs )
             => ( C24 != C23 ) ) )
         => ~ ! [C212: list(A)] :
                ( ( C13 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
               => ( C23 != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C24)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_4605_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls) )
     => ~ ! [L2: list(B),L6: list(B)] :
            ( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L6) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_4606_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
     => ~ ! [L2: list(B),L6: list(B)] :
            ( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L2),L6) )
           => ( ( aa(list(B),list(A),map(B,A,F3),L2) = Fl )
             => ( aa(list(B),list(A),map(B,A,F3),L6) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_4607_Misc_Oappend__eq__map__conv,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F3: fun(B,A),Ls: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) = aa(list(B),list(A),map(B,A,F3),Ls) )
    <=> ? [L4: list(B),L7: list(B)] :
          ( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L7) )
          & ( aa(list(B),list(A),map(B,A,F3),L4) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L7) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_4608_Misc_Omap__eq__append__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Ls) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Fl),Fl2) )
    <=> ? [L4: list(B),L7: list(B)] :
          ( ( Ls = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L4),L7) )
          & ( aa(list(B),list(A),map(B,A,F3),L4) = Fl )
          & ( aa(list(B),list(A),map(B,A,F3),L7) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_4609_neq__Nil__revE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [Ll2: list(A),E2: A] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))) ) ).

% neq_Nil_revE
tff(fact_4610_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,P2: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),nil(B))
     => ( ! [X: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),nil(B))
       => ( ! [Y3: B,Ys5: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,nil(A)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))))
         => ( ! [X: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),Ys5)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_4611_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
    <=> ? [Xs3: list(A),X5: A] : L = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),nil(A))) ) ).

% neq_Nil_rev_conv
tff(fact_4612_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P2: fun(list(A),fun(list(B),$o))] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ! [X: A,Y3: B] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))
         => ( ! [X: A,Xs2: list(A),Y3: B] :
                ( ( Xs2 != nil(A) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B))) )
           => ( ! [X: A,Y3: B,Ys5: list(B)] :
                  ( ( Ys5 != nil(B) )
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))) )
             => ( ! [X: A,Xs2: list(A),Y3: B,Ys5: list(B)] :
                    ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs2),Ys5)
                   => ( ( Xs2 != nil(A) )
                     => ( ( Ys5 != nil(B) )
                       => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys5),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y3),nil(B)))) ) ) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P2,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_4613_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) ) )
       => ~ ! [Ys4: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys4) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),Zs) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_4614_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xa: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) ) )
       => ~ ! [Ys4: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Ys4) )
             => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),Zs) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_4615_xy__in__set__cases,axiom,
    ! [A: $tType,Xa: A,L: list(A),Y: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),L))
     => ( member(A,Y,aa(list(A),set(A),set2(A),L))
       => ( ( ( Xa = Y )
           => ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) )
         => ( ( ( Xa != Y )
             => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
           => ~ ( ( Xa != Y )
               => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L32)))) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_4616_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),L))
     => ~ ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L22)) ) ).

% in_set_list_format
tff(fact_4617_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R22: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S1),R1) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),S22),R22) )
       => ? [R1p: list(A)] : R22 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),R1p),R1) ) ) ).

% list_rest_coinc
tff(fact_4618_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W2: list(A),L: list(A),F3: fun(B,A),Ww: list(B),A2: B] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),Ww)),L) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F3,A2)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),L)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_4619_set__union__code,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% set_union_code
tff(fact_4620_distinct__match,axiom,
    ! [A: $tType,Al: list(A),E3: A,Bl: list(A),Al2: list(A),Bl2: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)))
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Al2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Bl2)) )
      <=> ( ( Al = Al2 )
          & ( Bl = Bl2 ) ) ) ) ).

% distinct_match
tff(fact_4621_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X5: A] :
                ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa3: A] :
                    ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa3) ) ) ) ) ) ).

% sorted_append
tff(fact_4622_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),I,Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I,Xa)),Ys) ) ) ).

% list_update_append1
tff(fact_4623_butlast__eq__consE,axiom,
    ! [A: $tType,L: list(A),Xa: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) )
     => ~ ! [Xl: A] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl),nil(A)))) ) ).

% butlast_eq_consE
tff(fact_4624_butlast__eq__cons__conv,axiom,
    ! [A: $tType,L: list(A),Xa: A,Xs: list(A)] :
      ( ( butlast(A,L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs) )
    <=> ? [Xl2: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xl2),nil(A)))) ) ).

% butlast_eq_cons_conv
tff(fact_4625_foldl__rule__aux__P,axiom,
    ! [A: $tType,B: $tType,I5: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F3: fun(A,fun(B,A)),P2: fun(A,$o)] :
      ( aa(list(B),$o,aa(A,fun(list(B),$o),I5,Sigma_0),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(A,fun(list(B),$o),I5,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(A,fun(list(B),$o),I5,aa(B,A,aa(A,fun(B,A),F3,Sigma),X)),L22) ) )
       => ( ! [Sigma: A] :
              ( aa(list(B),$o,aa(A,fun(list(B),$o),I5,Sigma),nil(B))
             => aa(A,$o,P2,Sigma) )
         => aa(A,$o,P2,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),Sigma_0),L0)) ) ) ) ).

% foldl_rule_aux_P
tff(fact_4626_foldl__rule__aux,axiom,
    ! [A: $tType,B: $tType,I5: fun(A,fun(list(B),$o)),Sigma_0: A,L0: list(B),F3: fun(A,fun(B,A))] :
      ( aa(list(B),$o,aa(A,fun(list(B),$o),I5,Sigma_0),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(A,fun(list(B),$o),I5,Sigma),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(A,fun(list(B),$o),I5,aa(B,A,aa(A,fun(B,A),F3,Sigma),X)),L22) ) )
       => aa(list(B),$o,aa(A,fun(list(B),$o),I5,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),Sigma_0),L0)),nil(B)) ) ) ).

% foldl_rule_aux
tff(fact_4627_foldl__rule__P,axiom,
    ! [A: $tType,B: $tType,I5: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F3: fun(A,fun(B,A)),P2: fun(A,$o)] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,Sigma_0),nil(B)),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,aa(B,A,aa(A,fun(B,A),F3,Sigma),X)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))),L22) ) )
       => ( ! [Sigma: A] :
              ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,Sigma),L0),nil(B))
             => aa(A,$o,P2,Sigma) )
         => aa(A,$o,P2,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),Sigma_0),L0)) ) ) ) ).

% foldl_rule_P
tff(fact_4628_foldl__rule,axiom,
    ! [A: $tType,B: $tType,I5: fun(A,fun(list(B),fun(list(B),$o))),Sigma_0: A,L0: list(B),F3: fun(A,fun(B,A))] :
      ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,Sigma_0),nil(B)),L0)
     => ( ! [L1: list(B),L22: list(B),X: B,Sigma: A] :
            ( ( L0 = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22)) )
           => ( aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,Sigma),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),L22))
             => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,aa(B,A,aa(A,fun(B,A),F3,Sigma),X)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),L1),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)))),L22) ) )
       => aa(list(B),$o,aa(list(B),fun(list(B),$o),aa(A,fun(list(B),fun(list(B),$o)),I5,aa(list(B),A,aa(A,fun(list(B),A),foldl(A,B,F3),Sigma_0),L0)),L0),nil(B)) ) ) ).

% foldl_rule
tff(fact_4629_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R2))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_4630_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lex(A,R2))
      <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_4631_map__by__foldl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),L: list(B)] : aa(list(B),list(A),aa(list(A),fun(list(B),list(A)),foldl(list(A),B,aTP_Lamp_ml(fun(B,A),fun(list(A),fun(B,list(A))),F3)),nil(A)),L) = aa(list(B),list(A),map(B,A,F3),L) ).

% map_by_foldl
tff(fact_4632_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_4633_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Na) )
    <=> ? [Y5: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Na ) ) ) ).

% length_Suc_conv_rev
tff(fact_4634_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Na) )
    <=> ? [Ys3: list(A),Y5: A] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Na ) ) ) ).

% length_Suc_rev_conv
tff(fact_4635_length__compl__rev__induct,axiom,
    ! [A: $tType,P2: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P2,nil(A))
     => ( ! [L2: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P2,Ll) )
           => aa(list(A),$o,P2,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)))) )
       => aa(list(A),$o,P2,L) ) ) ).

% length_compl_rev_induct
tff(fact_4636_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Na),aa(nat,A,nth(A,Ys),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_4637_not__distinct__split__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ~ distinct(A,Xs)
     => ~ ! [Y3: A,Ys5: list(A)] :
            ( distinct(A,Ys5)
           => ( member(A,Y3,aa(list(A),set(A),set2(A),Ys5))
             => ! [Zs2: list(A)] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys5),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))),Zs2)) ) ) ) ).

% not_distinct_split_distinct
tff(fact_4638_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,Na: nat,Xa: A] : replicate(A,aa(nat,nat,suc,Na),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,Na,Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_4639_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Na: nat,Xa: A] :
      list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Na,Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,Na,Xa)),Ys),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),list_update(A,Ys,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)),Xa))) ).

% list_update_append
tff(fact_4640_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N)
              & ( concat(A,replicate(list(A),N,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_4641_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
     => ~ ! [X: A,Y3: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),R2)
           => ! [Us: list(A),Vs: list(A)] :
                ( ( 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)) ) ) ) ) ).

% listrel1E
tff(fact_4642_listrel1I,axiom,
    ! [A: $tType,Xa: A,Y: A,R2: set(product_prod(A,A)),Xs: list(A),Us2: list(A),Vs2: list(A),Ys: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2)
     => ( ( 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),Xa),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),Y),Vs2)) )
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2)) ) ) ) ).

% listrel1I
tff(fact_4643_snoc__eq__iff__butlast_H,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A),Xa: A] :
      ( ( Ys = 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),Xa),nil(A))) )
    <=> ( ( Ys != nil(A) )
        & ( butlast(A,Ys) = Xs )
        & ( last(A,Ys) = Xa ) ) ) ).

% snoc_eq_iff_butlast'
tff(fact_4644_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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),Xa)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_4645_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
       => $let(
            p: nat,
            p:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs)) ) ) ) ).

% slice_prepend
tff(fact_4646_sorted__append__bigger,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y) )
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ) ) ) ).

% sorted_append_bigger
tff(fact_4647_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys: list(A),Y: A,R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))),listrel1(A,R2))
    <=> ( ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),listrel1(A,R2))
          & ( Xa = Y ) )
        | ( ( Xs = Ys )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R2) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_4648_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A2) )
           => ( linorder_insort_key(A,A,aTP_Lamp_jo(A,A),A2,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),A2),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_4649_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_4650_take__update__last,axiom,
    ! [A: $tType,Na: nat,List: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),List))
     => ( list_update(A,take(A,aa(nat,nat,suc,Na),List),Na,Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Na,List)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ) ).

% take_update_last
tff(fact_4651_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Na: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( M != Na )
         => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Na,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_4652_pos__n__replace,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Na,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))),drop(A,aa(nat,nat,suc,Na),Xs)))) ) ) ).

% pos_n_replace
tff(fact_4653_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = Y )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xa),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_4654_drop0,axiom,
    ! [A: $tType,X3: list(A)] : drop(A,zero_zero(nat),X3) = X3 ).

% drop0
tff(fact_4655_drop__Suc__Cons,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,Na),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = drop(A,Na,Xs) ).

% drop_Suc_Cons
tff(fact_4656_drop__update__cancel,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( drop(A,M,list_update(A,Xs,Na,Xa)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_4657_drop__upd__irrelevant,axiom,
    ! [A: $tType,M: nat,Na: nat,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( drop(A,Na,list_update(A,L,M,Xa)) = drop(A,Na,L) ) ) ).

% drop_upd_irrelevant
tff(fact_4658_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_4659_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_4660_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_4661_drop__all,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na)
     => ( drop(A,Na,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_4662_drop__eq__Nil,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( drop(A,Na,Xs) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% drop_eq_Nil
tff(fact_4663_drop__eq__Nil2,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,Na,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% drop_eq_Nil2
tff(fact_4664_last__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( last(A,drop(A,Na,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_4665_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_4666_nth__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,drop(A,Na,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),I)) ) ) ).

% nth_drop
tff(fact_4667_drop__last__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ( drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(nat,nat,suc,zero_zero(nat))),L) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),last(A,L)),nil(A)) ) ) ).

% drop_last_conv
tff(fact_4668_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_4669_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_4670_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt(int,ord_less_eq(int),upto(M,Na)) ).

% sorted_upto
tff(fact_4671_drop__eq__ConsD,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Xa: A,Xs4: list(A)] :
      ( ( drop(A,Na,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs4) )
     => ( drop(A,aa(nat,nat,suc,Na),Xs) = Xs4 ) ) ).

% drop_eq_ConsD
tff(fact_4672_set__drop__subset,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Na,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_4673_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Na: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,Na,Xs)) ) ) ).

% sorted_drop
tff(fact_4674_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,Na,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_4675_drop__update__swap,axiom,
    ! [A: $tType,M: nat,Na: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( drop(A,M,list_update(A,Xs,Na,Xa)) = list_update(A,drop(A,M,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M),Xa) ) ) ).

% drop_update_swap
tff(fact_4676_drop__take__drop__unsplit,axiom,
    ! [A: $tType,I: nat,J: nat,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,I,take(A,J,L))),drop(A,J,L)) = drop(A,I,L) ) ) ).

% drop_take_drop_unsplit
tff(fact_4677_Misc_Oslice__def,axiom,
    ! [A: $tType,From: nat,To: nat,List: list(A)] : slice(A,From,To,List) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),To),From),drop(A,From,List)) ).

% Misc.slice_def
tff(fact_4678_Union__take__drop__id,axiom,
    ! [A: $tType,Na: nat,L: list(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),Na,L)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),take(set(A),Na,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),L)) ).

% Union_take_drop_id
tff(fact_4679_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_4680_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_4681_drop__Cons_H,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      drop(A,Na,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = $ite(Na = zero_zero(nat),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_4682_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
          & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ),
          ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
          & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_4683_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_4684_in__set__drop__conv__nth,axiom,
    ! [A: $tType,Xa: A,Na: nat,L: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),drop(A,Na,L)))
    <=> ? [I3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),L))
          & ( Xa = aa(nat,A,nth(A,L),I3) ) ) ) ).

% in_set_drop_conv_nth
tff(fact_4685_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_4686_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xa,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_4687_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_4688_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_4689_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs2: list(A),I: nat,J: nat] :
      ( distinct(A,Vs2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs2))),aa(list(A),set(A),set2(A),drop(A,J,Vs2))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_4690_foldl__list__update,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs: list(A),F3: fun(B,fun(A,B)),A2: B,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A2),list_update(A,Xs,Na,Xa)) = aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),aa(A,B,aa(B,fun(A,B),F3,aa(list(A),B,aa(B,fun(list(A),B),foldl(B,A,F3),A2),take(A,Na,Xs))),Xa)),drop(A,aa(nat,nat,suc,Na),Xs)) ) ) ).

% foldl_list_update
tff(fact_4691_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I,aa(int,int,aa(int,fun(int,int),minus_minus(int),J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_4692_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_4693_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A2: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_4694_Un__set__drop__extend,axiom,
    ! [A: $tType,J: nat,L: list(set(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(set(A)),nat,size_size(list(set(A))),L))
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(nat,set(A),nth(set(A),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),J,L)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(list(set(A)),set(set(A)),set2(set(A)),drop(set(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,zero_zero(nat))),L))) ) ) ) ).

% Un_set_drop_extend
tff(fact_4695_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_4696_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_4697_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_4698_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_4699_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ) ).

% numeral_le_iff
tff(fact_4700_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ) ).

% numeral_less_iff
tff(fact_4701_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),M) ) ) ).

% neg_numeral_le_iff
tff(fact_4702_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Na),M) ) ) ).

% neg_numeral_less_iff
tff(fact_4703_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(3)
tff(fact_4704_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(4)
tff(fact_4705_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)),aa(num,A,numeral_numeral(A),V3),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_4706_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) = A2 )
        <=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_4707_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2)) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_4708_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_4709_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_4710_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_4711_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W2))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_4712_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_4713_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)),aa(num,A,numeral_numeral(A),V3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_4714_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_4715_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_4716_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W2: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_4717_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W2: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A2 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_4718_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_4719_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_4720_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_4721_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W2: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_4722_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_le_numeral_iff
tff(fact_4723_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_4724_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_less_numeral_iff
tff(fact_4725_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_4726_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V3)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V3)),Xa) ) ) ).

% numeral_le_floor
tff(fact_4727_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),V3)) ) ) ).

% floor_less_numeral
tff(fact_4728_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(num,A,numeral_numeral(A),V3)) ) ) ).

% ceiling_le_numeral
tff(fact_4729_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V3)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V3)),Xa) ) ) ).

% numeral_less_ceiling
tff(fact_4730_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_4731_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_4732_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_4733_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_4734_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V3)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))),Xa) ) ) ).

% numeral_less_floor
tff(fact_4735_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_4736_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_4737_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V3)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(num,A,numeral_numeral(A),V3)),one_one(A))),Xa) ) ) ).

% numeral_le_ceiling
tff(fact_4738_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4739_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4740_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4741_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4742_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),Xa) ) ) ).

% neg_numeral_le_floor
tff(fact_4743_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) ) ) ).

% floor_less_neg_numeral
tff(fact_4744_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_4745_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),Xa) ) ) ).

% neg_numeral_less_ceiling
tff(fact_4746_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),Na))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_4747_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))),Xa) ) ) ).

% neg_numeral_less_floor
tff(fact_4748_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_4749_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V3: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_4750_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V3: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V3))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),one_one(A))),Xa) ) ) ).

% neg_numeral_le_ceiling
tff(fact_4751_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4752_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4753_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4754_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4755_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Na) ) ).

% zero_neq_numeral
tff(fact_4756_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_4757_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_4758_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_le_numeral
tff(fact_4759_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_less_numeral
tff(fact_4760_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_4761_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_le_numeral
tff(fact_4762_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_4763_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_neq_neg_numeral
tff(fact_4764_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_le_neg_numeral
tff(fact_4765_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_le_numeral
tff(fact_4766_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_less_neg_numeral
tff(fact_4767_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_less_numeral
tff(fact_4768_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_le_neg_numeral
tff(fact_4769_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_4770_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_4771_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_less_neg_numeral
tff(fact_4772_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3) = B2,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_4773_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) = aa(num,A,numeral_numeral(A),W2) )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_4774_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_4775_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_4776_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_4777_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_4778_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_4779_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_4780_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_4781_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_4782_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_4783_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_4784_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_4785_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_4786_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C3: A,W2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(C3 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_4787_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B2: A,C3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4788_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_4789_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_4790_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_4791_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_4792_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_4793_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B2),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_4794_distinct__concat_H,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),filter2(list(A),aTP_Lamp_mm(list(A),$o),Xs))
     => ( ! [Ys5: list(A)] :
            ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys5) )
       => ( ! [Ys5: list(A),Zs2: list(A)] :
              ( member(list(A),Ys5,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys5 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys5)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat'
tff(fact_4795_merge__list__correct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ls: list(list(A)),As3: list(list(A))] :
          ( ! [L2: list(A)] :
              ( member(list(A),L2,aa(list(list(A)),set(list(A)),set2(list(A)),Ls))
             => ( distinct(A,L2)
                & sorted_wrt(A,ord_less_eq(A),L2) ) )
         => ( ! [L2: list(A)] :
                ( member(list(A),L2,aa(list(list(A)),set(list(A)),set2(list(A)),As3))
               => ( distinct(A,L2)
                  & sorted_wrt(A,ord_less_eq(A),L2) ) )
           => ( distinct(A,merge_list(A,As3,Ls))
              & sorted_wrt(A,ord_less_eq(A),merge_list(A,As3,Ls))
              & ( aa(list(A),set(A),set2(A),merge_list(A,As3,Ls)) = aa(list(A),set(A),set2(A),concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),As3),Ls))) ) ) ) ) ) ).

% merge_list_correct
tff(fact_4796_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)) = 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_4797_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y))
        <=> ( zero_zero(word(A)) != aa(num,word(A),numeral_numeral(word(A)),Y) ) ) ) ).

% word_gt_0_no
tff(fact_4798_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_4799_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W2)),aa(nat,real,semiring_1_of_nat(real),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_4800_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(num,real,numeral_numeral(real),W2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_4801_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Na)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_4802_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W2) != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W2))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W2)),one_one(word(A)))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_4803_word__less__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W2) != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W2))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W2)),one_one(word(A)))) ) ) ) ).

% word_less_sub1_numberof
tff(fact_4804_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Na: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,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)),Na))),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_4805_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Na)) ) ) ).

% powr_numeral
tff(fact_4806_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_4807_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_4808_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_4809_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_4810_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( filter2(A,aTP_Lamp_mn(list(A),fun(A,$o),Ys),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_4811_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( filter2(A,aTP_Lamp_mo(list(A),fun(A,$o),Ys),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_4812_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( filter2(A,aTP_Lamp_mn(list(A),fun(A,$o),Xs),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_4813_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys))
       => ( filter2(A,aTP_Lamp_mo(list(A),fun(A,$o),Xs),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_4814_length__filter__le,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_4815_filter__is__subset,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),filter2(A,P2,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_4816_sorted__filter_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => sorted_wrt(A,ord_less_eq(A),filter2(A,P2,L)) ) ) ).

% sorted_filter'
tff(fact_4817_last__filter,axiom,
    ! [A: $tType,Xs: list(A),P2: fun(A,$o)] :
      ( ( Xs != nil(A) )
     => ( aa(A,$o,P2,last(A,Xs))
       => ( last(A,filter2(A,P2,Xs)) = last(A,Xs) ) ) ) ).

% last_filter
tff(fact_4818_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G3: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_mp(fun(list(A),A),fun(list(A),fun(A,$o)),G3),Xs),Xs)) ) ).

% sorted_same
tff(fact_4819_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,filter2(list(A),aTP_Lamp_mm(list(A),$o),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_4820_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) ).

% Cons_shuffles_subset2
tff(fact_4821_Cons__shuffles__subset1,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ys: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa)),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),Ys)) ).

% Cons_shuffles_subset1
tff(fact_4822_length__filter__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),P2: fun(A,$o)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P2,Xa)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_4823_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),filter2(B,P2,Xs))) ) ) ).

% sorted_filter
tff(fact_4824_merge__list_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( merge_list(A,nil(list(A)),nil(list(A))) = nil(A) ) ) ).

% merge_list.simps(1)
tff(fact_4825_merge__list_Osimps_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A)),L: list(A)] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23))) ) ).

% merge_list.simps(4)
tff(fact_4826_merge__list_Osimps_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [La2: list(A),Acc23: list(list(A))] : merge_list(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23),nil(list(A))) = merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La2),Acc23)) ) ).

% merge_list.simps(3)
tff(fact_4827_merge__list_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] : merge_list(A,nil(list(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L),nil(list(A)))) = L ) ).

% merge_list.simps(2)
tff(fact_4828_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G3: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_mq(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F3),G3),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_4829_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F3: fun(B,A),P2: fun(B,$o),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),filter2(B,P2,Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_mr(fun(B,A),fun(fun(B,$o),fun(B,A)),F3),P2)),Xs)) ) ).

% sum_list_map_filter'
tff(fact_4830_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),filter2(A,P2,Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))) ).

% sum_list_filter_le_nat
tff(fact_4831_filter__eq__snocD,axiom,
    ! [A: $tType,P2: fun(A,$o),L: list(A),L3: list(A),Xa: A] :
      ( ( filter2(A,P2,L) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),L))
        & aa(A,$o,P2,Xa) ) ) ).

% filter_eq_snocD
tff(fact_4832_filter__conv__foldr,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : filter2(A,P2,Xs) = foldr(A,list(A),aTP_Lamp_ms(fun(A,$o),fun(A,fun(list(A),list(A))),P2),Xs,nil(A)) ).

% filter_conv_foldr
tff(fact_4833_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Na))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),Na))) ) ) ).

% powr_neg_numeral
tff(fact_4834_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P2: fun(A,$o),F3: fun(A,B)] :
          ( ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P2,X)
               => ( aa(A,B,F3,X) = zero_zero(B) ) ) )
         => ( groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),filter2(A,P2,Xs))) = groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_4835_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P2: fun(B,$o),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( aa(B,$o,P2,Xa)
           => ( filter2(B,P2,linorder_insort_key(B,A,F3,Xa,Xs)) = linorder_insort_key(B,A,F3,Xa,filter2(B,P2,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_4836_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_mt(A,fun(A,$o),Y),Xs)) ).

% set_minus_filter_out
tff(fact_4837_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P3,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_mu(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_4838_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( member(list(A),Zs,shuffles(A,Xs,Ys))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_4839_filter__nth__ex__nth,axiom,
    ! [A: $tType,Na: nat,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),filter2(A,P2,Xs)))
     => ? [M3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,filter2(A,P2,Xs)),Na) = aa(nat,A,nth(A,Xs),M3) )
          & ( filter2(A,P2,take(A,M3,Xs)) = take(A,Na,filter2(A,P2,Xs)) ) ) ) ).

% filter_nth_ex_nth
tff(fact_4840_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_mv(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_mw(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_mx(list(B),$o),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_4841_mergesort__remdups__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : mergesort_remdups(A,Xs) = merge_list(A,nil(list(A)),aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_my(A,list(A))),Xs)) ) ).

% mergesort_remdups_def
tff(fact_4842_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(1)
tff(fact_4843_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(2)
tff(fact_4844_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
    <=> ( Na != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_4845_enat__less__induct,axiom,
    ! [P2: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M2),N)
             => aa(extended_enat,$o,P2,M2) )
         => aa(extended_enat,$o,P2,N) )
     => aa(extended_enat,$o,P2,Na) ) ).

% enat_less_induct
tff(fact_4846_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_4847_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_4848_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xa: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),Y),Z)) = aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),minus_minus(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_4849_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),zero_zero(extended_enat))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_4850_i0__lb,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Na) ).

% i0_lb
tff(fact_4851_insert_H__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_set_vebt(vEBT_VEBT_insert(T2,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),inf_inf(set(nat)),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat))))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ) ).

% insert'_correct
tff(fact_4852_setceilmax,axiom,
    ! [Sa: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Na: nat] :
      ( vEBT_invar_vebt(Sa,M)
     => ( ! [X: vEBT_VEBT] :
            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X,Na) )
       => ( ( M = aa(nat,nat,suc,Na) )
         => ( ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Sa)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),aa(vEBT_VEBT,fun(set(vEBT_VEBT),set(vEBT_VEBT)),insert(vEBT_VEBT),Sa),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_4853_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_mz(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_mm(list(A),$o),Xs)) ).

% transpose_max_length
tff(fact_4854_pow__sum,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_4855_mulcomm,axiom,
    ! [I: nat,Va2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),I) ).

% mulcomm
tff(fact_4856_two__realpow__ge__two,axiom,
    ! [Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Na))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ) ).

% two_realpow_ge_two
tff(fact_4857_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
     => ( vEBT_invar_vebt(Tree,Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% member_bound
tff(fact_4858_valid__pres__insert,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => vEBT_invar_vebt(vEBT_vebt_insert(T2,Xa),Na) ) ) ).

% valid_pres_insert
tff(fact_4859_misiz,axiom,
    ! [T2: vEBT_VEBT,Na: nat,M: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(T2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% misiz
tff(fact_4860_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_4861_two__powr__height__bound__deg,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% two_powr_height_bound_deg
tff(fact_4862_helpyd,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_succ(T2,Xa) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% helpyd
tff(fact_4863_helpypredd,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( vEBT_vebt_pred(T2,Xa) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% helpypredd
tff(fact_4864_valid__insert__both__member__options__add,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T2,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_4865_valid__insert__both__member__options__pres,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(T2),Xa)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(T2,Y)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_4866_post__member__pre__member,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(T2,Xa)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(T2),Y)
              | ( Xa = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_4867_log__ceil__idem,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),Xa)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xa)))) ) ) ).

% log_ceil_idem
tff(fact_4868_count__buildup,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na))) ).

% count_buildup
tff(fact_4869_delt__out__of__range,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% delt_out_of_range
tff(fact_4870_del__single__cont,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary) ) ) ) ).

% del_single_cont
tff(fact_4871_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(78)
tff(fact_4872_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(71)
tff(fact_4873_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,M: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na) ) ) ).

% set_n_deg_not_0
tff(fact_4874_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),one2) ).

% semiring_norm(75)
tff(fact_4875_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Na) ).

% semiring_norm(68)
tff(fact_4876_tdeletemimi_H,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),Xa)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_4877_count__buildup_H,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) ).

% count_buildup'
tff(fact_4878_heigt__uplog__rel,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ).

% heigt_uplog_rel
tff(fact_4879_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_4880_succ__min,axiom,
    ! [Deg: nat,Xa: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_4881_pred__max,axiom,
    ! [Deg: nat,Ma: nat,Xa: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_4882_cnt__bound_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% cnt_bound'
tff(fact_4883_delete__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_4884_bit__concat__def,axiom,
    ! [H2: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H2,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% bit_concat_def
tff(fact_4885_TBOUND__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND($o,vEBT_V854960066525838166emberi(T2,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)))) ).

% TBOUND_vebt_memberi
tff(fact_4886_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,T2: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(T2,Deg)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_4887_inrange,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(T2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% inrange
tff(fact_4888_Suc__numeral,axiom,
    ! [Na: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Na)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% Suc_numeral
tff(fact_4889_insert__correct,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) = vEBT_set_vebt(vEBT_vebt_insert(T2,Xa)) ) ) ) ).

% insert_correct
tff(fact_4890_set__vebt__insert,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( vEBT_set_vebt(vEBT_vebt_insert(T2,Xa)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) ) ) ) ).

% set_vebt_insert
tff(fact_4891_insert__corr,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_VEBT_set_vebt(T2)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat)))) = vEBT_VEBT_set_vebt(vEBT_vebt_insert(T2,Xa)) ) ) ) ).

% insert_corr
tff(fact_4892_zdiv__numeral__Bit0,axiom,
    ! [V3: num,W2: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V3))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V3)),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit0
tff(fact_4893_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Na)) ).

% semiring_norm(76)
tff(fact_4894_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_4895_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Ma)
          & ? [M3: nat] :
              ( ( aa(nat,option(nat),some(nat),M3) = vEBT_vebt_mint(Summary) )
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),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),Na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_4896_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_4897_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_4898_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_4899_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc'
tff(fact_4900_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc
tff(fact_4901_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_4902_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_4903_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_4904_zmod__numeral__Bit0,axiom,
    ! [V3: num,W2: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit0(V3))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V3)),aa(num,int,numeral_numeral(int),W2))) ).

% zmod_numeral_Bit0
tff(fact_4905_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),one2) ) ) ).

% numeral_le_one_iff
tff(fact_4906_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Na) ) ) ).

% one_less_numeral_iff
tff(fact_4907_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_4908_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_4909_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_4910_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xa = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_4911_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_4912_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_4913_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_4914_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_4915_card__doubleton__eq__2__iff,axiom,
    ! [A: $tType,A2: A,B2: A] :
      ( ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ( A2 != B2 ) ) ).

% card_doubleton_eq_2_iff
tff(fact_4916_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_4917_add__self__mod__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,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_4918_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_4919_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_4920_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_4921_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa) ) ) ).

% one_less_floor
tff(fact_4922_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_4923_card__bit1,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ( aa(set(numeral_bit1(A)),nat,finite_card(numeral_bit1(A)),top_top(set(numeral_bit1(A)))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ) ).

% card_bit1
tff(fact_4924_butlast__upd__last__eq,axiom,
    ! [A: $tType,L: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),L))
     => ( list_update(A,butlast(A,L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),L)),aa(num,nat,numeral_numeral(nat),bit0(one2))),L)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) ) ) ).

% butlast_upd_last_eq
tff(fact_4925_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)) ) ).

% self_le_ge2_pow
tff(fact_4926_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_4927_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_4928_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_4929_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = one_one(word(A)) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% p2_eq_1
tff(fact_4930_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_4931_double__not__eq__Suc__double,axiom,
    ! [M: nat,Na: 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))),Na)) ).

% double_not_eq_Suc_double
tff(fact_4932_Suc__double__not__eq__double,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_double_not_eq_double
tff(fact_4933_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q3))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(6)
tff(fact_4934_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(8)
tff(fact_4935_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_4936_pos__mod__bound2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_4937_n__less__equal__power__2,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% n_less_equal_power_2
tff(fact_4938_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xa),one2)
    <=> ( Xa = one2 ) ) ).

% le_num_One_iff
tff(fact_4939_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_4940_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_4941_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_4942_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_4943_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_4944_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xa = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_4945_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_4946_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_4947_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: 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),Na)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_4948_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: 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),Na)) != 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_4949_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_4950_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != 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),Na),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_4951_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_4952_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_4953_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_4954_nat__induct2,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( aa(nat,$o,P2,one_one(nat))
       => ( ! [N: nat] :
              ( aa(nat,$o,P2,N)
             => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P2,Na) ) ) ) ).

% nat_induct2
tff(fact_4955_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
     => ? [A4: A,B4: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B4),nil(A))) ) ).

% list_decomp_2
tff(fact_4956_two__le__card,axiom,
    ! [A: $tType] :
      ( card2(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% two_le_card
tff(fact_4957_card__2__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X5: A,Y5: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),bot_bot(set(A)))) )
          & ( X5 != Y5 ) ) ) ).

% card_2_iff
tff(fact_4958_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Na))) ) ).

% diff_le_diff_pow
tff(fact_4959_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_4960_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_4961_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K7)),binomial(Na,K)) ) ) ) ).

% binomial_antimono
tff(fact_4962_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),binomial(Na,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_4963_pos__mod__sign2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_4964_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_4965_binomial__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),binomial(Na,K7)) ) ) ).

% binomial_mono
tff(fact_4966_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),K)),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),Na)) ).

% binomial_maximum'
tff(fact_4967_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_4968_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% binomial_le_pow2
tff(fact_4969_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_4970_mod__exp__less__eq__exp,axiom,
    ! [A2: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% mod_exp_less_eq_exp
tff(fact_4971_ln__2__less__1,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_4972_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_4973_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_4974_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_4975_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_4976_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_4977_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xa != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_4978_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_4979_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),modulo_modulo(A),A2),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),modulo_modulo(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4980_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_4981_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_4982_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% zero_le_even_power'
tff(fact_4983_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: fun(A,fun(A,$o)),Xa: A] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X)
             => aa(A,$o,aa(A,fun(A,$o),P2,X),aa(nat,A,aa(A,fun(nat,A),power_power(A),X),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P2,aa(A,A,abs_abs(A),Xa)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_4984_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_4985_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_4986_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_4987_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Na: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_4988_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_4989_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_4990_nat__bit__induct,axiom,
    ! [P2: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P2,zero_zero(nat))
     => ( ! [N: nat] :
            ( aa(nat,$o,P2,N)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => aa(nat,$o,P2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) )
       => ( ! [N: nat] :
              ( aa(nat,$o,P2,N)
             => aa(nat,$o,P2,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(nat,$o,P2,Na) ) ) ) ).

% nat_bit_induct
tff(fact_4991_two__pow__div__gt__le,axiom,
    ! [V3: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),V3),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% two_pow_div_gt_le
tff(fact_4992_nat__add__offset__less,axiom,
    ! [Y: nat,Na: nat,Xa: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
       => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_4993_nat__power__less__diff,axiom,
    ! [Na: nat,Q3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Q3),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),M),Na))) ) ).

% nat_power_less_diff
tff(fact_4994_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( 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),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_4995_nat__le__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),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),M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_le_power_trans
tff(fact_4996_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K)),binomial(Na,aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_4997_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K)),binomial(Na,K7)) ) ) ).

% binomial_strict_mono
tff(fact_4998_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K7)),binomial(Na,K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_4999_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Na)),semiring_char_0_fact(real,Na))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% square_fact_le_2_fact
tff(fact_5000_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_5001_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% exp_half_le2
tff(fact_5002_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_5003_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_5004_triangle__def,axiom,
    ! [Na: nat] : nat_triangle(Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_5005_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_5006_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),A2),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),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_5007_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_5008_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_5009_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Na: 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))),Na))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_5010_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),Na),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_5011_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_5012_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_5013_less__two__pow__divD,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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),Na),M))) ) ) ).

% less_two_pow_divD
tff(fact_5014_less__two__pow__divI,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% less_two_pow_divI
tff(fact_5015_nat__less__power__trans,axiom,
    ! [Na: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),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),M),K)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_less_power_trans
tff(fact_5016_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q3: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) = zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_5017_num_Osize_I5_J,axiom,
    ! [X2: num] : aa(num,nat,size_size(num),bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_5018_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C3: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C3))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_5019_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))) ) ).

% plus_inverse_ge_2
tff(fact_5020_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_5021_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Na))) ) ).

% mask_eq_sum_exp
tff(fact_5022_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_5023_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_5024_Suc__nat__number__of__add,axiom,
    ! [V3: num,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V3)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V3),one2))),Na) ).

% Suc_nat_number_of_add
tff(fact_5025_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_na(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.in_pairs
tff(fact_5026_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),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))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_na(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Na)) ) ).

% sum.in_pairs_0
tff(fact_5027_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nb(fun(nat,A),fun(nat,A),G3)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.in_pairs
tff(fact_5028_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),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))),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nb(fun(nat,A),fun(nat,A),G3)),set_ord_atMost(nat,Na)) ) ).

% prod.in_pairs_0
tff(fact_5029_card__Pow,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow(A,A5)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A5)) ) ) ).

% card_Pow
tff(fact_5030_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),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),modulo_modulo(A),Xa),M) )
              | ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),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),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xa),M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_5031_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),modulo_modulo(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_5032_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_VEBT_insert(vEBT_Node(Info,Deg,TreeList,Summary),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)),Xa),vEBT_Node(Info,Deg,TreeList,Summary),vEBT_vebt_insert(vEBT_Node(Info,Deg,TreeList,Summary),Xa)) ).

% VEBT_internal.insert'.simps(2)
tff(fact_5033_nat__div__eq__Suc__0__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_5034_power__2__mult__step__le,axiom,
    ! [N2: nat,Na: nat,K7: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)),K7)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K7),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_5035_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_5036_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_5037_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_5038_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_5039_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_5040_exp__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_5041_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2)) ) ).

% sb_dec_lem'
tff(fact_5042_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_5043_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_5044_mask__eq__sum__exp__nat,axiom,
    ! [Na: 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))),Na)),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,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_5045_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_de(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Na))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_5046_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_5047_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xa: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Y) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_5048_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != vEBT_vebt_insert(vEBT_Leaf((A4),(B4)),Xaa) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xaa),vEBT_Node(Info2,Deg2,TreeList2,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg2,TreeList2,Summary2),Xaa)) ) ) ) ) ).

% VEBT_internal.insert'.elims
tff(fact_5049_card__UNIV__set,axiom,
    ! [A: $tType] :
      aa(set(set(A)),nat,finite_card(set(A)),top_top(set(set(A)))) = $ite(aa(set(A),nat,finite_card(A),top_top(set(A))) = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),top_top(set(A))))) ).

% card_UNIV_set
tff(fact_5050_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_less
tff(fact_5051_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem
tff(fact_5052_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A2),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem'
tff(fact_5053_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = 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))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_5054_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A2)) ) ).

% sb_dec_lem
tff(fact_5055_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_5056_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5057_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_5058_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: 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),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_5059_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,Na: 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_nc(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = 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),Na)),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))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2))) ) ).

% double_arith_series
tff(fact_5060_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),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),Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_5061_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_nd(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = 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,Na)),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))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),D2)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_5062_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( sinh(real,aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_5063_pred__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_5064_succ__bound__size__univ_H,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_5065_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),A2),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),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_5066_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Na)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_ne(int,int)),set_or1337092689740270186AtMost(int,M,Na)) = 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),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),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_5067_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_le
tff(fact_5068_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),exp(real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_5069_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: 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)),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_5070_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: 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),Na)) = 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),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_5071_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_nf(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = 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),Na)),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))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_5072_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5073_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_5074_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_5075_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)),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),Q3),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_5076_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: 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)),Na)) = 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),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_5077_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      binomial(Na,K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),binomial(Na,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),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),Na),K)),one_one(nat)),Na,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_5078_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_5079_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_5080_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ng(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),gbinomial(A,R2,aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_5081_arctan__double,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),arctan(Xa)) = arctan(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_5082_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Na)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))),aa(nat,real,semiring_1_of_nat(real),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),Na))) ) ).

% central_binomial_lower_bound
tff(fact_5083_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_5084_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,uminus_uminus(real),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_5085_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5086_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_5087_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_5088_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H2: A,Z: A,Na: nat] :
          ( ( H2 != 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),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H2)),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na))),H2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H2),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_ni(A,fun(A,fun(nat,fun(nat,A))),H2),Z),Na)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_5089_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_5090_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5091_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_mz(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_5092_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Na))),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,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nj(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))),Na)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_5093_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H2: A,Z: A,K4: real,Na: nat] :
          ( ( H2 != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H2))),K4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),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(A,A,aa(A,fun(A,A),plus_plus(A),Z),H2)),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Na))),H2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K4),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H2))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_5094_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_nl(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_5095_Tb__T__vebt__buildupi,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Na))),aa(int,int,aa(int,fun(int,int),minus_minus(int),vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi
tff(fact_5096_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_5097_Tb__T__vebt__buildupi_H,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),vEBT_V9176841429113362141ildupi(Na)),aa(int,int,aa(int,fun(int,int),minus_minus(int),vEBT_VEBT_Tb(Na)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi'
tff(fact_5098_htt__vebt__memberi__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T2),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_5099_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),vEBT_VEBT_Tb2(Na)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Tb_T_vebt_buildupi''
tff(fact_5100_htt__vebt__predi,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_predi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_au(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T2),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_predi
tff(fact_5101_Tb__Tb_H,axiom,
    ! [T2: nat] : vEBT_VEBT_Tb(T2) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(T2)) ).

% Tb_Tb'
tff(fact_5102_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_5103_tdeletemimi,axiom,
    ! [Deg: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg,TreeList,Summary),Xa)),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_5104_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(73)
tff(fact_5105_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(80)
tff(fact_5106_minNull__delete__time__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(T2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T2,Xa)),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_5107_Tb_H__cnt,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_VEBT_Tb2(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(Na)))) ).

% Tb'_cnt
tff(fact_5108_TBOUND__vebt__predi,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND(option(nat),vEBT_VEBT_vebt_predi(T2,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)))) ).

% TBOUND_vebt_predi
tff(fact_5109_TBOUND__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_TBOUND(option(nat),vEBT_VEBT_vebt_succi(T2,Tia,Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)))) ).

% TBOUND_vebt_succi
tff(fact_5110_T__vebt__buildupi__cnt_H,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V441764108873111860ildupi(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit0(one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na)))) ).

% T_vebt_buildupi_cnt'
tff(fact_5111_TBOUND__buildupi,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% TBOUND_buildupi
tff(fact_5112_delete__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit1(bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_5113_cnt__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),bit1(bit1(bit1(one2))))),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2)))))))) ) ).

% cnt_bound
tff(fact_5114_TBOUND__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      time_TBOUND(vEBT_VEBTi,vEBT_V3964819847710782039nserti(T2,Tia,Xa),
        $ite(vEBT_VEBT_minNull(T2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))))) ).

% TBOUND_vebt_inserti
tff(fact_5115_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_5116_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_5117_delete__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(bit1(bit1(bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_5118_time__vebt__delete,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(bit1(bit1(bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_delete
tff(fact_5119_time__vebt__pred,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(bit1(bit1(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit0(bit1(bit1(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_pred
tff(fact_5120_time__vebt__succ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit0(bit1(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit1(bit0(bit1(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)))))) ) ).

% time_vebt_succ
tff(fact_5121_htt__vebt__inserti,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_inserti(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)))) ).

% htt_vebt_inserti
tff(fact_5122_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(81)
tff(fact_5123_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(72)
tff(fact_5124_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit1(Na)) ).

% semiring_norm(77)
tff(fact_5125_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),one2) ).

% semiring_norm(70)
tff(fact_5126_TBOUND__vebt__deletei,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] :
      time_TBOUND(vEBT_VEBTi,vEBT_V1365221501068881998eletei(T2,Tia,Xa),
        $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(T2,Xa)),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))))) ).

% TBOUND_vebt_deletei
tff(fact_5127_htt__vebt__memberi,axiom,
    ! [T2: vEBT_VEBT,Tia: vEBT_VEBTi,Xa: nat] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),T2),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2)))) ).

% htt_vebt_memberi
tff(fact_5128_zdiv__numeral__Bit1,axiom,
    ! [V3: num,W2: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit1(V3))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V3)),aa(num,int,numeral_numeral(int),W2)) ).

% zdiv_numeral_Bit1
tff(fact_5129_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(79)
tff(fact_5130_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(74)
tff(fact_5131_htt__vebt__inserti__invar__vebt,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_inserti(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_5132_htt__vebt__deletei,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_deletei(Tia,Xa),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(bit0(one2)))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_deletei
tff(fact_5133_htt__vebt__succi,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Tia: vEBT_VEBTi,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,T2),Tia),vEBT_vebt_succi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_at(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),T2),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% htt_vebt_succi
tff(fact_5134_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V3: 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),V3)) = 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),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V3)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_5135_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Na: 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,Na)))) = 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),bit1(one2))),Na)) ).

% div_Suc_eq_div_add3
tff(fact_5136_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V3: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),aa(num,nat,numeral_numeral(nat),V3)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_5137_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na)) ).

% mod_Suc_eq_mod_add3
tff(fact_5138_zmod__numeral__Bit1,axiom,
    ! [V3: num,W2: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit1(V3))),aa(num,int,numeral_numeral(int),bit0(W2))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V3)),aa(num,int,numeral_numeral(int),W2)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_5139_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb2(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_5140_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb2(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_5141_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),bit1(Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Na))) ).

% eval_nat_numeral(3)
tff(fact_5142_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(10)
tff(fact_5143_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) ) ).

% cong_exp_iff_simps(12)
tff(fact_5144_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,Na: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_5145_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q3: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_5146_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),bit1(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_5147_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na) ).

% Suc3_eq_add_3
tff(fact_5148_maxt__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(T2)),aa(num,nat,numeral_numeral(nat),bit1(one2))) ).

% maxt_bound
tff(fact_5149_VEBT__internal_OTb_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb(zero_zero(nat)) = aa(num,int,numeral_numeral(int),bit1(one2)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_5150_mint__bound,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(T2)),aa(num,nat,numeral_numeral(nat),bit1(one2))) ).

% mint_bound
tff(fact_5151_num_Osize_I6_J,axiom,
    ! [X32: num] : aa(num,nat,size_size(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_5152_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(M))),aa(num,A,numeral_numeral(A),bit0(Q3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_5153_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,Na: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit1(Na))),aa(num,A,numeral_numeral(A),bit0(Q3))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_5154_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Na: 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)))),Na) = 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),bit1(one2))),M)),Na) ).

% Suc_div_eq_add3_div
tff(fact_5155_card__3__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
    <=> ? [X5: A,Y5: A,Z2: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z2),bot_bot(set(A))))) )
          & ( X5 != Y5 )
          & ( Y5 != Z2 )
          & ( X5 != Z2 ) ) ) ).

% card_3_iff
tff(fact_5156_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),M)),Na) ).

% Suc_mod_eq_add3_mod
tff(fact_5157_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),bit1(one2))) ).

% exp_le
tff(fact_5158_VEBT__internal_OTb_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb(aa(nat,nat,suc,zero_zero(nat))) = aa(num,int,numeral_numeral(int),bit1(one2)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_5159_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_5160_insersimp,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Y: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(T2),X_1)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,Y)),aa(num,nat,numeral_numeral(nat),bit1(one2))) ) ) ).

% insersimp
tff(fact_5161_insertsimp,axiom,
    ! [T2: vEBT_VEBT,Na: nat,L: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( vEBT_VEBT_minNull(T2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,L)),aa(num,nat,numeral_numeral(nat),bit1(one2))) ) ) ).

% insertsimp
tff(fact_5162_small__powers__of__2,axiom,
    ! [Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Xa)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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),Xa),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_5163_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq2
tff(fact_5164_pred__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(bit1(one2))))))) ) ).

% pred_bound_height
tff(fact_5165_insert__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit0(one2))))))) ) ).

% insert_bound_height
tff(fact_5166_succ__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(bit1(one2))))))) ) ).

% succ_bound_height
tff(fact_5167_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),exp(real,one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq1
tff(fact_5168_pred__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(bit1(bit1(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit0(bit1(bit1(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_5169_succ__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit0(bit1(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit1(bit0(bit1(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_5170_insert__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit1(bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit1(bit1(bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_5171_VEBT__internal_Ovebt__buildupi__rule,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% VEBT_internal.vebt_buildupi_rule
tff(fact_5172_vebt__buildupi__rule,axiom,
    ! [Na: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)),vEBT_vebt_buildupi(Na),vEBT_Intf_vebt_assn(Na,bot_bot(set(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% vebt_buildupi_rule
tff(fact_5173_vebt__memberi__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Xa: nat] : time_htt($o,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_memberi(Tia,Xa),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_nm(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Na),Sa),Tia),Xa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_memberi_rule
tff(fact_5174_vebt__predi__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Xa: nat,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_predi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_nn(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),Sa),Tia),Xa),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_predi_rule
tff(fact_5175_vebt__succi__rule,axiom,
    ! [Na: nat,Sa: set(nat),Tia: vEBT_VEBTi,Xa: nat,Y: nat] : time_htt(option(nat),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_succi(Tia,Xa),aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_no(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Na),Sa),Tia),Xa),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ).

% vebt_succi_rule
tff(fact_5176_vebt__inserti__rule,axiom,
    ! [Xa: nat,Na: nat,Sa: set(nat),Tia: vEBT_VEBTi] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Na,Sa),Tia),vEBT_vebt_inserti(Tia,Xa),vEBT_Intf_vebt_assn(Na,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),Sa),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Xa),bot_bot(set(nat))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),aa(int,nat,nat2,archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))))))) ) ).

% vebt_inserti_rule
tff(fact_5177_space__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),U)) ) ) ).

% space_bound
tff(fact_5178_vebt__space__linear__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% vebt_space_linear_bound
tff(fact_5179_space__2__pow__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(bit1(one2))))),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na)),one_one(real)))) ) ).

% space_2_pow_bound
tff(fact_5180_space__space_H,axiom,
    ! [T2: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,T2)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2)) ).

% space_space'
tff(fact_5181_space_H__bound,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),U)) ) ) ).

% space'_bound
tff(fact_5182_space__cnt,axiom,
    ! [T2: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,T2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit1(one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,T2))) ).

% space_cnt
tff(fact_5183_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_5184_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_5185_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_5186_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_5187_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeList,Summary)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_5188_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_5189_t__build__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),bit1(bit0(bit1(one2)))))) ).

% t_build_cnt
tff(fact_5190_t__buildup__cnt,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Na))),aa(num,real,numeral_numeral(real),bit1(bit0(bit1(one2)))))) ).

% t_buildup_cnt
tff(fact_5191_vebt__buildup__bound,axiom,
    ! [U: nat,Na: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(bit1(one2)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_5192_buildup__build__time,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_V8346862874174094_d_u_p(Na)),vEBT_V8646137997579335489_i_l_d(Na)) ).

% buildup_build_time
tff(fact_5193_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    vEBT_V8346862874174094_d_u_p(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_5194_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    vEBT_V8646137997579335489_i_l_d(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_5195_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_5196_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_5197_div__half__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)) = $let(
            q: nat,
            q:= 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),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)),
            $let(
              r2: nat,
              r2:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),r2),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),q),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),r2),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),q),r2)) ) ) ) ) ).

% div_half_nat
tff(fact_5198_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_5199_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xa) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),Xa)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B4))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList2),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Node(Info2,Deg2,TreeList2,Summary2)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_5200_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q3: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q3),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q3)),R2)) ) ).

% divmod_step_eq
tff(fact_5201_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_insert(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = vEBT_vebt_insert(vEBT_Leaf((A4),(B4)),Xaa) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary2) )
               => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)),Xaa),vEBT_Node(Info2,Deg2,TreeList2,Summary2),vEBT_vebt_insert(vEBT_Node(Info2,Deg2,TreeList2,Summary2),Xaa)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,Deg2,TreeList2,Summary2)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
tff(fact_5202_pred__list__to__short,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_5203_high__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_high(Xa,Na) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% high_def
tff(fact_5204_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Ma,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% high_bound_aux
tff(fact_5205_high__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Y ) ) ).

% high_inv
tff(fact_5206_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_5207_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5208_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_5209_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_5210_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image(code_integer,code_integer,aTP_Lamp_np(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_5211_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_5212_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_5213_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_5214_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_5215_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_5216_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_5217_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_5218_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,sgn_sgn(code_integer),K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_5219_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_5220_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_5221_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_5222_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,Va2: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( ( Na = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Mi)
         => ( ( Ma != Mi )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_5223_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_5224_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),Xa: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_5225_bit__split__inv,axiom,
    ! [Xa: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xa,D2),vEBT_VEBT_low(Xa,D2),D2) = Xa ).

% bit_split_inv
tff(fact_5226_low__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_low(Xa,Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% low_def
tff(fact_5227_low__inv,axiom,
    ! [Xa: nat,Na: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Xa ) ) ).

% low_inv
tff(fact_5228_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi != Ma )
       => ( the2(nat,vEBT_vebt_maxt(Summary)) = vEBT_VEBT_high(Ma,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_5229_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_5230_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or7035219750837199246ssThan(code_integer,L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_5231_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_5232_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or1337092689740270186AtMost(code_integer,L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_5233_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_5234_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_5235_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H2)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_5236_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H2)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_5237_del__x__not__mi,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H2),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,Newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H2)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_5238_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary,H2) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_5239_del__x__not__mia,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,H2,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,H2),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xa = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xa = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H2)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_5240_del__x__mia,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                xn: nat,
                xn:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(
                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary,h),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                    Ma ))),Deg,newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary) ) ) ),
                    vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_5241_del__x__mi__lets__in__minNull,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary,H2) )
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_5242_del__x__mi__lets__in,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H2,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H2),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,Newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H2)))),Ma))),Deg,Newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_5243_del__x__mi,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( Xa = Mi )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H2 )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H2),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,H2,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,H2),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H2),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),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H2)))),Ma))),Deg,newlist,Summary) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_5244_del__in__range,axiom,
    ! [Mi: nat,Xa: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                xn: nat,
                xn:= 
                  $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),Xa),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xa = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary,h),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                      Ma ))),Deg,newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                  $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg,newlist,Summary) ) ) ),
                      vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_5245_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_5246_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Xa)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_5247_pred__lesseq__max,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_5248_pred__less__length__list,axiom,
    ! [Deg: nat,Xa: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ma)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Xa) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_5249_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A5) = the2(A,none(A)) ) ) ) ).

% Max.infinite
tff(fact_5250_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = the2(A,none(A)) ) ) ) ).

% Min.infinite
tff(fact_5251_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xa,Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_5252_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        one_one(nat),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),Xa),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                    $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                  one_one(nat) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_5253_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        aa(nat,option(nat),some(nat),Mi),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summary,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_5254_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        aa(nat,option(nat),some(nat),Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summary,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa),aa(nat,option(nat),some(nat),Mi),none(nat)),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_5255_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
        $ite(
          ( ( Xa = Mi )
          & ( Xa = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
          $let(
            xn: nat,
            xn:= 
              $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),Xa),
            $let(
              minn: nat,
              minn:= 
                $ite(Xa = Mi,xn,Mi),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary,h),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                  Ma ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_5256_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),Sa: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,Sa),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_5257_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),one_one(nat))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),bit1(one2)))),
                      $ite(
                        ( ( minlow != none(nat) )
                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summary,h),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                            $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_5258_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summary,h),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_5259_vebt__delete_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B4)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B4: $o] : Xa = vEBT_Leaf((A4),(B4))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
                 => ( Y != vEBT_Leaf((A4),(B4)) ) ) )
           => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                          $let(
                                            newlist: list(vEBT_VEBT),
                                            newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                            $ite(
                                              vEBT_VEBT_minNull(newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                    aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                                        Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                    $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_5260_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                        one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_5261_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_5262_vebt__succ_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( ! [Uu: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((Uu),(B4)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xa = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)
               => ( Y != none(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            aa(nat,option(nat),some(nat),Mi2),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary2,h),
                                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_5263_vebt__pred_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw: $o] : Xa = vEBT_Leaf((A4),(Uw))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( Y != $ite(
                        (B4),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)
                 => ( Y != none(nat) ) )
               => ( ( ? [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              aa(nat,option(nat),some(nat),Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary2,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_5264_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_5265_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_5266_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw: $o] : Xa = vEBT_Leaf((A4),(Uw))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),bit1(one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                  $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_5267_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,B4: $o] : Xa = vEBT_Leaf((Uu),(B4))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xa = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary2,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_5268_vebt__member__code_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary)),Xa)
    <=> $ite(
          ( ( Deg = zero_zero(nat) )
          | ( Deg = aa(nat,nat,suc,zero_zero(nat)) ) ),
          $false,
          $ite(
            Xa = Mi,
            $true,
            $ite(
              Xa = Ma,
              $true,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
                $false,
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                  $false,
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% vebt_member_code(3)
tff(fact_5269_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
        $ite(
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
          | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa) ),
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
            $ite(
              ( ( Xa = Mi )
              & ( Xa = Ma ) ),
              aa(num,nat,numeral_numeral(nat),bit1(one2)),
              aa(nat,nat,
                aa(nat,fun(nat,nat),plus_plus(nat),
                  aa(nat,nat,
                    aa(nat,fun(nat,nat),plus_plus(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),
                        $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),one_one(nat)))),
                    one_one(nat))),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xa = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),the2(nat,vEBT_vebt_mint(Summary)))))),Xa),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $let(
                      h: nat,
                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l),
                            $let(
                              newlist: list(vEBT_VEBT),
                              newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                $ite(
                                  vEBT_VEBT_minNull(newnode),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary,h),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                        $ite(
                                          xn = Ma,
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                            $let(
                                              maxs: option(nat),
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                          one_one(nat) )) )),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                    $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                        one_one(nat) ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_5270_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
           => (Y) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_5271_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S3: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_5272_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa != vEBT_Node(Uu,zero_zero(nat),Uv,Uw)
         => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_5273_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_5274_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
             => ( ? [N: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),bit1(one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),one_one(nat)))),
                                          one_one(nat))),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(
                                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                  $let(
                                                    newlist: list(vEBT_VEBT),
                                                    newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                      $ite(
                                                        vEBT_VEBT_minNull(newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary2,h),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                  $let(
                                                                    maxs: option(nat),
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                      $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                                                one_one(nat) )) )),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                          $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                              one_one(nat) ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_5275_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_5276_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_5277_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Uu: $o,Uv: $o] : Xa != vEBT_Leaf((Uu),(Uv))
       => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_5278_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => (Y) )
       => ( ( ? [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_5279_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xa = Mi,
          zero_zero(nat),
          $ite(
            Xa = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              zero_zero(nat),
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
                  & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_5280_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_5281_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Xa,Mi)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_5282_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X: vEBT_VEBT] :
          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X: nat] :
                                    ( ( ( vEBT_VEBT_high(X,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_5283_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B4),$false) ) )
       => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
         => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)
           => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_5284_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => (Y) )
         => ( ( ? [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)
             => (Y) )
           => ( ( ? [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_5285_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi2,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_5286_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(bit0(one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
              & ~ ( ( Xa = Mi )
                  | ( Xa = Ma ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_5287_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] : Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] : Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_5288_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
    <=> ( ( ? [A7: $o,B7: $o] : A12 = vEBT_Leaf((A7),(B7))
          & ( A23 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList4,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X5,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList4,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X5,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_13)
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList4,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X5,N5) )
            & vEBT_invar_vebt(Summary3,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(Ma3,N5)) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N5) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(X5,N5)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X5)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma3) ) ) ) ) ) )
        | ? [TreeList4: list(vEBT_VEBT),N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A23,TreeList4,Summary3) )
            & ! [X5: vEBT_VEBT] :
                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
               => vEBT_invar_vebt(X5,N5) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList4) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList4))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi3 != Ma3 )
             => ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N5)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N5) = I3 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(Ma3,N5)) )
                    & ! [X5: nat] :
                        ( ( ( vEBT_VEBT_high(X5,N5) = I3 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList4),I3)),vEBT_VEBT_low(X5,N5)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X5)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_5289_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
     => ( ( ? [A4: $o,B4: $o] : A12 = vEBT_Leaf((A4),(B4))
         => ( A23 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
             => ( ( A23 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                     => vEBT_invar_vebt(X3,N) )
                 => ( vEBT_invar_vebt(Summary2,M3)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                     => ( ( M3 = N )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M3) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
               => ( ( A23 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                       => vEBT_invar_vebt(X3,N) )
                   => ( vEBT_invar_vebt(Summary2,M3)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                       => ( ( M3 = aa(nat,nat,suc,N) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M3) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_12)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_invar_vebt(X3,N) )
                     => ( vEBT_invar_vebt(Summary2,M3)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                         => ( ( M3 = N )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M3) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X3,N)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg2,TreeList2,Summary2) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_invar_vebt(X3,N) )
                       => ( vEBT_invar_vebt(Summary2,M3)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                           => ( ( M3 = aa(nat,nat,suc,N) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M3) )
                               => ( ! [I4: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(Ma2,N)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I4)),vEBT_VEBT_low(X3,N)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X3)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_5290_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xa = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B4)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts,S3) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) ) )
           => ( ! [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_5291_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_5292_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_5293_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] : Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] : Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(bit0(one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(
                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_5294_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw: $o] : Xa = vEBT_Leaf((A4),(Uw))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
             => ( ? [Va: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( Y != $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_5295_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( ( ? [Uu: $o,B4: $o] : Xa = vEBT_Leaf((Uu),(B4))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xa = vEBT_Leaf((Uv),(Uw))
           => ( ? [N: nat] : Xaa = aa(nat,nat,suc,N)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( Y != $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_5296_in__children__def,axiom,
    ! [Na: nat,TreeList: list(vEBT_VEBT),Xa: nat] :
      ( vEBT_V5917875025757280293ildren(Na,TreeList,Xa)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,Na))),vEBT_VEBT_low(Xa,Na)) ) ).

% in_children_def
tff(fact_5297_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                    | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),bit1(one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit1(one2))))),
                                                  $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),bit1(bit1(one2)))),one_one(nat)))),
                                              one_one(nat))),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(
                                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                      $let(
                                                        newlist: list(vEBT_VEBT),
                                                        newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                          $ite(
                                                            vEBT_VEBT_minNull(newnode),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary2,h),
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                      $let(
                                                                        maxs: option(nat),
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                          $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) )),
                                                                    one_one(nat) )) )),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_5298_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),
                                        $ite(
                                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_5299_vebt__succ_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B4),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                aa(nat,option(nat),some(nat),Mi2),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary2,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_5300_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite((B4),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)),Xaa)) ) )
                 => ( ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),bit1(one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary2,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_5301_vebt__pred_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = $ite(
                              (B4),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)),Xaa)) ) )
                 => ( ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = none(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  aa(nat,option(nat),some(nat),Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary2,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                                aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_5302_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_5303_vebt__delete_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B4)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B4: $o] :
                ( ( Xa = vEBT_Leaf((A4),(B4)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [N: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                     => ( ( Y = vEBT_Leaf((A4),(B4)) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,N)))) ) ) )
             => ( ! [Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary2)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2)
                                  | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),the2(nat,vEBT_vebt_mint(Summary2)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                              $let(
                                                newlist: list(vEBT_VEBT),
                                                newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                $ite(
                                                  vEBT_VEBT_minNull(newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary2,h),
                                                    vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                                            Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_5304_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(B4)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(B4))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((Uv),(Uw)) )
               => ! [N: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N))) ) ) )
           => ( ! [Ux: nat,Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc2,Vd)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                       => ( ( Y = $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_5305_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S3)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(bit0(one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_5306_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B4)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B4))) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts,S3) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S3)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Xaa,Mi2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_5307_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw: $o] :
                ( ( Xa = vEBT_Leaf((A4),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B4: $o] :
                  ( ( Xa = vEBT_Leaf((A4),(B4)) )
                 => ! [Va: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),aa(nat,nat,suc,aa(nat,nat,suc,Va)))) ) ) )
             => ( ! [Uy: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy,Uz2,Va3)),Xaa)) ) )
               => ( ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve)),Xaa)) ) )
                 => ( ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT] :
                        ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                          ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_5308_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts,S3) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts,S3)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S3)),Xaa)) ) )
             => ( ! [V: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList2,Summary2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_5309_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xaa)) ) )
           => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi2,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),Xaa)
                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_5310_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( ( ? [A4: $o,B4: $o] : Xa = vEBT_Leaf((A4),(B4))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary2: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $ite(
                            Xaa = Mi2,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_5311_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_5312_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_5313_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xa: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xa) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_5314_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(Xa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_5315_member__bound__height,axiom,
    ! [T2: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(T2,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,T2))),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(one2)))))) ) ).

% member_bound_height
tff(fact_5316_member__bound__size__univ,axiom,
    ! [T2: vEBT_VEBT,Na: nat,U: real,Xa: nat] :
      ( vEBT_invar_vebt(T2,Na)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Na) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(T2,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit1(bit1(one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit1(bit1(bit1(one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_5317_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),Xa) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xa = Mi,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xa = Ma,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                      one_one(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_5318_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xa,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xaa)) ) )
           => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              $ite(
                                Xaa = Mi2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_5319_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xaa)) ) )
           => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2) )
                 => ( ~ (Y)
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)),Xaa)) ) )
             => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_5320_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xaa)) )
           => ( ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz2)),Xaa)) )
             => ( ! [V: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_5321_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary2)),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_5322_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Xaa)) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_5323_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) ) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_5324_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xa = vEBT_Leaf((A4),(B4)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B4),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B4))),Xaa)) ) )
         => ( ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu,zero_zero(nat),Uv,Uw) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu,zero_zero(nat),Uv,Uw)),Xaa)) ) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList2: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList2,S3)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_5325_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xaa)) ) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)),Xaa)) ) )
               => ~ ! [V: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_5326_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xaa)) )
         => ( ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy)),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2) )
                   => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_5327_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V),TreeList2,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList2,Vd)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_5328_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T7)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),Xa)
            & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ky(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_5329_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ? [T7: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T7)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),Xa)
          & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ky(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_5330_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T7: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
      & ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_ky(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_5331_sin__cos__npi,axiom,
    ! [Na: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Na) ).

% sin_cos_npi
tff(fact_5332_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_5333_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_5334_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_5335_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_5336_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_5337_pi__less__4,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_5338_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_5339_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => ( ( sin(real,Xa) = zero_zero(real) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_5340_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),pi)
     => ( ( sin(real,Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_5341_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_less_two
tff(fact_5342_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_5343_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_5344_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_5345_m2pi__less__pi,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_5346_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_5347_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_5348_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_5349_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_5350_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_5351_sin__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xa) = sin(real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_5352_sin__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_5353_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_5354_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_5355_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Y)) ).

% arctan_lbound
tff(fact_5356_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_5357_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_5358_sin__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_5359_sin__monotone__2pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Y)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_5360_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% sin_total
tff(fact_5361_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_5362_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T7)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),Xa)
            & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_nq(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_5363_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T7)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),zero_zero(real))
            & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_nq(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_5364_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_nr(real,fun(nat,real),Xa)) ) ).

% monoseq_arctan_series
tff(fact_5365_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_5366_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_5367_cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),one_one(real)) ).

% cos_le_one
tff(fact_5368_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) = one_one(A) )
         => ( sin(A,Xa) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_5369_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_5370_cos__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_5371_cos__inj__pi,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( cos(real,Xa) = cos(real,Y) )
             => ( Xa = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_5372_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,Xa)) ).

% cos_ge_minus_one
tff(fact_5373_abs__cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,Xa))),one_one(real)) ).

% abs_cos_le_one
tff(fact_5374_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,X4,aa(nat,nat,suc,N)))
         => topological_monoseq(A,X4) ) ) ).

% mono_SucI1
tff(fact_5375_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,aa(nat,nat,suc,N))),aa(nat,A,X4,N))
         => topological_monoseq(A,X4) ) ) ).

% mono_SucI2
tff(fact_5376_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( topological_monoseq(A,X4)
        <=> ( ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N5)),aa(nat,A,X4,aa(nat,nat,suc,N5)))
            | ! [N5: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,aa(nat,nat,suc,N5))),aa(nat,A,X4,N5)) ) ) ) ).

% monoseq_Suc
tff(fact_5377_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( topological_monoseq(A,X4)
        <=> ( ! [M5: nat,N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,M5)),aa(nat,A,X4,N5)) )
            | ! [M5: nat,N5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N5)),aa(nat,A,X4,M5)) ) ) ) ) ).

% monoseq_def
tff(fact_5378_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [M3: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,X4,M3)) )
         => topological_monoseq(A,X4) ) ) ).

% monoI2
tff(fact_5379_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [M3: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,M3)),aa(nat,A,X4,N)) )
         => topological_monoseq(A,X4) ) ) ).

% monoI1
tff(fact_5380_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xa)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_5381_cos__mono__less__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_5382_cos__monotone__0__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_5383_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_5384_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_5385_cos__is__zero,axiom,
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X ) ) ) ).

% cos_is_zero
tff(fact_5386_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_5387_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Y)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_5388_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),pi)
            & ( cos(real,X) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X ) ) ) ) ) ).

% cos_total
tff(fact_5389_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y3),pi)
      & ( sin(real,Y3) = sin(real,Xa) )
      & ( cos(real,Y3) = cos(real,Xa) ) ) ).

% sincos_principal_value
tff(fact_5390_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xa)) ) ) ).

% monoseq_realpow
tff(fact_5391_sin__cos__le1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,Xa)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xa)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_5392_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_5393_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero
tff(fact_5394_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_5395_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_ge_zero
tff(fact_5396_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_ns(real,$o)) ).

% pi_half
tff(fact_5397_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),the(real,aTP_Lamp_ns(real,$o))) ).

% pi_def
tff(fact_5398_sincos__total__pi,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),pi)
            & ( Xa = cos(real,T7) )
            & ( Y = sin(real,T7) ) ) ) ) ).

% sincos_total_pi
tff(fact_5399_sin__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_5400_cos__expansion__lemma,axiom,
    ! [Xa: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_5401_sincos__total__pi__half,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xa = cos(real,T7) )
              & ( Y = sin(real,T7) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_5402_sincos__total__2pi__le,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T7: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xa = cos(real,T7) )
          & ( Y = sin(real,T7) ) ) ) ).

% sincos_total_2pi_le
tff(fact_5403_sincos__total__2pi,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xa = cos(real,T7) )
               => ( Y != sin(real,T7) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_5404_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T7: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
      & ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_nq(real,fun(nat,real),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi)))),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_5405_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = 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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_5406_lowi__hT,axiom,
    ! [Xa: nat,Na: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_nt(nat,fun(nat,fun(nat,assn)),Xa),Na),one_one(nat)) ).

% lowi_hT
tff(fact_5407_highi__hT,axiom,
    ! [Xa: nat,Na: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_nu(nat,fun(nat,fun(nat,assn)),Xa),Na),one_one(nat)) ).

% highi_hT
tff(fact_5408_TBOUND__highi,axiom,
    ! [Xa: nat,Na: nat] : time_TBOUND(nat,vEBT_VEBT_highi(Xa,Na),one_one(nat)) ).

% TBOUND_highi
tff(fact_5409_TBOUND__lowi,axiom,
    ! [Xa: nat,Na: nat] : time_TBOUND(nat,vEBT_VEBT_lowi(Xa,Na),one_one(nat)) ).

% TBOUND_lowi
tff(fact_5410_highi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_nu(nat,fun(nat,fun(nat,assn)),Xa),Na)) ).

% highi_h
tff(fact_5411_lowi__h,axiom,
    ! [Xa: nat,Na: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xa,Na),aa(nat,fun(nat,assn),aTP_Lamp_nt(nat,fun(nat,fun(nat,assn)),Xa),Na)) ).

% lowi_h
tff(fact_5412_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_5413_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_5414_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Na),K)) ).

% set_bit_greater_eq
tff(fact_5415_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_5416_set__encode__insert,axiom,
    ! [A5: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( ~ member(nat,Na,A5)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),Na),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))),Na)),aa(set(nat),nat,nat_set_encode,A5)) ) ) ) ).

% set_encode_insert
tff(fact_5417_sin__coeff__def,axiom,
    ! [X3: nat] :
      sin_coeff(X3) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X3),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X3),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X3))) ).

% sin_coeff_def
tff(fact_5418_ln__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xa) = suminf(real,aTP_Lamp_nv(real,fun(nat,real),Xa)) ) ) ) ).

% ln_series
tff(fact_5419_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_5420_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_5421_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_5422_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_5423_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_5424_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_5425_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),C3) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_5426_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_5427_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_5428_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_5429_dvd__prodI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),A2: A,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F3,A2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ).

% dvd_prodI
tff(fact_5430_dvd__prod__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A5: set(A),A2: A,B2: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( ( B2 = aa(A,B,F3,A2) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B2),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_5431_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_5432_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_5433_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_nw(nat,fun(nat,$o),Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_5434_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_5435_even__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,Na))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ).

% even_Suc
tff(fact_5436_even__Suc__Suc__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Na)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ).

% even_Suc_Suc_iff
tff(fact_5437_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(nat,A)] : suminf(A,aTP_Lamp_nx(fun(nat,A),fun(nat,A),F3)) = aa(nat,A,F3,zero_zero(nat)) ) ).

% powser_zero
tff(fact_5438_odd__Suc__div__two,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_5439_even__Suc__div__two,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_5440_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_5441_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_5442_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_5443_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_5444_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_5445_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ) ) ).

% even_diff_nat
tff(fact_5446_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),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))),Na)),one_one(A)))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_5447_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_5448_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2))
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_5449_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( 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)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_5450_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_5451_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_5452_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa)) ) ) ) ).

% minf(10)
tff(fact_5453_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa)) ) ) ) ).

% minf(9)
tff(fact_5454_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa)) ) ) ) ).

% pinf(10)
tff(fact_5455_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,Sa: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Sa)) ) ) ) ).

% pinf(9)
tff(fact_5456_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_5457_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ny(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ny(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_5458_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_5459_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_5460_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_5461_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
     => ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_5462_gcd__nat_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat))
        & ( A2 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_5463_gcd__nat_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
    <=> ( A2 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_5464_gcd__nat_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A2)
        & ( zero_zero(nat) != A2 ) ) ).

% gcd_nat.extremum_strict
tff(fact_5465_gcd__nat_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A2),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_5466_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_5467_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ny(A,fun(A,$o),A2))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ny(A,fun(A,$o),B2)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_5468_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_5469_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_5470_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_5471_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_5472_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Na: nat,B2: A,M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2) ) ) ) ).

% power_le_dvd
tff(fact_5473_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Y: A,Na: nat,M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_5474_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_5475_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_5476_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% dvd_minus_self
tff(fact_5477_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M) ) ) ) ).

% dvd_diffD
tff(fact_5478_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_5479_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_5480_set__encode__eq,axiom,
    ! [A5: set(nat),B5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(set(nat),$o,finite_finite2(nat),B5)
       => ( ( aa(set(nat),nat,nat_set_encode,A5) = aa(set(nat),nat,nat_set_encode,B5) )
        <=> ( A5 = B5 ) ) ) ) ).

% set_encode_eq
tff(fact_5481_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_5482_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),semiring_gcd_Gcd_fin(A,A5))
          <=> ! [X5: A] :
                ( member(A,X5,A5)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),X5) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_5483_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [B4: A] :
                ( member(A,B4,A5)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B4) )
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),semiring_gcd_Gcd_fin(A,A5)) ) ) ) ).

% Gcd_fin_greatest
tff(fact_5484_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P2: fun(A,$o),L: A] :
          ( ? [X5: A] : aa(A,$o,P2,aa(A,A,aa(A,fun(A,A),times_times(A),L),X5))
        <=> ? [X5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X5),zero_zero(A)))
              & aa(A,$o,P2,X5) ) ) ) ).

% unity_coeff_ex
tff(fact_5485_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C2: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ).

% unit_dvdE
tff(fact_5486_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_5487_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C3 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C3) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_5488_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B2: A,A2: A] :
          ( ( C3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C3))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_5489_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C3: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C3)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_5490_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C3: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C3 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_5491_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_5492_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_5493_even__set__encode__iff,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A5))
      <=> ~ member(nat,zero_zero(nat),A5) ) ) ).

% even_set_encode_iff
tff(fact_5494_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% dvd_imp_le
tff(fact_5495_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B),G3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F3,A4)),aa(A,B,G3,A4)) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),B5)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_5496_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B5)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_5497_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_5498_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_5499_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X: nat,Y3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),A2)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D3),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),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) ) ) ) ).

% bezout_add_strong_nat
tff(fact_5500_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_5501_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Q3) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_5502_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),semiring_char_0_fact(nat,Na)) ) ) ).

% dvd_fact
tff(fact_5503_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_nz(set(A),fun(fun(A,B),fun(A,$o)),A5),F3)))) ) ) ) ).

% even_sum_iff
tff(fact_5504_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_nw(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_5505_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_oa(nat,fun(nat,$o)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_5506_set__encode__inf,axiom,
    ! [A5: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A5)
     => ( aa(set(nat),nat,nat_set_encode,A5) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_5507_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_5508_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B4: A] :
                  ( ( B4 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B4),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B4 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B4) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B4) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C3),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B4) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_5509_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_5510_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_5511_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_5512_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,Xa: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            | ( Xa = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xa),Na)) ) ) ).

% dvd_power
tff(fact_5513_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_5514_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_5515_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),K)))),semiring_char_0_fact(A,Na)) ) ) ).

% choose_dvd
tff(fact_5516_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_5517_dvd__minus__add,axiom,
    ! [Q3: nat,Na: nat,R2: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),Q3))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)),Q3))) ) ) ) ).

% dvd_minus_add
tff(fact_5518_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),D2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),B2),D2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),D2),B2)) ) ) ).

% diff_mod_le
tff(fact_5519_mod__nat__eqI,axiom,
    ! [R2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),R2))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_5520_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_5521_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A5))
          <=> ? [X5: A] :
                ( member(A,X5,A5)
                & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F3,X5)) ) ) ) ) ).

% even_prod_iff
tff(fact_5522_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono_odd
tff(fact_5523_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% odd_pos
tff(fact_5524_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A5: set(A)] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A5))
     => ( A5 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_5525_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_5526_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_5527_dvd__partition,axiom,
    ! [A: $tType,C4: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4))
     => ( ! [X: set(A)] :
            ( member(set(A),X,C4)
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X)) )
       => ( ! [X: set(A)] :
              ( member(set(A),X,C4)
             => ! [Xa5: set(A)] :
                  ( member(set(A),Xa5,C4)
                 => ( ( X != Xa5 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4))) ) ) ) ).

% dvd_partition
tff(fact_5528_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A5: set(A),F3: fun(A,B),B2: B] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),A5)),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(B,fun(A,B),aTP_Lamp_ob(fun(A,B),fun(B,fun(A,B)),F3),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_oc(fun(A,B),fun(B,fun(A,$o)),F3),B2))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_od(fun(A,B),fun(B,fun(A,$o)),F3),B2))))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5529_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_5530_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_5531_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)) ) ) ).

% zero_le_even_power
tff(fact_5532_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_5533_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_5534_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Na)) ) ) ) ).

% power_mono_even
tff(fact_5535_central__binomial__odd,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( binomial(Na,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = binomial(Na,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_5536_Gcd__eq__Max,axiom,
    ! [M7: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M7)
     => ( ( M7 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M7)
         => ( gcd_Gcd(nat,M7) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),aTP_Lamp_oe(nat,set(nat))),M7))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5537_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_5538_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_5539_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Na)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_5540_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),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),Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_5541_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_5542_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),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),Na),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_5543_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_5544_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Na: nat] :
      vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),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(bit0(one2)))),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),divide_divide(nat),Na),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),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),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(bit0(bit0(one2))))),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),divide_divide(nat),Na),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(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_5545_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),Na),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_5546_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_5547_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),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),divide_divide(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),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),power_power(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,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,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),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),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),divide_divide(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(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),power_power(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)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_5548_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_5549_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,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),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,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),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,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),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,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),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,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),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_5550_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( cos(real,Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_5551_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( sin(real,Xa) = zero_zero(real) )
       => ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_5552_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Na: nat] :
      vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_5553_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Na: nat] :
      vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),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(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),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(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_5554_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_5555_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va2: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va2))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_5556_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_5557_VEBT__internal_OTb_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),bit1(one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),bit1(one2)) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_5558_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit1(one2)) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_5559_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Y != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_5560_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
               => ( Y != $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),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(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),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(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_5561_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( 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_of(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_even_sum
tff(fact_5562_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( 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_og(nat,fun(nat,A),Na)),set_ord_atMost(nat,Na))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_odd_sum
tff(fact_5563_arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( arctan(Xa) = suminf(real,aTP_Lamp_oh(real,fun(nat,real),Xa)) ) ) ).

% arctan_series
tff(fact_5564_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),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(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),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(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_5565_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit1(bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit1(one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_5566_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_5567_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),M),Na)
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_5568_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Na)
       => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Na),M) ) ) ).

% zdvd_not_zless
tff(fact_5569_abs__div,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Y),Xa)
     => ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),Xa)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_5570_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_5571_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_5572_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_5573_int__div__sub__1,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),Na),one_one(int))),M) = $ite(aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),M),Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),M)),one_one(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Na),M)) ) ) ).

% int_div_sub_1
tff(fact_5574_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_5575_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,Z)),M)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_5576_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_5577_eme1p,axiom,
    ! [Na: int,D2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),D2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)) ) ) ) ) ).

% eme1p
tff(fact_5578_emep1,axiom,
    ! [Na: int,D2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int))),D2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),D2)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_5579_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_5580_VEBT__internal_OTb_Opelims,axiom,
    ! [Xa: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),bit1(one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),bit1(one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),bit1(bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_5581_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
             => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_5582_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xa: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xa) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,N)) )
                 => ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),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),divide_divide(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),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),power_power(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,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,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),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),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),divide_divide(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(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,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),power_power(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,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_5583_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3)) ) ) ) ).

% suminf_geometric
tff(fact_5584_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_oi(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_5585_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N6)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N6)
               => ( aa(nat,A,F3,N) = zero_zero(A) ) )
           => ( suminf(A,F3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N6) ) ) ) ) ).

% suminf_finite
tff(fact_5586_vebt__delete__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_vebt_delete(vEBT_Node(Info,Deg,TreeList,Summary),Xa) = case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(Info,Deg,TreeList,Summary),aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_ok(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Info),Deg),TreeList),Summary),Xa),Info) ).

% vebt_delete_code(2)
tff(fact_5587_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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,Xa) = Y ) ) ) ) ).

% round_unique
tff(fact_5588_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na),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))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit0(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(7)
tff(fact_5589_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set(product_prod(A,B)),F3: fun(A,fun(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),A5)
     => member(C,aa(B,C,aa(A,fun(B,C),F3,A2),B2),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3)),A5)) ) ).

% pair_imageI
tff(fact_5590_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F3,A2),B2) ).

% case_prod_conv
tff(fact_5591_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_5592_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_5593_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,Na,M)) ) ) ).

% dvd_numeral_simp
tff(fact_5594_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,bit0(Na)) = 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_5595_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,bit1(Na)) = 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_5596_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit1(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),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),bit1(M))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit1(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(8)
tff(fact_5597_split__cong,axiom,
    ! [C: $tType,B: $tType,A: $tType,Q3: product_prod(A,B),F3: fun(A,fun(B,C)),G3: fun(A,fun(B,C)),P3: product_prod(A,B)] :
      ( ! [X: A,Y3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) = Q3 )
         => ( aa(B,C,aa(A,fun(B,C),F3,X),Y3) = aa(B,C,aa(A,fun(B,C),G3,X),Y3) ) )
     => ( ( P3 = Q3 )
       => ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G3),Q3) ) ) ) ).

% split_cong
tff(fact_5598_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,fun(C,A)),X1: B,X2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X2)) = aa(C,A,aa(B,fun(C,A),F3,X1),X2) ).

% old.prod.case
tff(fact_5599_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),G3: fun(product_prod(A,B),C)] :
      ( ! [X: A,Y3: B] : aa(B,C,aa(A,fun(B,C),F3,X),Y3) = aa(product_prod(A,B),C,G3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3))
     => ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) = G3 ) ) ).

% cond_case_prod_eta
tff(fact_5600_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_ol(fun(product_prod(A,B),C),fun(A,fun(B,C)),F3)) = F3 ).

% case_prod_eta
tff(fact_5601_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q2: fun(A,$o),P2: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P2),Z))
     => ~ ! [X: B,Y3: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y3) )
           => ~ aa(A,$o,Q2,aa(C,A,aa(B,fun(C,A),P2,X),Y3)) ) ) ).

% case_prodE2
tff(fact_5602_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,fun(D,A))),Xa: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F3),Xa),Y) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_om(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F3),Y)),Xa) ).

% nested_case_prod_simp
tff(fact_5603_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H2: fun(B,A),F3: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H2,aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_case_prod(C,D,B),F3),Prod)) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_on(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H2),F3)),Prod) ).

% prod.case_distrib
tff(fact_5604_uncurry__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C))] : uncurry(A,B,C,F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3) ).

% uncurry_def
tff(fact_5605_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_5606_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_oo(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_atMost(nat,Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_5607_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_oo(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_os(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_atMost(nat,Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_5608_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ot(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_lessThan(nat,Na)) ) ).

% sum.triangle_reindex
tff(fact_5609_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G3)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_ot(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_os(fun(nat,fun(nat,A)),fun(nat,A),G3)),set_ord_lessThan(nat,Na)) ) ).

% prod.triangle_reindex
tff(fact_5610_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_5611_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_round(A,Xa)) ) ).

% floor_le_round
tff(fact_5612_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% ceiling_ge_round
tff(fact_5613_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G3: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G3),Xs),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_ou(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G3))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_5614_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G3: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G3),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_ov(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G3))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_5615_foldl__snd__zip,axiom,
    ! [B: $tType,C: $tType,A: $tType,Ys: list(A),Xs: list(B),F3: fun(C,fun(A,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( aa(list(product_prod(B,A)),C,aa(C,fun(list(product_prod(B,A)),C),foldl(C,product_prod(B,A),aTP_Lamp_ox(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),F3)),B2),zip(B,A,Xs,Ys)) = aa(list(A),C,aa(C,fun(list(A),C),foldl(C,A,F3),B2),Ys) ) ) ).

% foldl_snd_zip
tff(fact_5616_divmod__int__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(int,M,Na) = 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),Na))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na))) ).

% divmod_int_def
tff(fact_5617_divmod_H__nat__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(nat,M,Na) = 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),Na))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na))) ).

% divmod'_nat_def
tff(fact_5618_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,M,Na) = 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),Na))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))) ) ).

% divmod_def
tff(fact_5619_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_5620_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_oy(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Max.eq_fold'
tff(fact_5621_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,M,Na) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,Na,unique8689654367752047608divmod(A,M,bit0(Na)))) ) ).

% divmod_divmod_step
tff(fact_5622_vebt__insert__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(Info,Deg,TreeList,Summary),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),vEBT_Node(Info,Deg,TreeList,Summary),case_option(vEBT_VEBT,product_prod(nat,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xa)),Deg,TreeList,Summary),aa(fun(nat,fun(nat,vEBT_VEBT)),fun(product_prod(nat,nat),vEBT_VEBT),product_case_prod(nat,nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_oz(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Deg),TreeList),Summary),Xa)),Info)) ).

% vebt_insert_code(2)
tff(fact_5623_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),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_5624_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),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,Xa))) ) ).

% of_int_round_ge
tff(fact_5625_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),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,Xa))) ) ).

% of_int_round_gt
tff(fact_5626_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),Xa))),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_5627_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),aa(int,A,ring_1_of_int(A),Na)))),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,Xa) = Na ) ) ) ).

% round_unique'
tff(fact_5628_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_round(A,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xa)),archimedean_ceiling(A,Xa),archim6421214686448440834_floor(A,Xa)) ) ).

% round_altdef
tff(fact_5629_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,semiring_1_of_nat(A),Na) = $ite(Na = zero_zero(nat),zero_zero(A),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_pa(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_5630_minus__one__div__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Na))) ).

% minus_one_div_numeral
tff(fact_5631_one__div__minus__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Na))) ).

% one_div_minus_numeral
tff(fact_5632_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P3: product_prod(A,B),Z: C,C3: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C3,A4),B4)) )
     => 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)),C3),P3)) ) ).

% mem_case_prodI2
tff(fact_5633_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,A2),B2))
     => 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)),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2))) ) ).

% mem_case_prodI
tff(fact_5634_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P3: product_prod(A,B),C3: fun(A,fun(B,fun(C,$o))),Xa: C] :
      ( ! [A4: A,B4: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = P3 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,A4),B4),Xa) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C3),P3),Xa) ) ).

% case_prodI2'
tff(fact_5635_split__part,axiom,
    ! [B: $tType,A: $tType,P2: $o,Q2: fun(A,fun(B,$o)),X3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_pb($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P2)),Q2)),X3)
    <=> ( (P2)
        & aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q2),X3) ) ) ).

% split_part
tff(fact_5636_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xa: A,Y: B] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_pc(A,fun(B,fun(A,fun(B,$o))),Xa),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) ).

% The_split_eq
tff(fact_5637_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),C3: fun(A,fun(B,$o))] :
      ( ! [A4: A,B4: B] :
          ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => aa(B,$o,aa(A,fun(B,$o),C3,A4),B4) )
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C3),P3) ) ).

% case_prodI2
tff(fact_5638_case__prodI,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F3,A2),B2)
     => aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_5639_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: $o] :
      aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_pd($o,fun(A,fun(B,$o)),(P2)))) = $ite((P2),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_5640_numeral__div__minus__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Na))) ).

% numeral_div_minus_numeral
tff(fact_5641_minus__numeral__div__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Na))) ).

% minus_numeral_div_numeral
tff(fact_5642_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,bit0(M),bit0(Na)) = 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_pe(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(5)
tff(fact_5643_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : unique8689654367752047608divmod(A,bit1(M),bit0(Na)) = 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_pf(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Na)) ) ).

% divmod_algorithm_code(6)
tff(fact_5644_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A5: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A5),B5)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A5))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B5))) ) ).

% Collect_case_prod_mono
tff(fact_5645_case__prodE,axiom,
    ! [A: $tType,B: $tType,C3: fun(A,fun(B,$o)),P3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),C3),P3)
     => ~ ! [X: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C3,X),Y3) ) ) ).

% case_prodE
tff(fact_5646_case__prodD,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),F3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),F3,A2),B2) ) ).

% case_prodD
tff(fact_5647_same__fst__def,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),R3: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P2,R3) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_ph(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),P2),R3)))) ).

% same_fst_def
tff(fact_5648_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C3: fun(B,fun(C,set(A))),P3: product_prod(B,C)] :
      ( 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)),C3),P3))
     => ~ ! [X: B,Y3: C] :
            ( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y3) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C3,X),Y3)) ) ) ).

% mem_case_prodE
tff(fact_5649_lex__prod__def,axiom,
    ! [A: $tType,B: $tType,Ra: set(product_prod(A,A)),Rb: set(product_prod(B,B))] : lex_prod(A,B,Ra,Rb) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),$o)),fun(product_prod(product_prod(A,B),product_prod(A,B)),$o),product_case_prod(product_prod(A,B),product_prod(A,B),$o),aa(fun(A,fun(B,fun(product_prod(A,B),$o))),fun(product_prod(A,B),fun(product_prod(A,B),$o)),product_case_prod(A,B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_pj(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Ra),Rb)))) ).

% lex_prod_def
tff(fact_5650_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,$o))),P3: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),C3),P3),Z)
     => ~ ! [X: A,Y3: B] :
            ( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C3,X),Y3),Z) ) ) ).

% case_prodE'
tff(fact_5651_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C3: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),aa(fun(A,fun(B,fun(C,$o))),fun(product_prod(A,B),fun(C,$o)),product_case_prod(A,B,fun(C,$o)),R3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C3)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R3,A2),B2),C3) ) ).

% case_prodD'
tff(fact_5652_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_pk(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_5653_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_pl(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_5654_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_pn(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_5655_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,T2: product_prod(A,B),F3: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A4: A,B4: B] :
          ( ( T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F3,A4),B4),aa(B,nat,aa(A,fun(B,nat),Bnd,A4),B4)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F3),T2),aa(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),Bnd),T2)) ) ).

% TBOUND_prod_case
tff(fact_5656_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_ep(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_5657_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xa: product_prod(A,B),P2: assn,F3: fun(A,fun(B,heap_Time_Heap(C))),Q2: fun(C,assn)] :
      ( ! [A4: A,B4: B] :
          ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) )
         => hoare_hoare_triple(C,P2,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F3,A4),B4),Q2) )
     => hoare_hoare_triple(C,P2,aa(product_prod(A,B),heap_Time_Heap(C),aa(fun(A,fun(B,heap_Time_Heap(C))),fun(product_prod(A,B),heap_Time_Heap(C)),product_case_prod(A,B,heap_Time_Heap(C)),F3),Xa),Q2) ) ).

% case_prod_rule
tff(fact_5658_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),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_po(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_5659_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),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_pp(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_5660_map__prod__fun__zip,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(C,A),G3: fun(D,B),Xs: list(C),Ys: list(D)] : aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_pq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G3))),zip(C,D,Xs,Ys)) = zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G3),Ys)) ).

% map_prod_fun_zip
tff(fact_5661_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xa),Y)
        <=> case_option($o,A,$true,aTP_Lamp_pr(option(A),fun(A,$o),Y),Xa) ) ) ).

% less_eq_option_def
tff(fact_5662_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less(option(A)),Xa),Y)
        <=> case_option($o,A,$false,aTP_Lamp_pt(option(A),fun(A,$o),Xa),Y) ) ) ).

% less_option_def
tff(fact_5663_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,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_pv(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_5664_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S2),X3),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Sup_SUP_eq2
tff(fact_5665_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_pw(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_5666_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S2: set(fun(A,fun(B,$o))),X3: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S2),X3),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S2)))) ) ).

% Inf_INT_eq2
tff(fact_5667_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P2: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_px(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P2))) ).

% rel_of_def
tff(fact_5668_trancl__insert2,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)),R2)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R2)),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_py(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),A2),B2),R2)))) ).

% trancl_insert2
tff(fact_5669_foldr__snd__zip,axiom,
    ! [B: $tType,A: $tType,C: $tType,Ys: list(A),Xs: list(B),F3: fun(A,fun(C,C)),B2: C] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(B),nat,size_size(list(B)),Xs))
     => ( foldr(product_prod(B,A),C,aa(fun(B,fun(A,fun(C,C))),fun(product_prod(B,A),fun(C,C)),product_case_prod(B,A,fun(C,C)),aTP_Lamp_pz(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),F3)),zip(B,A,Xs,Ys),B2) = foldr(A,C,F3,Ys,B2) ) ) ).

% foldr_snd_zip
tff(fact_5670_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_qa(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_5671_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_qb(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_5672_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_qc(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_5673_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_qd(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_5674_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_qe(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_5675_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_qf(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_5676_vebt__succ__code_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_succ(vEBT_Node(Info,Deg,TreeList,Summary),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Deg),one_one(nat)),none(nat),case_option(option(nat),product_prod(nat,nat),none(nat),aa(fun(nat,fun(nat,option(nat))),fun(product_prod(nat,nat),option(nat)),product_case_prod(nat,nat,option(nat)),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_qg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Deg),TreeList),Summary),Xa)),Info)) ).

% vebt_succ_code(2)
tff(fact_5677_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4080844693773952564amples(A,A_of_integer,Zero,I) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_qh(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),I)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
tff(fact_5678_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xa: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xa) = Y )
         => ( Y = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Xa),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_qh(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),Xa)),aa(code_integer,product_prod(A,A),A_of_integer,Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
tff(fact_5679_integer__of__int__code,axiom,
    ! [K: int] :
      aa(int,code_integer,code_integer_of_int,K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),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))))),
            $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_5680_integer__of__int__eq__of__int,axiom,
    code_integer_of_int = ring_1_of_int(code_integer) ).

% integer_of_int_eq_of_int
tff(fact_5681_divmod__integer_H__def,axiom,
    ! [M: num,Na: num] : unique8689654367752047608divmod(code_integer,M,Na) = 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),Na))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Na))) ).

% divmod_integer'_def
tff(fact_5682_one__integer__def,axiom,
    one_one(code_integer) = aa(int,code_integer,code_integer_of_int,one_one(int)) ).

% one_integer_def
tff(fact_5683_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Xb) ) ).

% less_integer.abs_eq
tff(fact_5684_divide__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xa),Xb)) ).

% divide_integer.abs_eq
tff(fact_5685_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),Xb)) ).

% modulo_integer.abs_eq
tff(fact_5686_integer__of__int__inject,axiom,
    ! [Xa: int,Y: int] :
      ( member(int,Xa,top_top(set(int)))
     => ( member(int,Y,top_top(set(int)))
       => ( ( aa(int,code_integer,code_integer_of_int,Xa) = aa(int,code_integer,code_integer_of_int,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% integer_of_int_inject
tff(fact_5687_integer__of__int__induct,axiom,
    ! [P2: fun(code_integer,$o),Xa: code_integer] :
      ( ! [Y3: int] :
          ( member(int,Y3,top_top(set(int)))
         => aa(code_integer,$o,P2,aa(int,code_integer,code_integer_of_int,Y3)) )
     => aa(code_integer,$o,P2,Xa) ) ).

% integer_of_int_induct
tff(fact_5688_integer__of__int__cases,axiom,
    ! [Xa: code_integer] :
      ~ ! [Y3: int] :
          ( ( Xa = aa(int,code_integer,code_integer_of_int,Y3) )
         => ~ member(int,Y3,top_top(set(int))) ) ).

% integer_of_int_cases
tff(fact_5689_zero__integer__def,axiom,
    zero_zero(code_integer) = aa(int,code_integer,code_integer_of_int,zero_zero(int)) ).

% zero_integer_def
tff(fact_5690_Code__Numeral_Oset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa),Xb)) ).

% Code_Numeral.set_bit_integer.abs_eq
tff(fact_5691_uminus__integer_Oabs__eq,axiom,
    ! [Xa: int] : aa(code_integer,code_integer,uminus_uminus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,code_integer,code_integer_of_int,aa(int,int,uminus_uminus(int),Xa)) ).

% uminus_integer.abs_eq
tff(fact_5692_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_5693_plus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),Xb)) ).

% plus_integer.abs_eq
tff(fact_5694_minus__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),minus_minus(int),Xa),Xb)) ).

% minus_integer.abs_eq
tff(fact_5695_times__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),times_times(int),Xa),Xb)) ).

% times_integer.abs_eq
tff(fact_5696_abs__integer_Oabs__eq,axiom,
    ! [Xa: int] : aa(code_integer,code_integer,abs_abs(code_integer),aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,code_integer,code_integer_of_int,aa(int,int,abs_abs(int),Xa)) ).

% abs_integer.abs_eq
tff(fact_5697_sgn__integer_Oabs__eq,axiom,
    ! [Xa: int] : aa(code_integer,code_integer,sgn_sgn(code_integer),aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,code_integer,code_integer_of_int,aa(int,int,sgn_sgn(int),Xa)) ).

% sgn_integer.abs_eq
tff(fact_5698_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xa: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xa) = Y )
         => ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xa)
           => ~ ( ( Y = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Xa),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_qh(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),Xa)),aa(code_integer,product_prod(A,A),A_of_integer,Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) )
               => ~ aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xa) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pelims
tff(fact_5699_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),I: code_integer,Zero: A] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I)
         => ( code_T4080844693773952564amples(A,A_of_integer,Zero,I) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I),aa(product_prod(A,A),list(A),aa(fun(A,fun(A,list(A))),fun(product_prod(A,A),list(A)),product_case_prod(A,A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_qi(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),A_of_integer),I),Zero)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.psimps
tff(fact_5700_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = $ite(Na = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_5701_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_5702_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_5703_signed__take__bit__Suc__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_5704_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_5705_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_5706_signed__take__bit__Suc__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(num,int,numeral_numeral(int),bit1(K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_5707_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_5708_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_5709_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_5710_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_5711_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_5712_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_5713_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_5714_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),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))),aa(nat,nat,suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_5715_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_5716_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_5717_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_5718_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_5719_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),A0: code_integer,P2: fun(code_integer,$o)] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),A0)
         => ( ! [I2: code_integer] :
                ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I2)
               => ( ! [X3: product_prod(A,A),Xa2: A,Y4: A] :
                      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),I2)
                     => ( ( X3 = aa(code_integer,product_prod(A,A),A_of_integer,I2) )
                       => ( ( aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa2),Y4) = X3 )
                         => aa(code_integer,$o,P2,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),I2),one_one(code_integer))) ) ) )
                 => aa(code_integer,$o,P2,I2) ) )
           => aa(code_integer,$o,P2,A0) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pinduct
tff(fact_5720_lowi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_lowi(Xa,Na) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% lowi_def
tff(fact_5721_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => summable(real,aTP_Lamp_oh(real,fun(nat,real),Xa)) ) ).

% summable_arctan_series
tff(fact_5722_highi__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_highi(Xa,Na) = heap_Time_return(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% highi_def
tff(fact_5723_highsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return(nat,vEBT_VEBT_high(Xa,Na)) = vEBT_VEBT_highi(Xa,Na) ).

% highsimp
tff(fact_5724_lowsimp,axiom,
    ! [Xa: nat,Na: nat] : heap_Time_return(nat,vEBT_VEBT_low(Xa,Na)) = vEBT_VEBT_lowi(Xa,Na) ).

% lowsimp
tff(fact_5725_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F3: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qj(nat,fun(fun(nat,A),fun(nat,A)),I),F3)) ) ).

% summable_single
tff(fact_5726_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_qk(nat,A)) ) ).

% summable_zero
tff(fact_5727_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ql(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_cmult_iff
tff(fact_5728_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qm(fun(nat,A),fun(A,fun(nat,A)),F3),C3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_divide_iff
tff(fact_5729_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,$o),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P2))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qn(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P2),F3)) ) ) ).

% summable_If_finite
tff(fact_5730_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A5: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A5)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qo(set(nat),fun(fun(nat,A),fun(nat,A)),A5),F3)) ) ) ).

% summable_If_finite_set
tff(fact_5731_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C3))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_5732_length__corresp,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(A)
     => ! [Tree_array: array(A),Tree_is: list(B)] :
          ( ( ex_assn(list(A),snga_assn(A,Tree_array)) = top_top(assn) )
         => ( heap_Time_return(nat,aa(list(B),nat,size_size(list(B)),Tree_is)) = array_len(A,Tree_array) ) ) ) ).

% length_corresp
tff(fact_5733_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_5734_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N6)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N6)
               => ( aa(nat,A,F3,N) = zero_zero(A) ) )
           => summable(A,F3) ) ) ) ).

% summable_finite
tff(fact_5735_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ql(A,fun(fun(nat,A),fun(nat,A)),C3),F3))
         => ( ( C3 != zero_zero(A) )
           => summable(A,F3) ) ) ) ).

% summable_mult_D
tff(fact_5736_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),aa(nat,real,G3,N)) )
         => ( summable(real,G3)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test
tff(fact_5737_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G3: fun(nat,real),N6: nat,F3: fun(nat,A)] :
          ( summable(real,G3)
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),aa(nat,real,G3,N)) )
           => summable(A,F3) ) ) ) ).

% summable_comparison_test'
tff(fact_5738_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F3))
        <=> summable(A,F3) ) ) ).

% summable_Suc_iff
tff(fact_5739_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C3: A] :
          ( summable(A,aTP_Lamp_qq(A,fun(nat,A),C3))
        <=> ( C3 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_5740_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A)] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,G3,N))
         => ( summable(A,F3)
           => ( summable(A,G3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),suminf(A,G3)) ) ) ) ) ).

% suminf_le
tff(fact_5741_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,A)),F3),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_qr(fun(nat,A),fun(A,fun(nat,real)),F3),Z)) ) ) ) ).

% powser_insidea
tff(fact_5742_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( ( suminf(A,F3) = zero_zero(A) )
            <=> ! [N5: nat] : aa(nat,A,F3,N5) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_5743_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_nonneg
tff(fact_5744_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,N))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ).

% suminf_pos
tff(fact_5745_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_qs(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_0_powser
tff(fact_5746_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_qt(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_zero_power'
tff(fact_5747_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qu(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_split_head
tff(fact_5748_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_qw(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% powser_split_head(3)
tff(fact_5749_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),aa(nat,real,G3,N)) )
         => ( summable(real,G3)
           => summable(real,aTP_Lamp_qx(fun(nat,A),fun(nat,real),F3)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_5750_summable__rabs__comparison__test,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ? [N9: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F3,N))),aa(nat,real,G3,N)) )
     => ( summable(real,G3)
       => summable(real,aTP_Lamp_qy(fun(nat,real),fun(nat,real),F3)) ) ) ).

% summable_rabs_comparison_test
tff(fact_5751_summable__rabs,axiom,
    ! [F3: fun(nat,real)] :
      ( summable(real,aTP_Lamp_qy(fun(nat,real),fun(nat,real),F3))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F3))),suminf(real,aTP_Lamp_qy(fun(nat,real),fun(nat,real),F3))) ) ).

% summable_rabs
tff(fact_5752_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3))
            <=> ? [I3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I3)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_5753_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I: nat] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F3)) ) ) ) ) ).

% suminf_pos2
tff(fact_5754_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xa: A] :
          ( summable(A,F3)
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_lessThan(nat,N))),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F3)),Xa) ) ) ) ).

% suminf_le_const
tff(fact_5755_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ) ).

% powser_inside
tff(fact_5756_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( suminf(A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F3)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,F3)),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_5757_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Xa: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_lessThan(nat,N))),Xa)
           => summable(A,F3) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_5758_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B5: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N))
         => ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A2),set_ord_atMost(nat,N))),B5)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_5759_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),C3)) ) ) ).

% summable_geometric
tff(fact_5760_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xa)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_5761_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_qz(fun(nat,A),fun(nat,real),F3))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F3))),suminf(real,aTP_Lamp_qz(fun(nat,A),fun(nat,real),F3))) ) ) ).

% summable_norm
tff(fact_5762_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I5: set(nat)] :
          ( summable(A,F3)
         => ( aa(set(nat),$o,finite_finite2(nat),I5)
           => ( ! [N: nat] :
                  ( member(nat,N,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),I5)),suminf(A,F3)) ) ) ) ) ).

% sum_le_suminf
tff(fact_5763_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Na: nat] :
          ( summable(A,F3)
         => ( ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,M3)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_lessThan(nat,Na))),suminf(A,F3)) ) ) ) ).

% sum_less_suminf
tff(fact_5764_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N8: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M5)
                 => ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or7035219750837199246ssThan(nat,M5,N5)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_5765_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F3,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_qw(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_5766_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_qw(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z) = aa(A,A,aa(A,fun(A,A),minus_minus(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_5767_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F3)
           => ? [N10: nat] :
              ! [N7: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N7)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(nat,fun(nat,A)),F3),N7)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_5768_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),E3: real] :
          ( summable(A,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ~ ! [N10: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),M2)
                     => ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_or1337092689740270186AtMost(nat,M2,N7)))),E3) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_5769_summable__power__series,axiom,
    ! [F3: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_rb(fun(nat,real),fun(real,fun(nat,real)),F3),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_5770_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A2: fun(nat,A),M7: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),R0)
           => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N))),M7)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_rc(real,fun(fun(nat,A),fun(nat,real)),R2),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_5771_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C3: real,N6: nat,F3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),one_one(real))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,aa(nat,nat,suc,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),C3),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N)))) )
           => summable(A,F3) ) ) ) ).

% summable_ratio_test
tff(fact_5772_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F3)
         => ( ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,M3)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F3,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),set_ord_lessThan(nat,Na))),suminf(A,F3)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_5773_suminf__eq__SUP__real,axiom,
    ! [X4: fun(nat,real)] :
      ( summable(real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,X4,I2))
       => ( suminf(real,X4) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_rd(fun(nat,real),fun(nat,real),X4)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_5774_sum__pos__lt__pair,axiom,
    ! [F3: fun(nat,real),K: nat] :
      ( summable(real,F3)
     => ( ! [D3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),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,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F3),set_ord_lessThan(nat,K))),suminf(real,F3)) ) ) ).

% sum_pos_lt_pair
tff(fact_5775_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_5776_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = 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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_5777_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( tanh(real,aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_5778_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_5779_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_5780_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_5781_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_5782_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( tanh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_5783_tanh__real__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xa)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% tanh_real_less_iff
tff(fact_5784_tanh__real__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xa)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% tanh_real_le_iff
tff(fact_5785_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_5786_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),tanh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% tanh_real_pos_iff
tff(fact_5787_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tanh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_5788_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_5789_flip__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa),Xb)) ).

% flip_bit_integer.abs_eq
tff(fact_5790_unset__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa),Xb)) ).

% unset_bit_integer.abs_eq
tff(fact_5791_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Na),K)),K) ).

% unset_bit_less_eq
tff(fact_5792_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),tanh(real,Xa)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_5793_tanh__real__gt__neg1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),tanh(real,Xa)) ).

% tanh_real_gt_neg1
tff(fact_5794_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_5795_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2))
        <=> ~ ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_5796_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_5797_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xa))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_5798_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T7),sin(real,T7)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_5799_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nl(nat,fun(list(A),$o),I),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_5800_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_5801_sorted__wrt__rev__linord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,aTP_Lamp_je(A,fun(A,$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,L)) ) ) ).

% sorted_wrt_rev_linord
tff(fact_5802_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_5803_rev__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rev(A,Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Na))) ) ) ).

% rev_nth
tff(fact_5804_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( rev(A,list_update(A,Xs,K,Y)) = list_update(A,rev(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_5805_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I3))),aa(nat,A,nth(A,Xs),I3)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_5806_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_5807_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I3)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_5808_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( foldr(A,A,ord_max(A),Xs,Y) = $ite(Xs = nil(A),Y,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Y)) ) ) ) ).

% foldr_max_sorted
tff(fact_5809_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,tan(real),X)) ) ) ).

% lemma_tan_total
tff(fact_5810_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_gt_zero
tff(fact_5811_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y ) ) ).

% lemma_tan_total1
tff(fact_5812_tan__mono__lt__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_5813_tan__monotone_H,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_5814_tan__monotone,axiom,
    ! [Y: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xa)) ) ) ) ).

% tan_monotone
tff(fact_5815_tan__total,axiom,
    ! [Y: real] :
    ? [X: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X) = Y )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y4) = Y ) )
         => ( Y4 = X ) ) ) ).

% tan_total
tff(fact_5816_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_5817_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X) = Y ) ) ) ).

% tan_total_pos
tff(fact_5818_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_5819_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_5820_tan__mono__le__eq,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_5821_tan__mono__le,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_5822_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xa))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_5823_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Y))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),arctan(Y)) = Y ) ) ).

% arctan
tff(fact_5824_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( arctan(aa(real,real,tan(real),Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_5825_arctan__unique,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xa) = Y )
         => ( arctan(Y) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_5826_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_5827_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_5828_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_5829_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ? [Z3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z3) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_5830_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)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_5831_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_nl(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_5832_arctan__def,axiom,
    ! [Y: real] : arctan(Y) = the(real,aTP_Lamp_re(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_5833_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_5834_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_nl(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_5835_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_rf(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_5836_time__array__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,F3: fun(A,A),P3: array(A),H2: heap_ext(product_unit)] :
          time_time(array(A),array_map_entry(A,I,F3,P3),H2) = $ite(time_fails(array(A),array_map_entry(A,I,F3,P3),H2),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_map_entry
tff(fact_5837_sin__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),Xa)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_5838_real__sqrt__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ).

% real_sqrt_less_iff
tff(fact_5839_real__sqrt__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ).

% real_sqrt_le_iff
tff(fact_5840_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_5841_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_5842_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_5843_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_5844_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_5845_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_5846_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_5847_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_5848_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_5849_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_5850_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_5851_real__sqrt__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_5852_real__sqrt__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_5853_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_5854_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,sqrt,Xa) = zero_zero(real) )
       => ( Xa = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_5855_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_5856_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_5857_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,sqrt,Xa)) = aa(real,real,sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_5858_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_5859_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_5860_arcsin__le__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_5861_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_5862_arcsin__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_5863_arcsin__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( aa(real,real,arcsin,Xa) = aa(real,real,arcsin,Y) )
        <=> ( Xa = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_5864_sqrt2__less__2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sqrt2_less_2
tff(fact_5865_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xa)),Xa) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_5866_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xa)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_5867_real__less__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_5868_real__le__rsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_5869_sqrt__le__D,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_5870_arcsin__less__arcsin,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_5871_arcsin__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_5872_real__le__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_5873_real__sqrt__unique,axiom,
    ! [Y: real,Xa: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xa) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_5874_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_5875_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_5876_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_5877_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C3: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_5878_sqrt__ge__absD,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_5879_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xa)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_5880_real__less__lsqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_5881_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_5882_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_5883_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_5884_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_5885_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_5886_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_5887_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xa)),Na) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_5888_arsinh__real__aux,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_5889_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Y: real,Xaa: real,Ya2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_5890_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_5891_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_5892_cos__x__y__le__one,axiom,
    ! [Xa: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_5893_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_5894_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,real,arcosh(real),Xa) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_5895_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_5896_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_5897_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_5898_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_5899_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_5900_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_5901_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa))
     => ( sin(real,Xa) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_5902_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_5903_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_5904_arcsin__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_5905_le__arcsin__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,arcsin,Xa))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_5906_cos__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,Xa) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_5907_time__array__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xa: A,P3: array(A),H2: heap_ext(product_unit),F3: fun(A,A)] :
          time_time(A,array_swap(A,I,Xa,P3),H2) = $ite(time_fails(array(A),array_map_entry(A,I,F3,P3),H2),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_swap
tff(fact_5908_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_5909_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xa)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_5910_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_5911_arccos__le__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_5912_arccos__eq__iff,axiom,
    ! [Xa: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( aa(real,real,arccos,Xa) = aa(real,real,arccos,Y) )
      <=> ( Xa = Y ) ) ) ).

% arccos_eq_iff
tff(fact_5913_arccos__le__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_5914_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_5915_arccos__less__arccos,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_5916_arccos__less__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_5917_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_5918_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,real,arccos,cos(real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_5919_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_5920_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_5921_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_5922_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_5923_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xa)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_5924_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xa)
       => ( aa(real,real,arccos,cos(real,Xa)) = aa(real,real,uminus_uminus(real),Xa) ) ) ) ).

% arccos_cos2
tff(fact_5925_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_5926_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_rg(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_5927_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_5928_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,aa(real,fun(real,real),minus_minus(real),pi),aa(real,real,arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_5929_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_5930_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => sums(A,aTP_Lamp_rh(A,fun(nat,A),Z),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(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_5931_remove__rev__alt__def,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : remove_rev(A,Xa,Xs) = filter2(A,aTP_Lamp_mt(A,fun(A,$o),Xa),rev(A,Xs)) ).

% remove_rev_alt_def
tff(fact_5932_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xa)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_5933_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_5934_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_qk(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_5935_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xa: A] :
          ( sums(A,aTP_Lamp_qs(fun(nat,A),fun(nat,A),A2),Xa)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xa ) ) ) ).

% powser_sums_zero_iff
tff(fact_5936_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F3: fun(nat,A)] : sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qj(nat,fun(fun(nat,A),fun(nat,A)),I),F3),aa(nat,A,F3,I)) ) ).

% sums_single
tff(fact_5937_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] :
          ( ! [N: nat] : aa(nat,A,F3,N) = zero_zero(A)
         => sums(A,F3,zero_zero(A)) ) ) ).

% sums_0
tff(fact_5938_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,A),Sa: A,T2: A] :
          ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,G3,N))
         => ( sums(A,F3,Sa)
           => ( sums(A,G3,T2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Sa),T2) ) ) ) ) ).

% sums_le
tff(fact_5939_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D2: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ri(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C3))
          <=> sums(A,F3,D2) ) ) ) ).

% sums_mult2_iff
tff(fact_5940_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F3: fun(nat,A),D2: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rj(A,fun(fun(nat,A),fun(nat,A)),C3),F3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2))
          <=> sums(A,F3,D2) ) ) ) ).

% sums_mult_iff
tff(fact_5941_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ql(A,fun(fun(nat,A),fun(nat,A)),C3),F3),A2)
         => ( ( C3 != zero_zero(A) )
           => sums(A,F3,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C3)) ) ) ) ).

% sums_mult_D
tff(fact_5942_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Sa: A] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F3),Sa)
           => sums(A,F3,Sa) ) ) ) ).

% sums_Suc_imp
tff(fact_5943_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),Sa: A] :
          ( sums(A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F3),Sa)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),Sa),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_5944_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_rk(fun(nat,A),fun(nat,A),F3),L)
         => sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_5945_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F3: fun(nat,A),Sa: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
             => ( aa(nat,A,F3,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rl(nat,fun(fun(nat,A),fun(nat,A)),Na),F3),Sa)
          <=> sums(A,F3,Sa) ) ) ) ).

% sums_zero_iff_shift
tff(fact_5946_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N6: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N6)
         => ( ! [N: nat] :
                ( ~ member(nat,N,N6)
               => ( aa(nat,A,F3,N) = zero_zero(A) ) )
           => sums(A,F3,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),N6)) ) ) ) ).

% sums_finite
tff(fact_5947_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P2: fun(nat,$o),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P2))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qn(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P2),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),aa(fun(nat,$o),set(nat),collect(nat),P2))) ) ) ).

% sums_If_finite
tff(fact_5948_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A5: set(nat),F3: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A5)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qo(set(nat),fun(fun(nat,A),fun(nat,A)),A5),F3),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F3),A5)) ) ) ).

% sums_If_finite_set
tff(fact_5949_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_rm(nat,fun(A,fun(nat,A)),M),Z),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)) ) ).

% powser_sums_if
tff(fact_5950_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_qs(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_5951_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G3: fun(nat,A),S2: A,A5: set(nat),S5: A,F3: fun(nat,A)] :
          ( sums(A,G3,S2)
         => ( aa(set(nat),$o,finite_finite2(nat),A5)
           => ( ( S5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),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_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G3),F3)),A5)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ro(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G3),A5),F3),S5) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_5952_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),Sa: A,K: nat] :
          ( sums(A,F3,Sa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_rp(fun(nat,A),fun(nat,fun(nat,A)),F3),K),Sa) ) ) ) ).

% sums_group
tff(fact_5953_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => sums(A,aa(A,fun(nat,A),power_power(A),C3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),minus_minus(A),one_one(A)),C3))) ) ) ).

% geometric_sums
tff(fact_5954_power__half__series,axiom,
    sums(real,aTP_Lamp_rq(nat,real),one_one(real)) ).

% power_half_series
tff(fact_5955_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xa)) ) ) ).

% cot_gt_zero
tff(fact_5956_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_rs(A,fun(A,fun(nat,A)),Xa),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_5957_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ru(A,fun(A,fun(nat,A)),Xa),Y),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_5958_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A,B2: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa) )
        <=> ( ( A2 = B2 )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_5959_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real] : aa(A,A,real_V8093663219630862766scaleR(A,A2),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_5960_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(real) )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_5961_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xa) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_5962_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,A2: real,B2: real] :
          ( ( Xa != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa) )
           => ( A2 = B2 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_5963_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C3)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_5964_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xa)) ) ) ) ).

% scaleR_right_mono
tff(fact_5965_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_5966_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_5967_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_5968_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_5969_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_5970_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V3: real,A2: A,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V3)),A2) = Xa )
        <=> $ite(V3 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V3),Xa)) ) ) ).

% vector_fraction_eq_iff
tff(fact_5971_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,U: real,V3: real,A2: A] :
          ( ( Xa = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V3)),A2) )
        <=> $ite(V3 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V3),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_5972_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),B2)),E3)),C3)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_5973_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C3: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),E3)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_5974_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_5975_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_5976_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_5977_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_5978_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_5979_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_5980_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_5981_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xa: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_5982_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C3: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_5983_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_5984_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)),Xa) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_5985_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C3)),set_or1337092689740270186AtMost(A,Xa,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C3),Xa),aa(A,A,real_V8093663219630862766scaleR(A,C3),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5986_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_5987_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_5988_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_5989_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_5990_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_5991_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_5992_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_5993_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_5994_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A2: real,Xa: A] :
          ( ( A2 != zero_zero(real) )
         => ( ( Xa != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xa)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A2)),aa(A,A,inverse_inverse(A),Xa)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_5995_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_5996_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_5997_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_5998_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_5999_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_6000_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_6001_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B2: A,A2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_6002_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A2: A,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_6003_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_rv(A,fun(nat,A),Xa))) ) ).

% exp_first_term
tff(fact_6004_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : sums(A,aTP_Lamp_rw(A,fun(nat,A),Xa),sinh(A,Xa)) ) ).

% sinh_converges
tff(fact_6005_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ry(A,fun(A,fun(nat,A)),Xa),Y),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_6006_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C3: fun(nat,A),Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_rz(fun(nat,A),fun(A,fun(nat,A)),C3),Xa))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_sa(fun(nat,A),fun(A,fun(nat,A)),C3),Xa),suminf(A,aa(A,fun(nat,A),aTP_Lamp_rz(fun(nat,A),fun(A,fun(nat,A)),C3),Xa))) ) ) ).

% diffs_equiv
tff(fact_6007_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : sums(A,aTP_Lamp_sb(A,fun(nat,A),Xa),cosh(A,Xa)) ) ).

% cosh_converges
tff(fact_6008_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cosh(A,zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_6009_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xa)) ).

% cosh_real_ge_1
tff(fact_6010_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_nonneg
tff(fact_6011_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_6012_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_6013_cosh__real__pos,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_pos
tff(fact_6014_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),cosh(real,Xa)) ).

% sinh_less_cosh_real
tff(fact_6015_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),cosh(real,Xa)) ).

% sinh_le_cosh_real
tff(fact_6016_cosh__real__strict__mono,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_6017_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_6018_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_6019_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,arcosh(real),cosh(real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_6020_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C3: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,C3),X3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X3))),aa(nat,A,C3,aa(nat,nat,suc,X3))) ) ).

% diffs_def
tff(fact_6021_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Y: A] :
          ( ( cosh(A,Xa) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( tanh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),tanh(A,Xa)),tanh(A,Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tanh(A,Xa)),tanh(A,Y)))) ) ) ) ) ).

% tanh_add
tff(fact_6022_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,K4: real,C3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),K4)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),K4)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),C3),X)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sd(A,fun(fun(nat,A),fun(nat,A)),Xa),C3)) ) ) ) ).

% termdiff_converges
tff(fact_6023_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cosh(A,Xa) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_6024_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( cosh(real,aa(real,real,ln_ln(real),Xa)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_6025_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),aa(nat,nat,suc,zero_zero(nat)))))),aa(word(A),int,ring_1_signed(A,int),W2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_6026_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_6027_signed__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,ring_1_signed(B,A),zero_zero(word(B))) = zero_zero(A) ) ) ).

% signed_0
tff(fact_6028_signed__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [W2: word(B)] :
          ( ( aa(word(B),A,ring_1_signed(B,A),W2) = zero_zero(A) )
        <=> ( W2 = zero_zero(word(B)) ) ) ) ).

% signed_eq_0_iff
tff(fact_6029_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),bit0(Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Na))) ).

% eval_nat_numeral(2)
tff(fact_6030_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),one_one(nat)))),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W2)),Xa) ) ) ).

% sint_above_size
tff(fact_6031_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,W2: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),one_one(nat)))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(word(A),int,ring_1_signed(A,int),W2)) ) ) ).

% sint_below_size
tff(fact_6032_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_se(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_6033_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_sf(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_6034_set__decode__0,axiom,
    ! [Xa: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xa))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) ) ).

% set_decode_0
tff(fact_6035_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))),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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_6036_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = zero_zero(A) )
        <=> ~ (P2) ) ) ).

% of_bool_eq_0_iff
tff(fact_6037_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_6038_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: $o,Q2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q2)))
        <=> ( (P2)
           => (Q2) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_6039_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P2: $o,Q2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),aa($o,A,zero_neq_one_of_bool(A),(Q2)))
        <=> ( ~ (P2)
            & (Q2) ) ) ) ).

% of_bool_less_iff
tff(fact_6040_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P2: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P2))) = aa($o,A,zero_neq_one_of_bool(A),(P2)) ) ).

% of_nat_of_bool
tff(fact_6041_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% zero_less_of_bool_iff
tff(fact_6042_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),one_one(A))
        <=> ~ (P2) ) ) ).

% of_bool_less_one_iff
tff(fact_6043_Suc__0__mod__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa($o,nat,zero_neq_one_of_bool(nat),Na != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_6044_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_6045_set__encode__inverse,axiom,
    ! [A5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A5)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A5)) = A5 ) ) ).

% set_encode_inverse
tff(fact_6046_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_6047_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6048_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_6049_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_6050_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% sgn_of_nat
tff(fact_6051_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),P2: fun(A,$o),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_sg(fun(A,$o),fun(fun(A,B),fun(A,B)),P2),F3)),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_6052_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),F3: fun(A,B),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_sh(fun(A,B),fun(fun(A,$o),fun(A,B)),F3),P2)),A5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_6053_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W2),aa($o,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))) ) ).

% mod_word_by_minus_1_eq
tff(fact_6054_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,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_6055_set__decode__Suc,axiom,
    ! [Na: nat,Xa: nat] :
      ( member(nat,aa(nat,nat,suc,Na),nat_set_decode(Xa))
    <=> member(nat,Na,nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_6056_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A5: set(A),P2: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),$o,finite_finite2(A),A5)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_si(fun(A,$o),fun(A,B),P2)),A5) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_6057_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_6058_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_6059_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_6060_finite__set__decode,axiom,
    ! [Na: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_6061_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P2))) ) ).

% zero_less_eq_of_bool
tff(fact_6062_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_6063_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,$o),P3: $o] :
          ( aa(A,$o,P2,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P2,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P2,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_6064_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: fun(A,$o),P3: $o] :
          ( aa(A,$o,P2,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P2,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P2,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_6065_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P2: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P2))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_6066_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_sj(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_6067_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% subset_decode_imp_le
tff(fact_6068_horner__sum__of__bool__2__less,axiom,
    ! [Bs2: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs2))) ).

% horner_sum_of_bool_2_less
tff(fact_6069_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_6070_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_6071_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),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),Na))) ) ).

% exp_div_exp_eq
tff(fact_6072_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: 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),Na))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M)))))) ) ).

% divide_int_unfold
tff(fact_6073_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      aa(int,int,aa(int,fun(int,int),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),Na))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        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)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na))))) ) ).

% modulo_int_unfold
tff(fact_6074_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,B,F3),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_sk(fun(A,B),fun(list(A),fun(A,$o)),F3),Xs),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_sl(fun(A,B),fun(list(A),fun(A,$o)),F3),Xs),Xs)),aa(list(A),list(A),linorder_sort_key(A,B,F3),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_sm(fun(A,B),fun(list(A),fun(A,$o)),F3),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_6075_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_jo(A,A)),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_jo(A,A)),filter2(A,aTP_Lamp_sn(list(A),fun(A,$o),Xs),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),filter2(A,aTP_Lamp_so(list(A),fun(A,$o),Xs),Xs)),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_jo(A,A)),filter2(A,aa(A,fun(A,$o),ord_less(A),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_6076_False__map2__or,axiom,
    ! [Xs: list($o),Ys: list($o)] :
      ( aa(set($o),$o,aa(set($o),fun(set($o),$o),ord_less_eq(set($o)),aa(list($o),set($o),set2($o),Xs)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$false),bot_bot(set($o))))
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(list($o),nat,size_size(list($o)),Xs) )
       => ( aa(list(product_prod($o,$o)),list($o),map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fdisj)),zip($o,$o,Xs,Ys)) = Ys ) ) ) ).

% False_map2_or
tff(fact_6077_False__map2__and,axiom,
    ! [Xs: list($o),Ys: list($o)] :
      ( aa(set($o),$o,aa(set($o),fun(set($o),$o),ord_less_eq(set($o)),aa(list($o),set($o),set2($o),Xs)),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$false),bot_bot(set($o))))
     => ( ( aa(list($o),nat,size_size(list($o)),Ys) = aa(list($o),nat,size_size(list($o)),Xs) )
       => ( aa(list(product_prod($o,$o)),list($o),map(product_prod($o,$o),$o,aa(fun($o,fun($o,$o)),fun(product_prod($o,$o),$o),product_case_prod($o,$o,$o),fconj)),zip($o,$o,Xs,Ys)) = Xs ) ) ) ).

% False_map2_and
tff(fact_6078_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_jo(A,A)),Xs)) ) ).

% sorted_sort
tff(fact_6079_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_jo(A,A)),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_6080_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),linorder_sort_key(B,A,F3),Xs))) ) ).

% sorted_sort_key
tff(fact_6081_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
          <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% word_2p_lem
tff(fact_6082_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_6083_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_6084_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_6085_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_6086_unsigned__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),zero_zero(word(B))) = zero_zero(A) ) ) ).

% unsigned_0
tff(fact_6087_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W2)) ) ).

% uint_nonnegative
tff(fact_6088_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),zero_zero(int))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),Xa) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_6089_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)) ) ).

% uint_ge_0
tff(fact_6090_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_6091_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_6092_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_le_no
tff(fact_6093_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_less_no
tff(fact_6094_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_6095_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_6096_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))
        <=> ( ( Na = zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_6097_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,zero_zero(nat))),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_6098_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_of_exp
tff(fact_6099_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_6100_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_iff_unsigned
tff(fact_6101_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W2: word(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W2)) ) ).

% unsigned_greater_eq
tff(fact_6102_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert($o),$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_6103_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_6104_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_less_def
tff(fact_6105_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_6106_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_6107_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),Q3)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_6108_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_6109_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_6110_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_6111_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_le_def
tff(fact_6112_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_6113_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_6114_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ).

% take_bit_nonnegative
tff(fact_6115_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_6116_take__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Xa),Xb)) ).

% take_bit_integer.abs_eq
tff(fact_6117_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),B2) )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_6118_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),bit_ri4674362597316999326ke_bit(A,M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_6119_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se8732182000553998342ip_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_6120_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_6121_unsigned__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [W2: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),W2) = zero_zero(A) )
        <=> ( W2 = zero_zero(word(B)) ) ) ) ).

% unsigned_eq_0_iff
tff(fact_6122_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_6123_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_6124_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_add_ge0
tff(fact_6125_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_mult_ge0
tff(fact_6126_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,Xa: word(A)] :
          ( ( aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,aa(word(A),int,semiring_1_unsigned(A,int),Xa))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_6127_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))) ) ).

% uint_add_le
tff(fact_6128_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple_iff
tff(fact_6129_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_plus_simple
tff(fact_6130_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Xa)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)) ) ) ).

% no_ulen_sub
tff(fact_6131_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_sub_lem
tff(fact_6132_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y))) ) ).

% uint_sub_ge
tff(fact_6133_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)),Xa)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_iff
tff(fact_6134_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_minus_simple_alt
tff(fact_6135_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Na: int,K4: word(A),N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_6136_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Na: int,K4: word(A),N2: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K4))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_6137_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q3: word(A),Na: int,K4: word(A),N2: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),Q3) ) ) ) ) ).

% udvd_incr0
tff(fact_6138_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q3: word(A),Na: int,K4: word(A),N2: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Q3),K4)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_6139_udvd__unfold__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( udvd(A,A2,B2)
        <=> ? [N5: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N5)
              & ( aa(word(A),int,semiring_1_unsigned(A,int),B2) = aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ) ) ) ).

% udvd_unfold_int
tff(fact_6140_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_6141_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_6142_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_less_exp
tff(fact_6143_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_6144_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_less_exp
tff(fact_6145_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q3: word(A),Ua: int,Na: int,K4: word(A),N2: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P3),K4)),Q3) ) ) ) ) ).

% udvd_incr'
tff(fact_6146_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Q3: word(A),Ua: int,Na: int,K4: word(A),N2: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),Q3)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N2),aa(word(A),int,semiring_1_unsigned(A,int),K4))) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Q3),K4)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_6147_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_6148_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: nat,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),L),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% bin_last_bintrunc
tff(fact_6149_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),Na),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_6150_take__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_6151_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_6152_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_6153_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_6154_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_6155_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_6156_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),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,Na))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_6157_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2) = 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(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_6158_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),W2))) ) ) ).

% uint_range_size
tff(fact_6159_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_2p
tff(fact_6160_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_6161_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),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))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_6162_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_6163_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_6164_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_6165_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% uint_plus_if_size
tff(fact_6166_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% uint_sub_if_size
tff(fact_6167_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),K)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_6168_take__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = 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(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_6169_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Na: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Na = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Na ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_sq(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Na)) ) ) ) ).

% transpose_rectangle
tff(fact_6170_take__upt,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),Na)
     => ( take(nat,M,upt(I,Na)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_6171_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_6172_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6173_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K) )
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_6174_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),one_one(nat)) ) ) ).

% last_upt
tff(fact_6175_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != zero_zero(word(A)) )
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_minus_one
tff(fact_6176_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_6177_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),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_6178_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( groups8242544230860333062m_list(nat,upt(M,Na)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_de(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) ) ) ).

% sum_list_upt
tff(fact_6179_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na))),nil(nat)) ).

% upt_rec_numeral
tff(fact_6180_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( upt(I,J) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Xs),Ys) )
      <=> ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),J)
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_6181_sorted__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,Na)) ).

% sorted_upt
tff(fact_6182_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C3: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B2) = C3 )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),C3)) ) ) ) ).

% unat_of_nat_less
tff(fact_6183_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = zero_zero(nat) )
        <=> ( Xa = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_6184_unat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = zero_zero(nat) ) ) ).

% unat_0
tff(fact_6185_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_less_nat_alt
tff(fact_6186_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),B2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% unat_mono
tff(fact_6187_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_le_nat_alt
tff(fact_6188_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y) ) ) ).

% le_ucast_ucast_le
tff(fact_6189_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)) = Y ) ) ) ).

% le_unat_uoi
tff(fact_6190_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_sr(nat,fun(nat,nat),Ofs)),upt(A2,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_6191_map__Suc__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(M,Na)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na)) ).

% map_Suc_upt
tff(fact_6192_map__add__upt,axiom,
    ! [Na: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_sr(nat,fun(nat,nat),Na)),upt(zero_zero(nat),M)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% map_add_upt
tff(fact_6193_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less(nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_6194_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6195_butlast__upt,axiom,
    ! [M: nat,Na: nat] : butlast(nat,upt(M,Na)) = upt(M,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ).

% butlast_upt
tff(fact_6196_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Na: nat,Ns: list(nat),Q3: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Na),Ns)) = upt(M,Q3) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Na),Ns) = upt(aa(nat,nat,suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_6197_filter__upt__take__conv,axiom,
    ! [A: $tType,P2: fun(A,$o),M: nat,L: list(A),Na: nat] : filter2(nat,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_ss(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),P2),M),L),upt(Na,M)) = filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_st(fun(A,$o),fun(list(A),fun(nat,$o)),P2),L),upt(Na,M)) ).

% filter_upt_take_conv
tff(fact_6198_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( Xa = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_6199_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ( Xa != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_6200_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6201_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(B),nat,semiring_1_unsigned(B,nat),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(B),int,semiring_1_unsigned(B,int),B2)) ) ) ).

% un_ui_le
tff(fact_6202_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_plus_simple
tff(fact_6203_unat__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ).

% unat_sub
tff(fact_6204_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Na) ) ) ).

% unat_less_helper
tff(fact_6205_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),Xa) ) ) ).

% word_of_nat_less
tff(fact_6206_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P2: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U4)
     => ( ! [I2: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I2)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),U4) )
           => ~ aa(nat,$o,P2,I2) )
       => ( filter2(nat,P2,upt(zero_zero(nat),U)) = filter2(nat,P2,upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_6207_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),Xa) ) ) ).

% word_of_nat_le
tff(fact_6208_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),B2) ) ) ).

% word_unat_less_le
tff(fact_6209_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_su(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_6210_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_6211_atLeastAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or1337092689740270186AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(Na,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_6212_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_6213_atLeast__upt,axiom,
    ! [Na: nat] : set_ord_lessThan(nat,Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Na)) ).

% atLeast_upt
tff(fact_6214_map__replicate__trivial,axiom,
    ! [A: $tType,Xa: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_sv(A,fun(nat,A),Xa)),upt(zero_zero(nat),I)) = replicate(A,I,Xa) ).

% map_replicate_trivial
tff(fact_6215_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Xa),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xa )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6216_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)) ) ) ).

% unat_1_0
tff(fact_6217_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% unat_max_word_pos
tff(fact_6218_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A))) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unatSuc2
tff(fact_6219_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),Na) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),Na)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unatSuc
tff(fact_6220_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xa)
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% Suc_unat_diff_1
tff(fact_6221_unat__Suc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          ( ( Na != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Na),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Na)) ) ) ) ).

% unat_Suc2
tff(fact_6222_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)) ) ) ) ).

% uno_simps(1)
tff(fact_6223_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A)] :
          ( ( P3 != zero_zero(word(A)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),P3),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P3)) ) ) ).

% measure_unat
tff(fact_6224_atMost__upto,axiom,
    ! [Na: nat] : set_ord_atMost(nat,Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Na))) ).

% atMost_upto
tff(fact_6225_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6226_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),Na: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,Na))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_sw(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),Na))) ).

% map_upt_Suc
tff(fact_6227_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_6228_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Na: nat,M: nat,F3: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(M,Na))),I) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_6229_upt__eq__lel__conv,axiom,
    ! [L: nat,H2: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H2) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H2) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),I)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),H2) ) ) ).

% upt_eq_lel_conv
tff(fact_6230_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6231_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_6232_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Na: nat,M: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(M,Na)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6233_map__nth__upt__drop__take__conv,axiom,
    ! [A: $tType,N6: nat,L: list(A),M7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(list(nat),list(A),map(nat,A,nth(A,L)),upt(M7,N6)) = drop(A,M7,take(A,N6,L)) ) ) ).

% map_nth_upt_drop_take_conv
tff(fact_6234_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,MaxBound: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_6235_filter__upt__last,axiom,
    ! [A: $tType,P2: fun(A,$o),L: list(A),Js: list(nat),J: nat,I: nat] :
      ( ( filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_st(fun(A,$o),fun(list(A),fun(nat,$o)),P2),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),I)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,P2,aa(nat,A,nth(A,L),I)) ) ) ) ).

% filter_upt_last
tff(fact_6236_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Na),M) = one_one(word(A)) )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),M))) ) ) ) ).

% word_div_eq_1_iff
tff(fact_6237_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))) ) ).

% unat_plus_if_size
tff(fact_6238_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_6239_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xa))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_6240_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Na: nat,F3: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,Na,F3),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_sx(nat,fun(fun(nat,A),fun(array(A),assn)),Na),F3)) ) ).

% make_rule
tff(fact_6241_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Na: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_6242_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_6243_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Na) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_6244_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit1(M))),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Na) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_6245_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_6246_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),zero_zero(int))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_6247_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W2)))),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),Na) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_6248_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(W2)))),aa(nat,nat,suc,Na))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),Na) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6249_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),zero_zero(nat))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% bit_0
tff(fact_6250_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_6251_bin__nth__minus__Bit0,axiom,
    ! [Na: nat,W2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit0(W2))),Na)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_6252_bin__nth__minus__Bit1,axiom,
    ! [Na: nat,W2: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit1(W2))),Na)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_6253_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_6254_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: $o,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),Na)
        <=> ( (B2)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_6255_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),Na)
        <=> ( Na = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_6256_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(nat,nat,suc,Na)) ) ).

% not_bit_1_Suc
tff(fact_6257_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [Na: nat,K: int,L: int] :
      ( ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),L) )
     => ( aa(list(nat),list($o),map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K)),upt(zero_zero(nat),Na)) = aa(list(nat),list($o),map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L)),upt(zero_zero(nat),Na)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_6258_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6259_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6260_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,Na))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Na) ) ) ).

% bit_Suc
tff(fact_6261_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),M2)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M2)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N),one_one(nat)))
              <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ) ).

% int_bit_bound
tff(fact_6262_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6263_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6264_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs2: list($o),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list($o),nat,size_size(list($o)),Bs2))
            & aa(nat,$o,nth($o,Bs2),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6265_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Na: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),Na)
          <=> $ite(Na = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6266_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> $ite(Na = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_6267_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list(nat),list($o),map(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2)),upt(zero_zero(nat),Na))) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6268_take__bit__Suc__from__most,axiom,
    ! [Na: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,suc,Na)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),K)) ).

% take_bit_Suc_from_most
tff(fact_6269_num_Osize__gen_I3_J,axiom,
    ! [X32: num] : size_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_6270_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a: fun(A,uint32),H_b: fun(B,uint32),Xa: A,Xaa: B] : hash_hash_code_prod(A,B,H_a,H_b,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xaa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a,Xa)),aa(num,uint32,numeral_numeral(uint32),bit0(bit1(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit1(one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(B,uint32,H_b,Xaa)),aa(num,uint32,numeral_numeral(uint32),bit1(bit1(bit1(bit1(bit0(bit1(bit0(bit0(bit0(bit0(bit1(bit1(bit0(bit0(bit1(bit1(bit1(bit0(bit1(bit0(bit0(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit0(bit1(one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit0(bit1(bit0(bit0(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one2))))))))))))))))))))))))))))))))) ).

% hash_code_prod_simps
tff(fact_6271_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Na)
        <=> ( Na = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_6272_test__bit__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2)) ) ) ).

% test_bit_size
tff(fact_6273_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V3: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V3),N) ) )
         => ( U = V3 ) ) ) ).

% word_eqI
tff(fact_6274_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_6275_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_6276_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2))) ) ).

% finite_bit_word
tff(fact_6277_bit__integer_Oabs__eq,axiom,
    ! [Xa: int] : aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xa) ).

% bit_integer.abs_eq
tff(fact_6278_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xa)),Na)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na) ) ) ).

% test_bit_over
tff(fact_6279_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Na)) ).

% not_bit_Suc_0_numeral
tff(fact_6280_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_6281_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V3: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N)
               => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V3),N) ) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),U),V3) ) ) ).

% word_leI
tff(fact_6282_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(int,nat,nat2,K)),Na)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ) ).

% bit_nat_iff
tff(fact_6283_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_6284_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),Na) ) ) ) ).

% test_bit_bin'
tff(fact_6285_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),Xa) ) ) ).

% bang_is_le
tff(fact_6286_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_6287_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_6288_num_Osize__gen_I2_J,axiom,
    ! [X2: num] : size_num(bit0(X2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X2)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_6289_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R3: set(product_prod(A,A))] : mlex_prod(A,F3,R3) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_sy(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F3),R3))) ).

% mlex_eq
tff(fact_6290_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F3),Xs) = aa(list(A),list(A),case_list(list(A),A,nil(A),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_td(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),F3),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_6291_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),Xa: A,Y: A,R3: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xa)),aa(A,nat,F3,Y))
     => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F3,R3)) ) ).

% mlex_less
tff(fact_6292_mlex__iff,axiom,
    ! [A: $tType,Xa: A,Y: A,F3: fun(A,nat),R3: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F3,R3))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F3,Xa)),aa(A,nat,F3,Y))
        | ( ( aa(A,nat,F3,Xa) = aa(A,nat,F3,Y) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R3) ) ) ) ).

% mlex_iff
tff(fact_6293_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),Xa: A,Y: A,R3: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F3,Xa)),aa(A,nat,F3,Y))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),R3)
       => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y),mlex_prod(A,F3,R3)) ) ) ).

% mlex_leq
tff(fact_6294_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_te(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_6295_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),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_tf(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Xa),Xs)),Ys) ).

% zip_Cons1
tff(fact_6296_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Pivot: B,Xa: A,Xs: list(A)] : linorder_part(A,B,F3,Pivot,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),aa(fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_th(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),F3),Pivot),Xa)),linorder_part(A,B,F3,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_6297_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F3,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),filter2(A,aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),F3),Pivot),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_tj(fun(A,B),fun(B,fun(A,$o)),F3),Pivot),Xs)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_tk(fun(A,B),fun(B,fun(A,$o)),F3),Pivot),Xs))) ) ).

% part_def
tff(fact_6298_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_mm(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_6299_Code__Numeral_Opositive__def,axiom,
    code_positive = numeral_numeral(code_integer) ).

% Code_Numeral.positive_def
tff(fact_6300_length__takeWhile__le,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_6301_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P2: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P2,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_6302_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P2,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_6303_nth__length__takeWhile,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_6304_drop__takeWhile,axiom,
    ! [A: $tType,I: nat,P2: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L)))
     => ( drop(A,I,takeWhile(A,P2,L)) = takeWhile(A,P2,drop(A,I,L)) ) ) ).

% drop_takeWhile
tff(fact_6305_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P2: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_6306_eq__len__takeWhile__conv,axiom,
    ! [A: $tType,I: nat,P2: fun(A,$o),L: list(A)] :
      ( ( I = aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(list(A),nat,size_size(list(A)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I)
           => aa(A,$o,P2,aa(nat,A,nth(A,L),J3)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,P2,aa(nat,A,nth(A,L),I)) ) ) ) ).

% eq_len_takeWhile_conv
tff(fact_6307_less__length__takeWhile__conv,axiom,
    ! [A: $tType,I: nat,P2: fun(A,$o),L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),takeWhile(A,P2,L)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),L))
        & ! [J3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J3),I)
           => aa(A,$o,P2,aa(nat,A,nth(A,L),J3)) ) ) ) ).

% less_length_takeWhile_conv
tff(fact_6308_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),P2: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P2,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
         => ~ aa(A,$o,P2,aa(nat,A,nth(A,Xs),Na)) )
       => ( takeWhile(A,P2,Xs) = take(A,Na,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_6309_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( filter2(B,aa(A,fun(B,$o),aTP_Lamp_tl(fun(B,A),fun(A,fun(B,$o)),F3),T2),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_tl(fun(B,A),fun(A,fun(B,$o)),F3),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_6310_concat__bit__Suc,axiom,
    ! [Na: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,Na),K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_concat_bit(Na,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))),L))) ).

% concat_bit_Suc
tff(fact_6311_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P3) = P3 ).

% case_prod_Pair_iden
tff(fact_6312_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_6313_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(Na,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6314_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_concat_bit(Na,K,L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_6315_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,fun(B,$o)),Q2: fun(A,fun(B,$o)),R3: $o,Xa: A,Y: B] :
      ( ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P2),Q2)
        & (R3) )
     => ( (R3)
        & ( aa(B,$o,aa(A,fun(B,$o),P2,Xa),Y)
         => aa(B,$o,aa(A,fun(B,$o),Q2,Xa),Y) ) ) ) ).

% predicate2D_conj
tff(fact_6316_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),bit_concat_bit(M,K,L)),Na)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6317_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F3: fun(fun(A,B),C),G3: C] :
          ( ! [X: fun(A,B)] : aa(fun(A,B),C,F3,X) = G3
         => ( aa(fun(A,B),C,F3,aTP_Lamp_tm(A,B)) = G3 ) ) ) ).

% fun_cong_unused_0
tff(fact_6318_eq__subset,axiom,
    ! [A: $tType,P2: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_tn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P2)) ).

% eq_subset
tff(fact_6319_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Y: list(A)] :
          ( ( linorder_quicksort(A,Xa) = Y )
         => ( ( ( Xa = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X: A,Xs2: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) )
                 => ( Y != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_quicksort(A,filter2(A,aTP_Lamp_to(A,fun(A,$o),X),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),linorder_quicksort(A,filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_6320_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] : linorder_quicksort(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_quicksort(A,filter2(A,aTP_Lamp_to(A,fun(A,$o),Xa),Xs))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))),linorder_quicksort(A,filter2(A,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_6321_sorted__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_quicksort(A,Xs)) ) ).

% sorted_quicksort
tff(fact_6322_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: list(A),Y: list(A)] :
          ( ( linorder_quicksort(A,Xa) = Y )
         => ( aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),Xa)
           => ( ( ( Xa = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A)) ) )
             => ~ ! [X: A,Xs2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) )
                   => ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),linorder_quicksort(A,filter2(A,aTP_Lamp_to(A,fun(A,$o),X),Xs2))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),linorder_quicksort(A,filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X),Xs2)))) )
                     => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_6323_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),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),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% xor_Suc_0_eq
tff(fact_6324_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),zero_zero(A)) = A2 ) ).

% xor.right_neutral
tff(fact_6325_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A2) = A2 ) ).

% xor.left_neutral
tff(fact_6326_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),A2) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_6327_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Xa) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_6328_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% xor_nat_numerals(1)
tff(fact_6329_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_6330_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% xor_nat_numerals(3)
tff(fact_6331_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xa)) ).

% xor_nat_numerals(4)
tff(fact_6332_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),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),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_6333_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_xor_eq
tff(fact_6334_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),A2)),one_one(nat))))),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2))) ) ).

% sdiv_word_min
tff(fact_6335_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),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),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_6336_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A2) = A2 ) ).

% or.left_neutral
tff(fact_6337_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),zero_zero(A)) = A2 ) ).

% or.right_neutral
tff(fact_6338_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_6339_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_6340_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% or_nat_numerals(2)
tff(fact_6341_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% or_nat_numerals(4)
tff(fact_6342_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% or_nat_numerals(3)
tff(fact_6343_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),bit1(Y)) ).

% or_nat_numerals(1)
tff(fact_6344_xor__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Xb)) ).

% xor_integer.abs_eq
tff(fact_6345_XOR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Y)) ) ) ).

% XOR_lower
tff(fact_6346_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_6347_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),zero_zero(A)) = Xa ) ).

% bit.disj_zero_right
tff(fact_6348_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_6349_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% or_Suc_0_eq
tff(fact_6350_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_or_eq
tff(fact_6351_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_6352_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C3: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C3) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ( ! [N7: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N7)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N7),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),N7) ) )
            & ! [M2: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split
tff(fact_6353_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C3: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C3) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ! [N7: nat,M2: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N7)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N7),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),N7) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M2)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_6354_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_6355_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_6356_le__word__or2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)) ) ).

% le_word_or2
tff(fact_6357_le__word__or1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C3: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),C3)) ) ).

% le_word_or1
tff(fact_6358_OR__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)) ) ) ).

% OR_lower
tff(fact_6359_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_6360_or__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Xb)) ).

% or_integer.abs_eq
tff(fact_6361_OR__upper,axiom,
    ! [Xa: int,Na: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_6362_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C3: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C3) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
        <=> ( ! [N5: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N5)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),N5) ) )
            & ! [M5: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M5)
              <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M5),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_6363_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [T2: word(A),U: word(B),V3: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),T2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V3)))
         => ( ( aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V3) = word_split(A,B,C,T2) )
           => ( T2 = word_cat(B,C,A,U,V3) ) ) ) ) ).

% word_cat_split_size
tff(fact_6364_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W2: word(A),U: word(B),V3: word(C)] :
          ( ( W2 = word_cat(B,C,A,U,V3) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V3))),aa(word(A),nat,size_size(word(A)),W2))
           => ( word_split(A,B,C,W2) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V3) ) ) ) ) ).

% word_split_cat_alt
tff(fact_6365_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [A2: word(B),B2: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,A2,B2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,A2,B2)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(C),nat,size_size(word(C)),B2)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),B2),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(word(C),nat,size_size(word(C)),B2)))) ) ) ) ).

% test_bit_cat
tff(fact_6366_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W2: word(A),U: word(B),V3: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V3)))
         => ( ( word_split(A,B,C,W2) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V3) )
           => ( word_cat(B,C,A,U,V3) = W2 ) ) ) ) ).

% word_cat_split_alt
tff(fact_6367_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),Na: nat,C3: word(B),B2: word(C)] :
          ( ( A2 = slice2(B,A,Na,C3) )
         => ( ( B2 = slice2(B,C,zero_zero(nat),C3) )
           => ( ( Na = aa(word(C),nat,size_size(word(C)),B2) )
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),C3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(C),nat,size_size(word(C)),B2)))
               => ( word_cat(A,C,B,A2,B2) = C3 ) ) ) ) ) ) ).

% cat_slices
tff(fact_6368_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
         => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_6369_at__within__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] : topolo174197925503356063within(A,A2,bot_bot(set(A))) = bot_bot(filter(A)) ) ).

% at_within_empty
tff(fact_6370_has__real__derivative__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xa)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_6371_has__real__derivative__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6))),aa(real,real,F3,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_6372_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A,Sa: set(A),G3: fun(A,A),E6: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G3,E6,topolo174197925503356063within(A,Xa,Sa))
           => ( ( aa(A,A,G3,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_tp(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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),D5),aa(A,A,G3,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,Xa)),E6))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G3,Xa)),aa(A,A,G3,Xa))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% DERIV_divide
tff(fact_6373_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,Sa))
         => ( ( aa(A,A,F3,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_tq(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,F3,Xa))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,F3,Xa)))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_inverse'
tff(fact_6374_MVT2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F8: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => has_field_derivative(real,F3,aa(real,real,F8,X),topolo174197925503356063within(real,X,top_top(set(real)))) ) )
       => ? [Z3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),Z3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z3),B2)
            & ( aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),A2)),aa(real,real,F8,Z3)) ) ) ) ) ).

% MVT2
tff(fact_6375_DERIV__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6))),aa(real,real,F3,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_6376_DERIV__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xa)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_6377_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_6378_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,A2)),aa(real,real,F3,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_6379_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_6380_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G3: fun(real,real),G6: fun(real,real)] :
      ( ! [X: real] :
          ( member(real,X,set_or1337092689740270186AtMost(real,A2,B2))
         => has_field_derivative(real,G3,aa(real,real,G6,X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ( ! [X: real] :
            ( member(real,X,set_or1337092689740270186AtMost(real,A2,B2))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G6,X)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G3,A2)),aa(real,real,G3,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_6381_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,B2)),aa(real,real,F3,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_6382_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,A2)),aa(real,real,F3,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_6383_DERIV__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xa)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_6384_DERIV__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6))),aa(real,real,F3,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_6385_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [T2: word(A)] : slice2(A,A,zero_zero(nat),T2) = T2 ) ).

% slice_id
tff(fact_6386_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F4: filter(A)] : has_field_derivative(A,aTP_Lamp_tr(A,fun(A,A),K),zero_zero(A),F4) ) ).

% DERIV_const
tff(fact_6387_has__real__derivative__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6))),aa(real,real,F3,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_6388_has__real__derivative__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,S2))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6),S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H6),D3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,Xa)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_6389_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Sa: set(A),T2: set(A),Xa: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Sa),T2)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),topolo174197925503356063within(A,Xa,Sa)),topolo174197925503356063within(A,Xa,T2)) ) ) ).

% at_le
tff(fact_6390_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Xa: A,Sa: set(A),T2: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,Xa,Sa))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),Sa)
           => has_field_derivative(A,F3,Y,topolo174197925503356063within(A,Xa,T2)) ) ) ) ).

% has_field_derivative_subset
tff(fact_6391_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F8: A,Xa: A,Sa: set(A),T2: set(A)] :
          ( has_field_derivative(A,F3,F8,topolo174197925503356063within(A,Xa,Sa))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),Sa)
           => has_field_derivative(A,F3,F8,topolo174197925503356063within(A,Xa,T2)) ) ) ) ).

% DERIV_subset
tff(fact_6392_DERIV__local__const,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => ( aa(real,real,F3,Xa) = aa(real,real,F3,Y3) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_6393_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ts(fun(A,A),fun(nat,fun(A,A)),F3),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Na))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xa)),Na))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_power_Suc
tff(fact_6394_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A,Sa: set(A),Na: nat] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,Sa))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_tt(fun(A,A),fun(nat,fun(A,A)),F3),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_power
tff(fact_6395_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xa: A,Sa: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ).

% DERIV_inverse
tff(fact_6396_DERIV__local__max,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Y3)),aa(real,real,F3,Xa)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_6397_DERIV__local__min,axiom,
    ! [F3: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),Y3))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F3,Xa)),aa(real,real,F3,Y3)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_6398_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_6399_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,Sa: set(real)] : has_field_derivative(real,aTP_Lamp_tu(nat,fun(real,real),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xa,Sa)) ).

% DERIV_pow
tff(fact_6400_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xa: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B2)
           => ( topolo174197925503356063within(A,Xa,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xa,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_6401_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_6402_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D2: A,Xa: A,Sa: set(A),G3: fun(A,A),E3: A] :
          ( has_field_derivative(A,F3,D2,topolo174197925503356063within(A,Xa,Sa))
         => ( has_field_derivative(A,G3,E3,topolo174197925503356063within(A,Xa,Sa))
           => ( ( aa(A,A,G3,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_tp(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),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),D2),aa(A,A,G3,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F3,Xa)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G3,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% DERIV_quotient
tff(fact_6403_trivial__limit__at__left__bot,axiom,
    ! [A: $tType] :
      ( ( order_bot(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,bot_bot(A),set_ord_lessThan(A,bot_bot(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_left_bot
tff(fact_6404_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D2: A,Xa: A,Sa: set(A)] :
          ( has_field_derivative(A,F3,D2,topolo174197925503356063within(A,Xa,Sa))
         => ( ( aa(A,A,F3,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_tq(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F3,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_6405_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C3: fun(nat,A),F3: fun(A,A),F8: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K4)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),C3),Z3),aa(A,A,F3,Z3)) )
         => ( has_field_derivative(A,F3,F8,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),C3),Z),F8) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_6406_ucast__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W2: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W2) = slice2(B,A,zero_zero(nat),W2) ) ).

% ucast_slice
tff(fact_6407_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_tw(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_6408_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A2: word(B),T2: word(A)] : slice2(A,A,zero_zero(nat),word_cat(B,A,A,A2,T2)) = T2 ) ).

% slice_cat2
tff(fact_6409_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_tx(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
               => has_field_derivative(A,aTP_Lamp_ty(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),C3),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_6410_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
           => has_field_derivative(A,aTP_Lamp_ty(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),C3),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_6411_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C3: fun(nat,A),Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z3)),K4)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4)
           => has_field_derivative(A,aTP_Lamp_ty(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),C3),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_6412_DERIV__fun__powr,axiom,
    ! [G3: fun(real,real),M: real,Xa: real,R2: real] :
      ( has_field_derivative(real,G3,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G3,Xa))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_tz(fun(real,real),fun(real,fun(real,real)),G3),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G3,Xa),aa(real,real,aa(real,fun(real,real),minus_minus(real),R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_6413_DERIV__log,axiom,
    ! [Xa: real,B2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,log(B2),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B2)),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_6414_DERIV__powr,axiom,
    ! [G3: fun(real,real),M: real,Xa: real,F3: fun(real,real),R2: real] :
      ( has_field_derivative(real,G3,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G3,Xa))
       => ( has_field_derivative(real,F3,R2,topolo174197925503356063within(real,Xa,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_ua(fun(real,real),fun(fun(real,real),fun(real,real)),G3),F3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G3,Xa),aa(real,real,F3,Xa))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G3,Xa)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F3,Xa))),aa(real,real,G3,Xa)))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_6415_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_6416_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_6417_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_6418_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G3: fun(A,A),Xa: A,Db: A,Sa: set(A)] :
          ( ( cosh(A,aa(A,A,G3,Xa)) != zero_zero(A) )
         => ( has_field_derivative(A,G3,Db,topolo174197925503356063within(A,Xa,Sa))
           => has_field_derivative(A,aTP_Lamp_ub(fun(A,A),fun(A,A),G3),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)),aa(nat,A,aa(A,fun(nat,A),power_power(A),tanh(A,aa(A,A,G3,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_6419_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D5: real] :
      ( ( Xa != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
           => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D5,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_6420_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A5: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xa,A5)) ) ).

% arcosh_real_has_field_derivative
tff(fact_6421_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A5: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xa,A5)) ) ).

% artanh_real_has_field_derivative
tff(fact_6422_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_6423_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_6424_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_6425_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xa: real,Na: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M3: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X),topolo174197925503356063within(real,X,top_top(set(real))))
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F3,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_uc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_6426_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Xa: real,Na: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
        & ! [M3: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X),topolo174197925503356063within(real,X,top_top(set(real)))) )
     => ? [T7: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
          & ( aa(real,real,F3,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_uc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_6427_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( Xa != zero_zero(real) )
       => has_field_derivative(real,root(Na),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_6428_Maclaurin,axiom,
    ! [H2: real,Na: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M3: nat,T7: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H2) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
           => ? [T7: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),H2)
                & ( aa(real,real,F3,H2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ud(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H2),Diff)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H2),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_6429_Maclaurin2,axiom,
    ! [H2: real,Diff: fun(nat,fun(real,real)),F3: fun(real,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H2)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M3: nat,T7: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ? [T7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T7)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H2)
              & ( aa(real,real,F3,H2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ud(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H2),Diff)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H2),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_6430_Maclaurin__minus,axiom,
    ! [H2: real,Na: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H2),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M3: nat,T7: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H2),T7)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
           => ? [T7: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H2),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),zero_zero(real))
                & ( aa(real,real,F3,H2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ud(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H2),Diff)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H2),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_6431_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Na: nat,Xa: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( Xa != zero_zero(real) )
         => ( ! [M3: nat,X: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X),topolo174197925503356063within(real,X,top_top(set(real))))
           => ? [T7: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T7))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
                & ( aa(real,real,F3,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_uc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_6432_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),Na: nat,Xa: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M3: nat,T7: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
       => ? [T7: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T7)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F3,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aTP_Lamp_uc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_6433_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M3: nat,T7: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xa)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),B2)
                 => ( ( Xa != C3 )
                   => ? [T7: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),C3),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T7)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),C3) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T7)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),Xa) ) )
                        & ( aa(real,real,F3,Xa) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ue(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C3),Xa)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Xa),C3)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_6434_Taylor__up,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M3: nat,T7: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B2)
             => ? [T7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T7)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),B2)
                  & ( aa(real,real,F3,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_uf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C3)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),B2),C3)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_6435_Taylor__down,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M3: nat,T7: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),T7)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),B2) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B2)
             => ? [T7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),T7)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T7),C3)
                  & ( aa(real,real,F3,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_uf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C3)),set_ord_lessThan(nat,Na))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,Na),T7)),semiring_char_0_fact(real,Na))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),A2),C3)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_6436_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A2)),aa(word(B),nat,size_size(word(B)),B2))),aa(word(C),nat,size_size(word(C)),word_cat(A,B,C,A2,B2)))
         => ( slice2(C,A,aa(word(B),nat,size_size(word(B)),B2),word_cat(A,B,C,A2,B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_6437_Maclaurin__lemma2,axiom,
    ! [Na: nat,H2: real,Diff: fun(nat,fun(real,real)),K: nat,B5: real] :
      ( ! [M3: nat,T7: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H2) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7),topolo174197925503356063within(real,T7,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M2: nat,T9: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T9)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T9),H2) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_uh(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B5),M2),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T9)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ui(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T9)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B5),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T9),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T9,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_6438_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W2: word(C),U: word(A),V3: word(B)] :
          ( ( word_split(C,A,B,W2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),U),V3) )
         => ( ( U = slice2(C,A,aa(word(B),nat,size_size(word(B)),V3),W2) )
            & ( V3 = slice2(C,B,zero_zero(nat),W2) ) ) ) ) ).

% split_slices
tff(fact_6439_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_uj(real,real),suminf(real,aTP_Lamp_uk(real,fun(nat,real),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_6440_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D5: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( Xa != zero_zero(real) )
       => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
               => ( D5 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
               => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Na),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Na),D5,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_6441_DERIV__power__series_H,axiom,
    ! [R3: real,F3: fun(nat,real),X0: real] :
      ( ! [X: real] :
          ( member(real,X,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R3),R3))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_ul(fun(nat,real),fun(real,fun(nat,real)),F3),X)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R3),R3))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
         => has_field_derivative(real,aTP_Lamp_un(fun(nat,real),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),aTP_Lamp_ul(fun(nat,real),fun(real,fun(nat,real)),F3),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_6442_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xa: A,G6: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xa)),one_one(real))
           => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xa,Sa))
             => has_derivative(A,real,aTP_Lamp_uo(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_up(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xa),G6),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_6443_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_6444_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_6445_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_6446_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_6447_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioo_iff
tff(fact_6448_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xa,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_6449_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,Xa)) = Xa ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_6450_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xa,Y)) = Xa ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_6451_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,Xa)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_6452_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,Sa: set(A),T2: set(A)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),Sa)
           => has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,T2)) ) ) ) ).

% has_derivative_subset
tff(fact_6453_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A2,B2)) ) ) ).

% infinite_Ioo
tff(fact_6454_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C3: B,F4: filter(A)] : has_derivative(A,B,aTP_Lamp_uq(B,fun(A,B),C3),aTP_Lamp_ur(A,B),F4) ) ).

% has_derivative_const
tff(fact_6455_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F4: fun(A,B),Xa: A] :
          ( has_derivative(A,B,aTP_Lamp_ur(A,B),F4,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => ! [X3: A] : aa(A,B,F4,X3) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_6456_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T2: set(A),G3: fun(A,B),G6: fun(A,fun(A,B)),F3: fun(C,A),Sa: set(C),Xa: C,F8: fun(C,A)] :
          ( ! [X: A] :
              ( member(A,X,T2)
             => has_derivative(A,B,G3,aa(A,fun(A,B),G6,X),topolo174197925503356063within(A,X,T2)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F3),Sa)),T2)
           => ( member(C,Xa,Sa)
             => ( has_derivative(C,A,F3,F8,topolo174197925503356063within(C,Xa,Sa))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_us(fun(A,B),fun(fun(C,A),fun(C,B)),G3),F3),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_ut(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G6),F3),Xa),F8),topolo174197925503356063within(C,Xa,Sa)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_6457_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_6458_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_6459_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_6460_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_6461_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_6462_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_6463_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_6464_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6465_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_6466_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or5935395276787703475ssThan(A,L,U)) = set_ord_lessThan(A,U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_6467_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,S2: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,S2))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xa,S2))
           => ( ( aa(A,B,G3,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uu(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_uv(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F8),Xa),G3),G6),topolo174197925503356063within(A,Xa,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_6468_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,S2: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_uw(A,fun(A,A),Xa),topolo174197925503356063within(A,Xa,S2)) ) ) ).

% has_derivative_inverse'
tff(fact_6469_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(B,A),Xa: B,F8: fun(B,A),S2: set(B)] :
          ( ( aa(B,A,F3,Xa) != zero_zero(A) )
         => ( has_derivative(B,A,F3,F8,topolo174197925503356063within(B,Xa,S2))
           => has_derivative(B,A,aTP_Lamp_ux(fun(B,A),fun(B,A),F3),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_uy(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F3),Xa),F8),topolo174197925503356063within(B,Xa,S2)) ) ) ) ).

% has_derivative_inverse
tff(fact_6470_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,Xa: real,Y: real,F3: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),B2)
     => ( member(real,Xa,set_or5935395276787703475ssThan(real,A2,B2))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A2,B2))
         => ( ! [X: real] :
                ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X,top_top(set(real)))) )
           => ( aa(real,real,F3,Xa) = aa(real,real,F3,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_6471_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6472_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6473_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xa: A,G6: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xa))
         => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aTP_Lamp_uz(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_va(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xa),G6),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_ln
tff(fact_6474_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,S2: set(A),G3: fun(A,B),G6: fun(A,B)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,S2))
         => ( has_derivative(A,B,G3,G6,topolo174197925503356063within(A,Xa,S2))
           => ( ( aa(A,B,G3,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vb(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_vc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F3),F8),Xa),G3),G6),topolo174197925503356063within(A,Xa,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_6475_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F3: fun(A,fun(B,C)),F8: fun(A,fun(B,C)),Xa: B,S2: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F3,I2),aa(A,fun(B,C),F8,I2),topolo174197925503356063within(B,Xa,S2)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ve(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F3),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_vg(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F3),F8),Xa),topolo174197925503356063within(B,Xa,S2)) ) ) ).

% has_derivative_prod
tff(fact_6476_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),G6: fun(A,real),Xa: A,X4: set(A),F3: fun(A,real),F8: fun(A,real)] :
          ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xa,X4))
         => ( has_derivative(A,real,F3,F8,topolo174197925503356063within(A,Xa,X4))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xa))
             => ( member(A,Xa,X4)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vh(fun(A,real),fun(fun(A,real),fun(A,real)),G3),F3),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G3),G6),Xa),F3),F8),topolo174197925503356063within(A,Xa,X4)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_6477_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xa: A,G6: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G3,Xa))
         => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xa),G6),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_6478_DERIV__series_H,axiom,
    ! [F3: fun(real,fun(nat,real)),F8: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_vl(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F3),N),aa(nat,real,aa(real,fun(nat,real),F8,X0),N),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X: real] :
            ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
           => summable(real,aa(real,fun(nat,real),F3,X)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A2,B2))
         => ( summable(real,aa(real,fun(nat,real),F8,X0))
           => ( summable(real,L5)
             => ( ! [N: nat,X: real,Y3: real] :
                    ( member(real,X,set_or5935395276787703475ssThan(real,A2,B2))
                   => ( member(real,Y3,set_or5935395276787703475ssThan(real,A2,B2))
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,aa(real,fun(nat,real),F3,X),N)),aa(nat,real,aa(real,fun(nat,real),F3,Y3),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),X),Y3)))) ) )
               => has_field_derivative(real,aTP_Lamp_vm(fun(real,fun(nat,real)),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),F8,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_6479_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G3: fun(A,real),Xa: A,G6: fun(A,real),Sa: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G3,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G3,Xa)),one_one(real))
           => ( has_derivative(A,real,G3,G6,topolo174197925503356063within(A,Xa,Sa))
             => has_derivative(A,real,aTP_Lamp_vn(fun(A,real),fun(A,real),G3),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vo(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G3),Xa),G6),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_6480_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_tx(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_vq(fun(nat,A),fun(A,fun(A,A)),C3),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6481_int__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,int,code_int_of_integer,K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_vr(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_6482_int__of__integer__max,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),K),L)) = aa(int,int,aa(int,fun(int,int),ord_max(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).

% int_of_integer_max
tff(fact_6483_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U)) ).

% finite_greaterThanLessThan
tff(fact_6484_finite__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or5935395276787703475ssThan(code_integer,L,U)) ).

% finite_greaterThanLessThan_integer
tff(fact_6485_int__of__integer__of__nat,axiom,
    ! [Na: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,semiring_1_of_nat(code_integer),Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ).

% int_of_integer_of_nat
tff(fact_6486_integer__of__int__int__of__integer,axiom,
    ! [K: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,K)) = K ).

% integer_of_int_int_of_integer
tff(fact_6487_int__of__integer__integer__of__int,axiom,
    ! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,K)) = K ).

% int_of_integer_integer_of_int
tff(fact_6488_int__of__integer__inverse,axiom,
    ! [Xa: code_integer] : aa(int,code_integer,code_integer_of_int,aa(code_integer,int,code_int_of_integer,Xa)) = Xa ).

% int_of_integer_inverse
tff(fact_6489_of__int__integer__of,axiom,
    ! [K: code_integer] : aa(int,code_integer,ring_1_of_int(code_integer),aa(code_integer,int,code_int_of_integer,K)) = K ).

% of_int_integer_of
tff(fact_6490_int__of__integer__of__int,axiom,
    ! [K: int] : aa(code_integer,int,code_int_of_integer,aa(int,code_integer,ring_1_of_int(code_integer),K)) = K ).

% int_of_integer_of_int
tff(fact_6491_zero__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_6492_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_6493_int__of__integer__numeral,axiom,
    ! [K: num] : aa(code_integer,int,code_int_of_integer,aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_6494_one__integer_Orep__eq,axiom,
    aa(code_integer,int,code_int_of_integer,one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_6495_uminus__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Xa)) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,Xa)) ).

% uminus_integer.rep_eq
tff(fact_6496_plus__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% plus_integer.rep_eq
tff(fact_6497_minus__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% minus_integer.rep_eq
tff(fact_6498_times__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% times_integer.rep_eq
tff(fact_6499_abs__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,abs_abs(code_integer),Xa)) = aa(int,int,abs_abs(int),aa(code_integer,int,code_int_of_integer,Xa)) ).

% abs_integer.rep_eq
tff(fact_6500_divide__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% divide_integer.rep_eq
tff(fact_6501_modulo__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% modulo_integer.rep_eq
tff(fact_6502_sgn__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,sgn_sgn(code_integer),Xa)) = aa(int,int,sgn_sgn(int),aa(code_integer,int,code_int_of_integer,Xa)) ).

% sgn_integer.rep_eq
tff(fact_6503_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vs(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_6504_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F3: fun(B,A),L: A,F4: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vt(A,fun(fun(B,A),fun(B,A)),C3),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F4)
          <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F4) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_6505_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vu(nat,fun(fun(A,real),fun(A,real)),Na),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% power_tendsto_0_iff
tff(fact_6506_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vv(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6507_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vv(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6508_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_vv(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6509_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(A,real),A2: A,G3: fun(A,real)] :
          ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X: A] :
                ( ( X != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,G3,X)) )
           => ( ! [X: A] :
                  ( ( X != A2 )
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,G3,X)),aa(A,real,F3,X)) )
             => filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6510_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L: B,A2: A,G3: fun(A,C),M: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X: A] :
                ( ( X != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,aa(C,fun(C,C),minus_minus(C),aa(A,C,G3,X)),M))),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X)),L))) )
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6511_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A2: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_vw(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A2,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_6512_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(A,B),L: filter(B),Xa: A,S2: set(A),T3: set(A)] :
          ( filterlim(A,B,F3,L,topolo174197925503356063within(A,Xa,S2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S2)
           => filterlim(A,B,F3,L,topolo174197925503356063within(A,Xa,T3)) ) ) ) ).

% tendsto_within_subset
tff(fact_6513_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,L),U) = set_or5935395276787703475ssThan(nat,L,U) ).

% atLeastSucLessThan_greaterThanLessThan
tff(fact_6514_or__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% or_integer.rep_eq
tff(fact_6515_bit__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),Xa) = aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(code_integer,int,code_int_of_integer,Xa)) ).

% bit_integer.rep_eq
tff(fact_6516_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vx(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_add_zero
tff(fact_6517_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vy(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_mult_zero
tff(fact_6518_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_left_zero
tff(fact_6519_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_mult_right_zero
tff(fact_6520_int__of__integer__induct,axiom,
    ! [Y: int,P2: fun(int,$o)] :
      ( member(int,Y,top_top(set(int)))
     => ( ! [X: code_integer] : aa(int,$o,P2,aa(code_integer,int,code_int_of_integer,X))
       => aa(int,$o,P2,Y) ) ) ).

% int_of_integer_induct
tff(fact_6521_int__of__integer__cases,axiom,
    ! [Y: int] :
      ( member(int,Y,top_top(set(int)))
     => ~ ! [X: code_integer] : Y != aa(code_integer,int,code_int_of_integer,X) ) ).

% int_of_integer_cases
tff(fact_6522_int__of__integer,axiom,
    ! [Xa: code_integer] : member(int,aa(code_integer,int,code_int_of_integer,Xa),top_top(set(int))) ).

% int_of_integer
tff(fact_6523_type__definition__integer,axiom,
    type_definition(code_integer,int,code_int_of_integer,code_integer_of_int,top_top(set(int))) ).

% type_definition_integer
tff(fact_6524_integer__of__int__inverse,axiom,
    ! [Y: int] :
      ( member(int,Y,top_top(set(int)))
     => ( aa(code_integer,int,code_int_of_integer,aa(int,code_integer,code_integer_of_int,Y)) = Y ) ) ).

% integer_of_int_inverse
tff(fact_6525_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( cos(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_wb(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A2)),F4) ) ) ) ).

% tendsto_tan
tff(fact_6526_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B),B2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,B2),F4)
           => ( ( B2 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wc(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A2),B2)),F4) ) ) ) ) ).

% tendsto_divide
tff(fact_6527_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,B)),F3),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_divide_zero
tff(fact_6528_int__of__integer__less__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Y))
    <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xa),Y) ) ).

% int_of_integer_less_iff
tff(fact_6529_less__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xa),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% less_integer.rep_eq
tff(fact_6530_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_iff
tff(fact_6531_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F3: fun(B,fun(A,C)),F4: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_we(fun(B,fun(A,C)),fun(A,fun(B,C)),F3),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wf(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ).

% tendsto_null_sum
tff(fact_6532_tendsto__log,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
         => ( ( A2 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wg(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F4) ) ) ) ) ) ).

% tendsto_log
tff(fact_6533_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,B),F4: filter(A),Na: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_wh(fun(A,B),fun(nat,fun(A,B)),F3),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_null_power
tff(fact_6534_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),G3: fun(A,B),F4: filter(A),A2: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wi(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
          <=> filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform_eq
tff(fact_6535_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_cancel
tff(fact_6536_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wi(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform2
tff(fact_6537_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G3: fun(A,B),A2: B,F4: filter(A),F3: fun(A,B)] :
          ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wk(fun(A,B),fun(fun(A,B),fun(A,B)),G3),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4) ) ) ) ).

% Lim_transform
tff(fact_6538_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% LIM_zero_iff
tff(fact_6539_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,B)),F3),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% LIM_zero
tff(fact_6540_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => filterlim(A,real,aTP_Lamp_wl(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_norm_zero
tff(fact_6541_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_wl(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
        <=> filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_iff
tff(fact_6542_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_wl(fun(A,B),fun(A,real),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_6543_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A2)),F4) ) ) ) ).

% tendsto_inverse
tff(fact_6544_int__of__integer__inject,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( ( aa(code_integer,int,code_int_of_integer,Xa) = aa(code_integer,int,code_int_of_integer,Y) )
    <=> ( Xa = Y ) ) ).

% int_of_integer_inject
tff(fact_6545_integer__eqI,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) )
     => ( K = L ) ) ).

% integer_eqI
tff(fact_6546_integer__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( K = L )
    <=> ( aa(code_integer,int,code_int_of_integer,K) = aa(code_integer,int,code_int_of_integer,L) ) ) ).

% integer_eq_iff
tff(fact_6547_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_wn(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F4) ) ) ) ).

% tendsto_sgn
tff(fact_6548_tendsto__artanh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A2),one_one(real))
         => filterlim(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F4) ) ) ) ).

% tendsto_artanh
tff(fact_6549_tendsto__arcosh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A2)
       => filterlim(A,real,aTP_Lamp_wp(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ).

% tendsto_arcosh
tff(fact_6550_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(A,A),A2: A,F4: filter(A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,A2),F4)
         => ( ( sin(A,A2) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_wq(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A2)),F4) ) ) ) ).

% tendsto_cot
tff(fact_6551_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( cosh(B,A2) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_wr(fun(A,B),fun(A,B),F3),topolo7230453075368039082e_nhds(B,tanh(B,A2)),F4) ) ) ) ).

% tendsto_tanh
tff(fact_6552_tendsto__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F4: filter(A),F6: filter(A),F3: fun(A,B),L: B] :
          ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F6)
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% tendsto_mono
tff(fact_6553_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F24: filter(B),F13: filter(A),F25: filter(B),F15: filter(A)] :
      ( filterlim(A,B,F3,F24,F13)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F24),F25)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F15),F13)
         => filterlim(A,B,F3,F25,F15) ) ) ) ).

% filterlim_mono
tff(fact_6554_less__eq__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),Xa),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_6555_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ) ).

% integer_less_eq_iff
tff(fact_6556_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,C),B2: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,B2),F4)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ws(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F3),G3),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)),F4) ) ) ) ).

% tendsto_Pair
tff(fact_6557_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_wt(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6558_xor__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% xor_integer.rep_eq
tff(fact_6559_take__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2584673776208193580ke_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% take_bit_integer.rep_eq
tff(fact_6560_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R3: real,A2: A,F3: fun(A,B),G3: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
         => ( ! [X: A] :
                ( ( X != A2 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),R3)
                 => ( aa(A,B,F3,X) = aa(A,B,G3,X) ) ) )
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6561_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X5),A2))),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X5)),L5))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_6562_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X)),L5))),R) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6563_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X3: A] :
                    ( ( ( X3 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X3),A2))),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,X3)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_6564_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,A),A2: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wu(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wv(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6565_unset__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se2638667681897837118et_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% unset_bit_integer.rep_eq
tff(fact_6566_LIM__fun__less__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X3))),R) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F3,X3)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6567_LIM__fun__not__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X3))),R) )
               => ( aa(real,real,F3,X3) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6568_LIM__fun__gt__zero,axiom,
    ! [F3: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
            & ! [X3: real] :
                ( ( ( X3 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),C3),X3))),R) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F3,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6569_Code__Numeral_Oset__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se5668285175392031749et_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% Code_Numeral.set_bit_integer.rep_eq
tff(fact_6570_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G3: fun(B,C),C3: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),X),A2))),D4) )
                     => ( aa(A,B,F3,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ww(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6571_flip__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se8732182000553998342ip_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se8732182000553998342ip_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% flip_bit_integer.rep_eq
tff(fact_6572_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),F3),Xa),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6573_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A] :
          ( has_field_derivative(A,F3,D5,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),F3),Xa),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6574_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_wy(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6575_greaterThanLessThan__upt,axiom,
    ! [Na: nat,M: nat] : set_or5935395276787703475ssThan(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),M)) ).

% greaterThanLessThan_upt
tff(fact_6576_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F3: fun(A,B),K4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H: A] :
                ( ( H != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,H))),aa(real,real,aa(real,fun(real,real),times_times(real),K4),real_V7770717601297561774m_norm(A,H))) ) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6577_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D5: A,Xa: A] :
          ( has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),F3),Xa),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6578_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F4: filter(B),A2: A] :
          ( filterlim(A,B,F3,F4,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(A,fun(A,B)),F3),A2),F4,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6579_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or5935395276787703475ssThan(code_integer,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_6580_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_6581_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Sa)
         => ( ! [X: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),Sa)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),A2),X),aa(A,A,F3,X)) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6582_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Sa: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Sa)
         => ( ! [X: A] :
                ( ( X != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X)),Sa)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),A2),X),aa(A,A,F3,X)) ) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6583_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F3: fun(nat,real),G3: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F3)
           => ( ! [H: A,N: nat] :
                  ( ( H != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G3,H),N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F3,N)),real_V7770717601297561774m_norm(A,H))) ) )
             => filterlim(A,B,aTP_Lamp_xa(fun(A,fun(nat,B)),fun(A,B),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_6584_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_6585_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)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L)) ).

% divmod_integer_def
tff(fact_6586_summable__Leibniz_I2_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat)))
         => ! [N7: nat] : member(real,suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6587_summable__Leibniz_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real))
         => ! [N7: nat] : member(real,suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6588_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xc(A,fun(fun(nat,A),fun(nat,A)),C3),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_6589_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xd(A,fun(fun(nat,A),fun(nat,A)),C3),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_6590_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xe(A,fun(fun(nat,A),fun(nat,A)),C3),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_6591_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ? [U5: fun(nat,A)] :
              ( ! [N7: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,U5,N7))
              & filterlim(nat,A,U5,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_6592_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ? [U5: fun(nat,A)] :
              ( ! [N7: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U5,N7)),Xa)
              & filterlim(nat,A,U5,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_6593_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_xf(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6594_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_xf(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_6595_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6596_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F4: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_sw(fun(nat,A),fun(nat,A),F3),F4,at_top(nat))
    <=> filterlim(nat,A,F3,F4,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6597_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,A2: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( ? [N9: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),A2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A2) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6598_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,A2: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( ? [N9: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,X4,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),Xa) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6599_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,N6: nat,C4: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C4),aa(nat,A,F3,N)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C4),L) ) ) ) ).

% Lim_bounded2
tff(fact_6600_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,M7: nat,C4: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F3,N)),C4) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C4) ) ) ) ).

% Lim_bounded
tff(fact_6601_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N9: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,Y6,N)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6602_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N6: nat,X4: fun(nat,A),Y6: fun(nat,A),Xa: A,Y: A] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,Y6,N)) )
         => ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) ) ) ) ) ).

% lim_mono
tff(fact_6603_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),Sa: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),Sa)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),Sa)) ) ) ) ).

% Sup_lim
tff(fact_6604_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),Sa: set(A),A2: A] :
          ( ! [N: nat] : member(A,aa(nat,A,B2,N),Sa)
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),Sa)),A2) ) ) ) ).

% Inf_lim
tff(fact_6605_Inf__as__limit,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A5: set(A)] :
          ( ( A5 != bot_bot(set(A)) )
         => ? [U5: fun(nat,A)] :
              ( ! [N7: nat] : member(A,aa(nat,A,U5,N7),A5)
              & filterlim(nat,A,U5,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),A5)),at_top(nat)) ) ) ) ).

% Inf_as_limit
tff(fact_6606_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_6607_mult__nat__left__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_6608_mult__nat__right__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aTP_Lamp_xg(nat,fun(nat,nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6609_monoseq__convergent,axiom,
    ! [X4: fun(nat,real),B5: real] :
      ( topological_monoseq(real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X4,I2))),B5)
       => ~ ! [L8: real] : ~ filterlim(nat,real,X4,topolo7230453075368039082e_nhds(real,L8),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_6610_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),Xa: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
           => ( ( ! [N7: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N7)),Xa)
                & ! [M2: nat,N7: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N7)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N7)) ) )
              | ( ! [N7: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,A2,N7))
                & ! [M2: nat,N7: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N7)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,N7)),aa(nat,A,A2,M2)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6611_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_xh(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6612_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X4: fun(nat,A),Xa: A,L: nat] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xi(fun(nat,A),fun(nat,fun(nat,A)),X4),L),topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6613_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xj(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_6614_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_xk(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable'
tff(fact_6615_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => summable(A,aTP_Lamp_xl(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable
tff(fact_6616_nested__sequence__unique,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,F3,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,G3,aa(nat,nat,suc,N))),aa(nat,real,G3,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,G3,N))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_xm(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N7)),L2)
                & filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(nat,real,G3,N7))
                & filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6617_LIMSEQ__inverse__zero,axiom,
    ! [X4: fun(nat,real)] :
      ( ! [R: real] :
        ? [N9: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R),aa(nat,real,X4,N)) )
     => filterlim(nat,real,aTP_Lamp_xn(fun(nat,real),fun(nat,real),X4),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6618_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_xo(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6619_LIMSEQ__root__const,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
     => filterlim(nat,real,aTP_Lamp_xp(real,fun(nat,real),C3),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6620_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_xq(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6621_increasing__LIMSEQ,axiom,
    ! [F3: fun(nat,real),L: real] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),aa(nat,real,F3,aa(nat,nat,suc,N)))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F3,N)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,N7)),E2)) )
         => filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6622_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xr(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6623_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xs(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6624_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_xt(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6625_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_xl(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_6626_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_xk(fun(nat,A),fun(nat,A),F3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,F3,zero_zero(nat))),C3)) ) ) ).

% telescope_sums'
tff(fact_6627_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_6628_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_xu(real,fun(real,fun(nat,real)),Xa),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6629_LIMSEQ__abs__realpow__zero,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),C3)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_6630_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_6631_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aTP_Lamp_xv(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6632_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),Sa: A] :
          ( sums(A,F3,Sa)
        <=> filterlim(nat,A,aTP_Lamp_xw(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,Sa),at_top(nat)) ) ) ).

% sums_def'
tff(fact_6633_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_xx(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6634_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),Xa: real] :
          ( filterlim(nat,real,aTP_Lamp_xy(fun(nat,A),fun(nat,real),F3),topolo7230453075368039082e_nhds(real,Xa),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => summable(A,F3) ) ) ) ).

% root_test_convergence
tff(fact_6635_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No2: nat] :
                ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,N5)),L5))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6636_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L5: A] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,N)),L5))),R) ) )
         => filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6637_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N7: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N7)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,X4,N7)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6638_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_6639_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,nat),F4: filter(A),Xa: B] :
          ( filterlim(A,nat,F3,at_top(nat),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xa)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_xz(fun(A,nat),fun(B,fun(A,B)),F3),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_power_zero
tff(fact_6640_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_ya(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6641_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6642_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Df: A,Z: A,Sa: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F3,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,Sa,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N: nat] : aa(nat,A,Sa,N) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_yb(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F3),Z),Sa),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6643_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_yc(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6644_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xa))
         => filterlim(nat,A,aTP_Lamp_yd(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6645_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => summable(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6646_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_nr(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6647_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6648_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_ye(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6649_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7)))),L2)
              & filterlim(nat,real,aTP_Lamp_ye(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L2),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N7)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_yf(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6650_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6651_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N))
       => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N))),aa(nat,real,A2,N))
         => filterlim(nat,real,aTP_Lamp_yf(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6652_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_yg(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_6653_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F8),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6654_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_ur(A,B)) ) ).

% bounded_linear_zero
tff(fact_6655_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K8: real] :
            ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_6656_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(C,A),F4: filter(C)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( filterlim(C,A,G3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F4)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yi(fun(A,B),fun(fun(C,A),fun(C,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_6657_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6658_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G3: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F3,G3,topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G3) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_6659_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6660_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),K4: real] :
          ( ! [X: A,Y3: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X)),aa(A,B,F3,Y3))
         => ( ! [R: real,X: A] : aa(A,B,F3,aa(A,A,real_V8093663219630862766scaleR(A,R),X)) = aa(B,B,real_V8093663219630862766scaleR(B,R),aa(A,B,F3,X))
           => ( ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K4))
             => real_V3181309239436604168linear(A,B,F3) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6661_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F8: fun(A,B),Xa: A,F3: fun(A,B),Sa: set(A)] :
          ( real_V3181309239436604168linear(A,B,F8)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_yj(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F8),Xa),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa))
           => has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivativeI
tff(fact_6662_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,Sa: set(A)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F8),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_at_within
tff(fact_6663_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & ? [E4: fun(A,B)] :
                ( ! [H4: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xa)),aa(A,B,F8,H4))),aa(A,B,E4,H4))
                & filterlim(A,real,aTP_Lamp_yl(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_6664_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,Sa: set(A)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa))
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F8),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,Sa)) ) ) ) ).

% has_derivative_within
tff(fact_6665_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),D5: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F3,D5,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ym(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),D5),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6666_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),F4: filter(A)] :
          ( has_derivative(A,B,F3,F8,F4)
        <=> ( real_V3181309239436604168linear(A,B,F8)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F3),F8),F4),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% has_derivative_def
tff(fact_6667_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_yp(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_6668_nat__of__integer__of__nat,axiom,
    ! [Na: nat] : code_nat_of_integer(aa(nat,code_integer,semiring_1_of_nat(code_integer),Na)) = Na ).

% nat_of_integer_of_nat
tff(fact_6669_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),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_6670_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),code_nat_of_integer(K)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_max(code_integer),zero_zero(code_integer)),K) ).

% of_nat_of_integer
tff(fact_6671_nat__of__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : code_nat_of_integer(Xa) = aa(int,nat,nat2,aa(code_integer,int,code_int_of_integer,Xa)) ).

% nat_of_integer.rep_eq
tff(fact_6672_nat__of__integer_Oabs__eq,axiom,
    ! [Xa: int] : code_nat_of_integer(aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,nat,nat2,Xa) ).

% nat_of_integer.abs_eq
tff(fact_6673_nat__of__integer__code__post_I1_J,axiom,
    code_nat_of_integer(zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_6674_nat__of__integer__less__iff,axiom,
    ! [Xa: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Xa)
     => ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Y)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),code_nat_of_integer(Xa)),code_nat_of_integer(Y))
        <=> aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xa),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_6675_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),U)
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image(nat,code_integer,semiring_1_of_nat(code_integer)),set_ord_lessThan(nat,code_nat_of_integer(U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_6676_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xa: A,S2: set(A),F3: fun(A,B),F8: fun(A,B)] :
          ( member(A,Xa,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,S2))
            <=> ( real_V3181309239436604168linear(A,B,F8)
                & ? [E4: fun(A,B)] :
                    ( ! [H4: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4),S2)
                       => ( aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,Xa)),aa(A,B,F8,H4))),aa(A,B,E4,H4)) ) )
                    & filterlim(A,real,aTP_Lamp_yl(fun(A,B),fun(A,real),E4),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_6677_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E3: real,F8: fun(A,B),Sa: set(A),Xa: A,F3: fun(A,B),H7: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
         => ( real_V3181309239436604168linear(A,B,F8)
           => ( ! [Y3: A] :
                  ( member(A,Y3,Sa)
                 => ( ( Y3 != Xa )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y3,Xa)),E3)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,F3,Y3)),aa(A,B,F3,Xa))),aa(A,B,F8,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xa))))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Y3),Xa)))),aa(A,real,H7,Y3)) ) ) )
             => ( filterlim(A,real,H7,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xa,Sa))
               => has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6678_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_6679_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : real_V557655796197034286t_dist(A,zero_zero(A),Xa) = real_V7770717601297561774m_norm(A,Xa) ) ).

% dist_0_norm
tff(fact_6680_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Y))
        <=> ( Xa != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_6681_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Y)),zero_zero(real))
        <=> ( Xa = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_6682_open__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A5: set(A),B5: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => topolo1002775350975398744n_open(B,aa(A,set(B),B5,X)) )
           => topolo1002775350975398744n_open(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B5),A5))) ) ) ) ).

% open_INT
tff(fact_6683_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_6684_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => ? [U6: set(A),V4: set(A)] :
              ( topolo1002775350975398744n_open(A,U6)
              & topolo1002775350975398744n_open(A,V4)
              & member(A,Xa,U6)
              & member(A,Y,V4)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U6),V4) = bot_bot(set(A)) ) ) ) ) ).

% hausdorff
tff(fact_6685_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
        <=> ? [U7: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U7)
              & topolo1002775350975398744n_open(A,V5)
              & member(A,Xa,U7)
              & member(A,Y,V5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U7),V5) = bot_bot(set(A)) ) ) ) ) ).

% separation_t2
tff(fact_6686_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Y)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle
tff(fact_6687_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A,Z: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Y,Z))) ) ).

% dist_triangle2
tff(fact_6688_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A,A2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,Xa)),real_V557655796197034286t_dist(A,A2,Y))) ) ).

% dist_triangle3
tff(fact_6689_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Y,Z))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Y)),E3) ) ) ).

% dist_triangle_le
tff(fact_6690_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E1: real,X2: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_6691_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Y,Z))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),E3) ) ) ).

% dist_triangle_lt
tff(fact_6692_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xa: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A5),A5) ) ) ) ).

% Sup_notin_open
tff(fact_6693_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A5: set(A),Xa: A] :
          ( topolo1002775350975398744n_open(A,A5)
         => ( ! [X: A] :
                ( member(A,X,A5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A5),A5) ) ) ) ).

% Inf_notin_open
tff(fact_6694_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : real_V7770717601297561774m_norm(A,Xa) = real_V557655796197034286t_dist(A,Xa,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_6695_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,Xa: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xa)),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),E3) ) ) ).

% dist_commute_lessI
tff(fact_6696_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,D2: real] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_yq(A,fun(real,fun(A,$o)),Xa),D2))) ) ).

% open_ball
tff(fact_6697_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X5: A] :
              ( member(A,X5,S2)
             => ? [E4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
                  & ! [Y5: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y5,X5)),E4)
                     => member(A,Y5,S2) ) ) ) ) ) ).

% open_dist
tff(fact_6698_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Y)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_6699_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] :
          ( ( Xa != Y )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Y)) ) ) ).

% dist_pos_lt
tff(fact_6700_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Y)) ) ).

% zero_le_dist
tff(fact_6701_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [X: A] :
              ( member(A,X,S2)
             => ? [T10: set(A)] :
                  ( topolo1002775350975398744n_open(A,T10)
                  & member(A,X,T10)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T10),S2) ) )
         => topolo1002775350975398744n_open(A,S2) ) ) ).

% openI
tff(fact_6702_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X5: A] :
              ( member(A,X5,S2)
             => ? [T11: set(A)] :
                  ( topolo1002775350975398744n_open(A,T11)
                  & member(A,X5,T11)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T11),S2) ) ) ) ) ).

% open_subopen
tff(fact_6703_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xa: A] :
        ? [A6: fun(nat,set(A))] :
          ( ! [I4: nat] :
              ( member(A,Xa,aa(nat,set(A),A6,I4))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A6,I4)) )
          & ! [S8: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S8)
                & member(A,Xa,S8) )
             => ? [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),A6,I2)),S8) ) ) ) ).

% first_countable_basis
tff(fact_6704_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A),T3: set(A)] :
          ( member(A,A2,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T3)
             => ( topolo174197925503356063within(A,A2,T3) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_6705_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xa: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xa,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B4)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xa,B4)),S2) ) ) ) ) ) ).

% open_right
tff(fact_6706_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),real_V557655796197034286t_dist(A,A2,B2)),real_V557655796197034286t_dist(A,B2,C3)))),real_V557655796197034286t_dist(A,A2,C3)) ) ).

% abs_dist_diff_le
tff(fact_6707_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F8: A,A2: A,S2: set(A),D2: real,G3: fun(A,A)] :
          ( has_field_derivative(A,F3,F8,topolo174197925503356063within(A,A2,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,A2,S2)
             => ( ! [X: A] :
                    ( member(A,X,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D2)
                     => ( aa(A,A,F3,X) = aa(A,A,G3,X) ) ) )
               => has_field_derivative(A,G3,F8,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_6708_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F8: fun(A,B),Xa: A,Sa: set(A),D2: real,G3: fun(A,B)] :
          ( has_derivative(A,B,F3,F8,topolo174197925503356063within(A,Xa,Sa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,Xa,Sa)
             => ( ! [X8: A] :
                    ( member(A,X8,Sa)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                     => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) )
               => has_derivative(A,B,G3,F8,topolo174197925503356063within(A,Xa,Sa)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_6709_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M12: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M3)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M3),aa(nat,A,X4,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% metric_CauchyI
tff(fact_6710_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => ? [M10: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M2)
               => ! [N7: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N7)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M2),aa(nat,A,X4,N7))),E3) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6711_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Sa: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,Sa)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [N8: nat] :
                ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,Sa,N5),aa(nat,A,Sa,N8))),E4) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6712_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M11: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
                 => ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N5)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M5),aa(nat,A,X4,N5))),E4) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6713_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( topolo1002775350975398744n_open(A,S9)
             => ( member(A,F0,S9)
               => ? [N8: nat] :
                  ! [N5: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N5)
                   => member(A,aa(nat,A,F3,N5),S9) ) ) ) ) ) ).

% lim_explicit
tff(fact_6714_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X1: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X1)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,X2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_r
tff(fact_6715_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,Y: A,E3: real,X2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),E3) ) ) ) ).

% dist_triangle_half_l
tff(fact_6716_metric__LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L: B,A2: A,G3: fun(A,C),M: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X: A] :
                ( ( X != A2 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,G3,X),M)),real_V557655796197034286t_dist(B,aa(A,B,F3,X),L)) )
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_6717_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X1: A,X2: A,E3: real,X32: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X2)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,X32)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),aa(real,real,aa(real,fun(real,real),divide_divide(real),E3),aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X1,X42)),E3) ) ) ) ) ).

% dist_triangle_third
tff(fact_6718_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L: B,Xa: A,S2: set(A),D2: real,G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S2))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X8: A] :
                  ( member(A,X8,S2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xa))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                     => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) ) )
             => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_6719_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S2: set(A),T3: set(A),U2: set(A)] :
          ( member(A,Xa,S2)
         => ( topolo1002775350975398744n_open(A,S2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),S2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xa,T3) = topolo174197925503356063within(A,Xa,U2) ) ) ) ) ) ).

% at_within_nhd
tff(fact_6720_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G3: fun(A,B),G5: filter(B),Xa: A,S2: set(A),F4: filter(B),D2: real,F3: fun(A,B)] :
          ( filterlim(A,B,G3,G5,topolo174197925503356063within(A,Xa,S2))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G5),F4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X8: A] :
                    ( member(A,X8,S2)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X8,Xa))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X8,Xa)),D2)
                       => ( aa(A,B,F3,X8) = aa(A,B,G3,X8) ) ) ) )
               => filterlim(A,B,F3,F4,topolo174197925503356063within(A,Xa,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_6721_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => ? [M12: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M12),M3)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M3),aa(nat,A,X4,N))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% CauchyI'
tff(fact_6722_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F3)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [M11: nat] :
                ! [M5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
                 => ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M5),N5)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F3,M5),aa(nat,A,F3,N5))),E4) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6723_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A] :
          ( ( topolo174197925503356063within(A,A2,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_6724_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G3: fun(A,B),L: B,A2: A,R3: real,F3: fun(A,B)] :
          ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
           => ( ! [X: A] :
                  ( ( X != A2 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),R3)
                   => ( aa(A,B,F3,X) = aa(A,B,G3,X) ) ) )
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_6725_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X),L5)),R) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_6726_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R2: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
                & ! [X3: A] :
                    ( ( ( X3 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S3) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X3),L5)),R2) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_6727_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S6: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S6)
                  & ! [X5: A] :
                      ( ( ( X5 != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),S6) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X5),L5)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_6728_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No2: nat] :
                ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N5),L5)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_6729_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L5: A] :
          ( ! [R: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N),L5)),R) ) )
         => filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6730_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N7: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N7)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N7),L5)),R2) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6731_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [J3: nat] :
            ? [M11: nat] :
            ! [M5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M5)
             => ! [N5: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N5)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M5),aa(nat,A,X4,N5))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6732_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G3: fun(B,C),C3: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,B2),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( filterlim(B,C,G3,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B2,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X: A] :
                      ( ( ( X != A2 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A2)),D4) )
                     => ( aa(A,B,F3,X) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_yr(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G3),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_6733_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A2: B,S2: set(B),F3: fun(B,C),L5: C] :
          ( nO_MATCH(A,B,zero_zero(A),A2)
         => ( member(B,A2,S2)
           => ( topolo1002775350975398744n_open(B,S2)
             => ( filterlim(B,C,F3,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,S2))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_wt(B,fun(fun(B,C),fun(B,C)),A2),F3),topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6734_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No2)
                  & ! [N5: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N5)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N5),L5)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6735_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [K3: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),K3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_yt(real,fun(A,set(A)),E4)),K3))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_6736_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_yu(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_6737_totally__bounded__empty,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => topolo6688025880775521714ounded(A,bot_bot(set(A))) ) ).

% totally_bounded_empty
tff(fact_6738_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A),T3: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S2)
           => topolo6688025880775521714ounded(A,T3) ) ) ) ).

% totally_bounded_subset
tff(fact_6739_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_6740_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( ( C3 != zero_zero(B) )
           => ( filterlim(A,B,G3,at_infinity(B),F4)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),at_infinity(B),F4) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_6741_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),F4: filter(A),Na: nat] :
          ( filterlim(A,B,F3,at_infinity(B),F4)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_yw(fun(A,B),fun(nat,fun(A,B)),F3),Na),at_infinity(B),F4) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6742_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( filterlim(A,B,G3,at_infinity(B),F4)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yx(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F4) ) ) ) ).

% tendsto_divide_0
tff(fact_6743_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_6744_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),G3),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F4)
        <=> filterlim(A,B,G3,at_infinity(B),F4) ) ) ).

% filterlim_inverse_at_iff
tff(fact_6745_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),C3: A,F4: filter(A),G3: fun(A,A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,C3),F4)
         => ( filterlim(A,A,G3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F4)
           => ( ( C3 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_tp(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G3),at_infinity(A),F4) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6746_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xa))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xa),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_6747_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C3: fun(nat,A),K: nat,Na: nat,B5: real] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_yy(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C3),Na),B5),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6748_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yz(nat,fun(fun(real,real),fun(real,real)),Na),F3),at_bot(real),F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6749_eventually__sequentially__Suc,axiom,
    ! [P2: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_io(fun(nat,$o),fun(nat,$o),P2),at_top(nat))
    <=> eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6750_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_za(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,F3,at_bot(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_6751_at__bot__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_bot(real)),at_infinity(real)) ).

% at_bot_le_at_infinity
tff(fact_6752_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_za(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,aTP_Lamp_zb(fun(A,real),fun(A,real),F3),at_bot(real),F4) ) ) ).

% filterlim_inverse_at_bot
tff(fact_6753_eventually__sequentiallyI,axiom,
    ! [C3: nat,P2: fun(nat,$o)] :
      ( ! [X: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),X)
         => aa(nat,$o,P2,X) )
     => eventually(nat,P2,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6754_eventually__sequentially,axiom,
    ! [P2: fun(nat,$o)] :
      ( eventually(nat,P2,at_top(nat))
    <=> ? [N8: nat] :
        ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N5)
         => aa(nat,$o,P2,N5) ) ) ).

% eventually_sequentially
tff(fact_6755_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N8),N5)
             => aa(A,$o,P2,N5) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6756_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P2: fun(A,$o)] :
          ( ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X)
             => aa(A,$o,P2,X) )
         => eventually(A,P2,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6757_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less_eq(A),C3),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6758_le__sequentially,axiom,
    ! [F4: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F4),at_top(nat))
    <=> ! [N8: nat] : eventually(nat,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),F4) ) ).

% le_sequentially
tff(fact_6759_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),ord_less(A),C3),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6760_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N8),N5)
             => aa(A,$o,P2,N5) ) ) ) ).

% eventually_at_top_dense
tff(fact_6761_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N5),N8)
             => aa(A,$o,P2,N5) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6762_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C3: A] : eventually(A,aTP_Lamp_zc(A,fun(A,$o),C3),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6763_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : eventually(A,aa(A,fun(A,$o),aTP_Lamp_je(A,fun(A,$o)),C3),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6764_le__filter__def,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
    <=> ! [P7: fun(A,$o)] :
          ( eventually(A,P7,F6)
         => eventually(A,P7,F4) ) ) ).

% le_filter_def
tff(fact_6765_filter__leI,axiom,
    ! [A: $tType,F6: filter(A),F4: filter(A)] :
      ( ! [P: fun(A,$o)] :
          ( eventually(A,P,F6)
         => eventually(A,P,F4) )
     => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6) ) ).

% filter_leI
tff(fact_6766_filter__leD,axiom,
    ! [A: $tType,F4: filter(A),F6: filter(A),P2: fun(A,$o)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
     => ( eventually(A,P2,F6)
       => eventually(A,P2,F4) ) ) ).

% filter_leD
tff(fact_6767_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_bot(A))
        <=> ? [N8: A] :
            ! [N5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N5),N8)
             => aa(A,$o,P2,N5) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6768_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F4: filter(B),G5: filter(A),F6: filter(B),G7: filter(A),F8: fun(A,B)] :
      ( filterlim(A,B,F3,F4,G5)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F4),F6)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G7),G5)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zd(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),F8),G7)
           => filterlim(A,B,F8,F6,G7) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6769_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z7),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_ze(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6770_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_ze(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ).

% filterlim_at_bot
tff(fact_6771_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z7),C3)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zf(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6772_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_zg(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ).

% filterlim_at_bot_dense
tff(fact_6773_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),top_top(A))
         => ( eventually(A,P2,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),top_top(A))
                & ! [Z2: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Z2)
                   => aa(A,$o,P2,Z2) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6774_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q2: fun(A,$o),F3: fun(A,B),P2: fun(B,$o),G3: fun(B,A)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,Q2,X)
             => ( aa(A,$o,Q2,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P2,X)
               => ( aa(A,B,F3,aa(B,A,G3,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P2,X)
                 => aa(A,$o,Q2,aa(B,A,G3,X)) )
             => ( eventually(A,Q2,at_top(A))
               => ( eventually(B,P2,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6775_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xa: A,P2: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( eventually(A,P2,topolo174197925503356063within(A,Xa,set_ord_lessThan(A,Xa)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xa)
                & ! [Y5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y5)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
                     => aa(A,$o,P2,Y5) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6776_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P2: fun(A,$o),Xa: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,Xa,set_ord_lessThan(A,Xa)))
        <=> ? [B7: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xa)
              & ! [Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),Xa)
                   => aa(A,$o,P2,Y5) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6777_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B7: real] :
            ! [X5: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X5))
             => aa(A,$o,P2,X5) ) ) ) ).

% eventually_at_infinity
tff(fact_6778_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),G3: fun(A,B),Net: filter(A),H2: fun(A,B),C3: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zh(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zh(fun(A,B),fun(fun(A,B),fun(A,$o)),G3),H2),Net)
           => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),Net)
             => ( filterlim(A,B,H2,topolo7230453075368039082e_nhds(B,C3),Net)
               => filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,C3),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6779_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Xa: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xa),F4)
        <=> ( ! [L4: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L4),Xa)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),F3),L4),F4) )
            & ! [U3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xa),U3)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),F3),U3),F4) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6780_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F3: fun(B,A),F4: filter(B)] :
          ( ! [A4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A4),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_zk(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
         => ( ! [A4: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),A4)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,A),fun(A,fun(B,$o)),F3),A4),F4) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Y),F4) ) ) ) ).

% order_tendstoI
tff(fact_6781_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A2),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(1)
tff(fact_6782_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),Y: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Y),F4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y),A2)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4) ) ) ) ).

% order_tendstoD(2)
tff(fact_6783_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,B)] :
          ( filterlim(A,B,F3,at_top(B),F4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zm(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
           => filterlim(A,B,G3,at_top(B),F4) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6784_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zn(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6785_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_zn(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ).

% filterlim_at_top
tff(fact_6786_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_zo(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ).

% filterlim_at_top_dense
tff(fact_6787_real__tendsto__sandwich,axiom,
    ! [A: $tType,F3: fun(A,real),G3: fun(A,real),Net: filter(A),H2: fun(A,real),C3: real] :
      ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_zp(fun(A,real),fun(fun(A,real),fun(A,$o)),F3),G3),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_zp(fun(A,real),fun(fun(A,real),fun(A,$o)),G3),H2),Net)
       => ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),Net)
         => ( filterlim(A,real,H2,topolo7230453075368039082e_nhds(real,C3),Net)
           => filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,C3),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_6788_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xa: A] :
          ~ ! [A6: fun(nat,set(A))] :
              ( ! [I4: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A6,I4))
             => ( ! [I4: nat] : member(A,Xa,aa(nat,set(A),A6,I4))
               => ~ ! [S8: set(A)] :
                      ( topolo1002775350975398744n_open(A,S8)
                     => ( member(A,Xa,S8)
                       => eventually(nat,aa(set(A),fun(nat,$o),aTP_Lamp_zq(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A6),S8),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_6789_eventually__Inf__base,axiom,
    ! [A: $tType,B5: set(filter(A)),P2: fun(A,$o)] :
      ( ( B5 != bot_bot(set(filter(A))) )
     => ( ! [F5: filter(A)] :
            ( member(filter(A),F5,B5)
           => ! [G4: filter(A)] :
                ( member(filter(A),G4,B5)
               => ? [X3: filter(A)] :
                    ( member(filter(A),X3,B5)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X3),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F5),G4)) ) ) )
       => ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
        <=> ? [X5: filter(A)] :
              ( member(filter(A),X5,B5)
              & eventually(A,P2,X5) ) ) ) ) ).

% eventually_Inf_base
tff(fact_6790_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,$o),F4: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),A5)))
      <=> ? [Q8: fun(A,fun(B,$o))] :
            ( ! [X5: A] :
                ( member(A,X5,A5)
               => eventually(B,aa(A,fun(B,$o),Q8,X5),aa(A,filter(B),F4,X5)) )
            & ! [Y5: B] :
                ( ! [X5: A] :
                    ( member(A,X5,A5)
                   => aa(B,$o,aa(A,fun(B,$o),Q8,X5),Y5) )
               => aa(B,$o,P2,Y5) ) ) ) ) ).

% eventually_INF_finite
tff(fact_6791_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B2),A2)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_zr(real,fun(real,fun(real,$o)),B2),A2),topolo174197925503356063within(real,A2,set_ord_lessThan(real,A2))) ) ).

% eventually_at_left_real
tff(fact_6792_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X5: A] :
                  ( member(A,X5,S2)
                 => ( ( ( X5 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),D6) )
                   => aa(A,$o,P2,X5) ) ) ) ) ) ).

% eventually_at
tff(fact_6793_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( eventually(A,P2,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X5: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,A2)),D6)
                 => aa(A,$o,P2,X5) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_6794_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P2: fun(A,$o)] :
          ( ! [X: A] :
              ( member(A,X,set_or5935395276787703475ssThan(A,A2,B2))
             => aa(A,$o,P2,X) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => eventually(A,P2,topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6795_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o),A2: A] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_zs(fun(A,$o),fun(A,fun(A,$o)),P2),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6796_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F3: fun(A,B),F4: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zt(B,fun(fun(A,B),fun(A,$o)),L),F3),F4)
         => ( ! [X: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),F3),X),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% decreasing_tendsto
tff(fact_6797_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_zu(fun(A,B),fun(B,fun(A,$o)),F3),L),F4)
         => ( ! [X: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),F3),X),F4) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ) ).

% increasing_tendsto
tff(fact_6798_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F4: filter(A),C3: B] :
          ( filterlim(A,B,F3,at_top(B),F4)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C3),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_zv(fun(A,B),fun(B,fun(A,$o)),F3),Z7),F4) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6799_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F4: filter(A),F3: fun(A,B),Xa: B,G3: fun(A,B),Y: B] :
          ( ( F4 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xa),F4)
           => ( filterlim(A,B,G3,topolo7230453075368039082e_nhds(B,Y),F4)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_zw(fun(A,B),fun(fun(A,B),fun(A,$o)),F3),G3),F4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xa) ) ) ) ) ) ).

% tendsto_le
tff(fact_6800_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xa: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xa),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_zx(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A2),Xa) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6801_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: fun(A,B),Xa: B,F4: filter(A),A2: B] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,Xa),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_zy(fun(A,B),fun(B,fun(A,$o)),F3),A2),F4)
           => ( ( F4 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),A2) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6802_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),A2: B,F4: filter(A),G3: fun(A,C),B2: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( eventually(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_zz(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F3),A2),G3),B2),F4)
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,B2),F4) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_6803_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),F4: fun(B,filter(A)),B5: set(B)] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F4),B5)))
    <=> ? [X9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X9),B5)
          & aa(set(B),$o,finite_finite2(B),X9)
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F4),X9))) ) ) ).

% eventually_INF
tff(fact_6804_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P2: fun(A,$o),A2: A,S2: set(A)] :
          ( eventually(A,P2,topolo174197925503356063within(A,A2,S2))
        <=> ? [D6: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
              & ! [X5: A] :
                  ( member(A,X5,S2)
                 => ( ( ( X5 != A2 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X5,A2)),D6) )
                   => aa(A,$o,P2,X5) ) ) ) ) ) ).

% eventually_at_le
tff(fact_6805_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,$o)] :
          ( eventually(A,P3,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B7)
              & ! [X5: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X5))
                 => aa(A,$o,P3,X5) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_6806_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F4)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),F3),L5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F4) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6807_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A),E3: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
           => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aaa(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E3),F4) ) ) ) ).

% tendstoD
tff(fact_6808_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( ! [E2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aaa(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E2),F4) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4) ) ) ).

% tendstoI
tff(fact_6809_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aaa(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),L),E4),F4) ) ) ) ).

% tendsto_iff
tff(fact_6810_eventually__Inf,axiom,
    ! [A: $tType,P2: fun(A,$o),B5: set(filter(A))] :
      ( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B5))
    <=> ? [X9: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X9),B5)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X9)
          & eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X9)) ) ) ).

% eventually_Inf
tff(fact_6811_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aab(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( summable(real,G3)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test_ev
tff(fact_6812_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A2)
       => ( eventually(A,aTP_Lamp_aac(fun(A,real),fun(A,$o),F3),F4)
         => filterlim(A,real,aTP_Lamp_wp(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F4) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6813_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q2: fun(A,$o),F3: fun(A,B),P2: fun(B,$o),G3: fun(B,A),A2: A] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,Q2,X)
             => ( aa(A,$o,Q2,Y3)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y3)) ) ) )
         => ( ! [X: B] :
                ( aa(B,$o,P2,X)
               => ( aa(A,B,F3,aa(B,A,G3,X)) = X ) )
           => ( ! [X: B] :
                  ( aa(B,$o,P2,X)
                 => aa(A,$o,Q2,aa(B,A,G3,X)) )
             => ( eventually(A,Q2,topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2)))
               => ( ! [B4: A] :
                      ( aa(A,$o,Q2,B4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),A2) )
                 => ( eventually(B,P2,at_top(B))
                   => filterlim(A,B,F3,at_top(B),topolo174197925503356063within(A,A2,set_ord_lessThan(A,A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6814_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B5: set(A),F4: fun(A,filter(B)),P2: fun(B,$o)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( ! [A4: A] :
            ( member(A,A4,B5)
           => ! [B4: A] :
                ( member(A,B4,B5)
               => ? [X3: A] :
                    ( member(A,X3,B5)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F4,X3)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F4,A4)),aa(A,filter(B),F4,B4))) ) ) )
       => ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F4),B5)))
        <=> ? [X5: A] :
              ( member(A,X5,B5)
              & eventually(B,P2,aa(A,filter(B),F4,X5)) ) ) ) ) ).

% eventually_INF_base
tff(fact_6815_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: fun(A,B),C3: B,F4: filter(A),A5: set(B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C3),F4)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aad(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F3),C3),A5),F4)
           => filterlim(A,B,F3,topolo174197925503356063within(B,C3,A5),F4) ) ) ) ).

% filterlim_at_withinI
tff(fact_6816_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F4: filter(A),G3: fun(A,C),K4: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F4)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_aae(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F3),G3),K4),F4)
           => filterlim(A,C,G3,topolo7230453075368039082e_nhds(C,zero_zero(C)),F4) ) ) ) ).

% tendsto_0_le
tff(fact_6817_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_bot(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aaf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6818_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aag(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_floor_less
tff(fact_6819_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F4)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_aah(fun(A,B),fun(B,fun(A,$o)),F3),L),F4) ) ) ) ).

% eventually_less_ceiling
tff(fact_6820_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C3: real,F3: fun(A,B),F4: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
         => ( filterlim(A,B,F3,at_infinity(B),F4)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),R5)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_aai(fun(A,B),fun(real,fun(A,$o)),F3),R5),F4) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_6821_tendsto__zero__powrI,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_aaj(fun(A,real),fun(A,$o),F3),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aak(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_6822_tendsto__powr2,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( eventually(A,aTP_Lamp_aaj(fun(A,real),fun(A,$o),F3),F4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aak(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ) ).

% tendsto_powr2
tff(fact_6823_tendsto__powr_H,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,B2),F4)
       => ( ( ( A2 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B2)
              & eventually(A,aTP_Lamp_aaj(fun(A,real),fun(A,$o),F3),F4) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aak(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F4) ) ) ) ).

% tendsto_powr'
tff(fact_6824_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),B2)
         => ? [Y4: real] :
              ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y4) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F3,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6825_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aal(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_Cauchy'
tff(fact_6826_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
        <=> ? [Y5: B,K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & eventually(A,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aam(fun(A,B),fun(B,fun(real,fun(A,$o))),F3),Y5),K5),F4) ) ) ) ).

% Bfun_metric_def
tff(fact_6827_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),F3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_6828_at__top__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_top(real)),at_infinity(real)) ).

% at_top_le_at_infinity
tff(fact_6829_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),K4: real] :
          ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N))),K4)
         => bfun(nat,A,X4,at_top(nat)) ) ) ).

% BseqI'
tff(fact_6830_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_top(A))
         => eventually(A,aTP_Lamp_aan(fun(A,$o),fun(A,$o),P2),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6831_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B),D2: B] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_aao(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A5),B5),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_6832_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ql(A,fun(fun(nat,A),fun(nat,A)),C3),F3),at_top(nat))
          <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_6833_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P2: fun(A,$o)] : ord_Least(A,P2) = the(A,aTP_Lamp_aap(fun(A,$o),fun(A,$o),P2)) ) ).

% Least_def
tff(fact_6834_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(nat,A),G3: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_aaq(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),F3),G3),at_top(nat))
         => ( bfun(nat,B,G3,at_top(nat))
           => bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_6835_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P2: fun(A,$o)] : order_Greatest(A,P2) = the(A,aTP_Lamp_aar(fun(A,$o),fun(A,$o),P2)) ) ).

% Greatest_def
tff(fact_6836_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F3: fun(A,real),F4: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vu(nat,fun(fun(A,real),fun(A,real)),Na),F3),at_top(real),F4) ) ) ).

% filterlim_pow_at_top
tff(fact_6837_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F4)
     => ( eventually(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,F3,at_top(real),F4) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_6838_Bseq__eq__bounded,axiom,
    ! [F3: fun(nat,real),A2: real,B2: real] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),aa(set(nat),set(real),image(nat,real,F3),top_top(set(nat)))),set_or1337092689740270186AtMost(real,A2,B2))
     => bfun(nat,real,F3,at_top(nat)) ) ).

% Bseq_eq_bounded
tff(fact_6839_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N5))),K5) ) ) ) ).

% Bseq_def
tff(fact_6840_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K4: real,X4: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K4)
         => ( ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N))),K4)
           => bfun(nat,A,X4,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_6841_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N7))),K8) ) ) ) ).

% BseqE
tff(fact_6842_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N7: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N7))),K8) ) ) ) ).

% BseqD
tff(fact_6843_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [N8: nat] :
            ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% Bseq_iff1a
tff(fact_6844_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [N8: nat] :
            ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N8))) ) ) ).

% Bseq_iff
tff(fact_6845_Bseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => bfun(nat,real,aa(real,fun(nat,real),power_power(real),Xa),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_6846_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aat(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6847_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aaf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6848_tendsto__neg__powr,axiom,
    ! [A: $tType,Sa: real,F3: fun(A,real),F4: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Sa),zero_zero(real))
     => ( filterlim(A,real,F3,at_top(real),F4)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aau(real,fun(fun(A,real),fun(A,real)),Sa),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% tendsto_neg_powr
tff(fact_6849_LIM__at__top__divide,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A2)
       => ( filterlim(A,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
         => ( eventually(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),G3),F4)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_top(real),F4) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_6850_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4)
     => ( eventually(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),F3),F4)
       => filterlim(A,real,aTP_Lamp_zb(fun(A,real),fun(A,real),F3),at_top(real),F4) ) ) ).

% filterlim_inverse_at_top
tff(fact_6851_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),F3),F4)
     => ( filterlim(A,real,aTP_Lamp_zb(fun(A,real),fun(A,real),F3),at_top(real),F4)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_6852_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C3: real,F4: filter(A),G3: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C3),F4)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
       => ( filterlim(A,real,G3,at_top(real),F4)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aaf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G3),at_bot(real),F4) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6853_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),K4: real,F4: filter(A)] :
          ( eventually(A,aa(real,fun(A,$o),aTP_Lamp_aaw(fun(A,B),fun(real,fun(A,$o)),F3),K4),F4)
         => bfun(A,B,F3,F4) ) ) ).

% BfunI
tff(fact_6854_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X5: A] :
                ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X4,N5)),aa(A,A,uminus_uminus(A),X5)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_6855_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N8: nat] :
                ! [N5: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X4,N5)),aa(A,A,uminus_uminus(A),aa(nat,A,X4,N8))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_6856_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),X)
         => ? [Y4: real] :
              ( has_field_derivative(real,F3,Y4,topolo174197925503356063within(real,X,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y4),zero_zero(real)) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F3,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6857_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),A2: B,F4: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F4)
         => ( ( A2 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),F3),F4) ) ) ) ).

% Bfun_inverse
tff(fact_6858_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F3: fun(real,real),F4: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F3,at_bot(real),F4)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yz(nat,fun(fun(real,real),fun(real,real)),Na),F3),at_top(real),F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6859_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
         => ~ ! [B8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B8)
               => ~ eventually(A,aa(real,fun(A,$o),aTP_Lamp_aaw(fun(A,B),fun(real,fun(A,$o)),F3),B8),F4) ) ) ) ).

% BfunE
tff(fact_6860_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F4: filter(A)] :
          ( bfun(A,B,F3,F4)
        <=> ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & eventually(A,aa(real,fun(A,$o),aTP_Lamp_aaw(fun(A,B),fun(real,fun(A,$o)),F3),K5),F4) ) ) ) ).

% Bfun_def
tff(fact_6861_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(nat,A),G3: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aax(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F3),G3),at_top(nat))
         => ( filterlim(nat,real,G3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_bounded_partials
tff(fact_6862_max__ext__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : max_ext(A,R3) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_aay(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R3))) ).

% max_ext_eq
tff(fact_6863_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U)) ).

% finite_greaterThanAtMost
tff(fact_6864_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_6865_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_6866_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_6867_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_6868_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% infinite_Ioc_iff
tff(fact_6869_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xa,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_6870_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,Xa)) = Xa ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_6871_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xa,Y)) = Xa ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_6872_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,Xa)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_6873_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_6874_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_6875_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C3,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3) )
            | ( ( A2 = C3 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_6876_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_6877_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,L),U) = set_or3652927894154168847AtMost(nat,L,U) ).

% atLeastSucAtMost_greaterThanAtMost
tff(fact_6878_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A2,B2)) ) ) ).

% infinite_Ioc
tff(fact_6879_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D2)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),A2) ) ) ) ).

% Ioc_disjoint
tff(fact_6880_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_6881_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),Xa: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( member(A,Xa,S2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa)
             => ? [B4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B4),Xa)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B4,Xa)),S2) ) ) ) ) ) ).

% open_left
tff(fact_6882_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_6883_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = set_ord_atMost(A,U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_6884_ivl__disj__int__one_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_ord_atMost(A,L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_6885_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_6886_greaterThanAtMost__upt,axiom,
    ! [Na: nat,M: nat] : set_or3652927894154168847AtMost(nat,Na,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,Na),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_6887_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% sum.head
tff(fact_6888_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% prod.head
tff(fact_6889_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_6890_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_6891_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6892_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_6893_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_6894_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6895_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6896_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I)),linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_6897_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6898_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6899_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),U)
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6900_max__extp__max__ext__eq,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X3: set(A),Xa2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),max_extp(A,aTP_Lamp_aaz(set(product_prod(A,A)),fun(A,fun(A,$o)),R3)),X3),Xa2)
    <=> member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X3),Xa2),max_ext(A,R3)) ) ).

% max_extp_max_ext_eq
tff(fact_6901_max__ext_Ocases,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R3: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A12),A23),max_ext(A,R3))
     => ~ ( aa(set(A),$o,finite_finite2(A),A12)
         => ( aa(set(A),$o,finite_finite2(A),A23)
           => ( ( A23 != bot_bot(set(A)) )
             => ~ ! [X3: A] :
                    ( member(A,X3,A12)
                   => ? [Xa5: A] :
                        ( member(A,Xa5,A23)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa5),R3) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_6902_max__ext_Osimps,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R3: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A12),A23),max_ext(A,R3))
    <=> ( aa(set(A),$o,finite_finite2(A),A12)
        & aa(set(A),$o,finite_finite2(A),A23)
        & ( A23 != bot_bot(set(A)) )
        & ! [X5: A] :
            ( member(A,X5,A12)
           => ? [Xa3: A] :
                ( member(A,Xa3,A23)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3),R3) ) ) ) ) ).

% max_ext.simps
tff(fact_6903_max__ext_Omax__extI,axiom,
    ! [A: $tType,X4: set(A),Y6: set(A),R3: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(A),$o,finite_finite2(A),Y6)
       => ( ( Y6 != bot_bot(set(A)) )
         => ( ! [X: A] :
                ( member(A,X,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,Y6)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa2),R3) ) )
           => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X4),Y6),max_ext(A,R3)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_6904_max__ext__def,axiom,
    ! [A: $tType,X3: set(product_prod(A,A))] : max_ext(A,X3) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),max_extp(A,aTP_Lamp_aaz(set(product_prod(A,A)),fun(A,fun(A,$o)),X3)))) ).

% max_ext_def
tff(fact_6905_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_6906_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xa,Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X5,n2) )
                          & vEBT_VEBT_valid(Summary2,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                              & ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_6907_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xa,Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m: nat,
                        m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                        ( ! [X5: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_VEBT_valid(X5,n2) )
                        & vEBT_VEBT_valid(Summary2,m)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                            & ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                            aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_6908_finite__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),$o,finite_finite2(code_integer),set_or3652927894154168847AtMost(code_integer,L,U)) ).

% finite_greaterThanAtMost_integer
tff(fact_6909_ball__empty,axiom,
    ! [A: $tType,P2: fun(A,$o),X3: A] :
      ( member(A,X3,bot_bot(set(A)))
     => aa(A,$o,P2,X3) ) ).

% ball_empty
tff(fact_6910_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),Q2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_abb(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P2),Q2)))
      <=> ! [Y5: A] :
            ( aa(A,$o,P2,Y5)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_iv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q2),Y5))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_6911_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: B,B5: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_abc(fun(B,A),fun(B,fun(set(B),fun(A,$o))),F3),A2),B5)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,F3,A2)),aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_abd(fun(B,A),fun(set(B),fun(A,$o)),F3),B5))) ).

% eq_or_mem_image_simp
tff(fact_6912_ball__UNIV,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,top_top(set(A)))
         => aa(A,$o,P2,X5) )
    <=> ! [X_13: A] : aa(A,$o,P2,X_13) ) ).

% ball_UNIV
tff(fact_6913_bex__UNIV,axiom,
    ! [A: $tType,P2: fun(A,$o)] :
      ( ? [X5: A] :
          ( member(A,X5,top_top(set(A)))
          & aa(A,$o,P2,X5) )
    <=> ? [X_13: A] : aa(A,$o,P2,X_13) ) ).

% bex_UNIV
tff(fact_6914_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X: A] :
            ( member(A,X,A5)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_iv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P2),X),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_abe(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),P2),Net) ) ) ).

% eventually_ball_finite
tff(fact_6915_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A5: set(A),P2: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_abe(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A5),P2),Net)
      <=> ! [X5: A] :
            ( member(A,X5,A5)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_iv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P2),X5),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_6916_listrel1__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : listrel1(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_abf(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% listrel1_def
tff(fact_6917_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: fun(A,$o),Q2: fun(B,$o),F3: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q2))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_abg(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P2),Q2),F3))) ) ) ).

% finite_image_set2
tff(fact_6918_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,$o),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P2))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_abh(fun(A,$o),fun(fun(A,B),fun(B,$o)),P2),F3))) ) ).

% finite_image_set
tff(fact_6919_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_abi(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_6920_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_abj(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_6921_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_abk(product_prod(A,A),$o))) ) ).

% open_diagonal_complement
tff(fact_6922_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_abd(fun(B,A),fun(set(B),fun(A,$o)),F3),A5)) = aa(set(B),set(A),image(B,A,F3),A5) ).

% Setcompr_eq_image
tff(fact_6923_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_abl(fun(B,A),fun(fun(B,$o),fun(A,$o)),F3),P2)) = aa(set(B),set(A),image(B,A,F3),aa(fun(B,$o),set(B),collect(B),P2)) ).

% setcompr_eq_image
tff(fact_6924_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abm(fun(B,A),fun(A,$o),F3)) = aa(set(B),set(A),image(B,A,F3),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_6925_Un__interval,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [B1: A,B22: A,B32: A,F3: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B1),B22)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B22),B32)
           => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_abn(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B22),F3))),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_abn(A,fun(A,fun(fun(A,B),fun(B,$o))),B22),B32),F3))) = aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_abn(A,fun(A,fun(fun(A,B),fun(B,$o))),B1),B32),F3)) ) ) ) ) ).

% Un_interval
tff(fact_6926_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A5) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abo(set(A),fun(A,$o),A5))) ) ).

% Inf_eq_Sup
tff(fact_6927_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A5) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abp(set(A),fun(A,$o),A5))) ) ).

% Sup_eq_Inf
tff(fact_6928_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_abq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_6929_Ball__fold,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ! [X5: A] :
            ( member(A,X5,A5)
           => aa(A,$o,P2,X5) )
      <=> finite_fold(A,$o,aTP_Lamp_abr(fun(A,$o),fun(A,fun($o,$o)),P2),$true,A5) ) ) ).

% Ball_fold
tff(fact_6930_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_abs(fun(A,option(B)),fun(product_prod(A,B),$o),M)) ).

% graph_def
tff(fact_6931_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abt(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_6932_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or1337092689740270186AtMost(code_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),L),one_one(code_integer)),U) = set_or3652927894154168847AtMost(code_integer,L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_6933_set__drop__conv,axiom,
    ! [A: $tType,Na: nat,L: list(A)] : aa(list(A),set(A),set2(A),drop(A,Na,L)) = aa(fun(A,$o),set(A),collect(A),aa(list(A),fun(A,$o),aTP_Lamp_abu(nat,fun(list(A),fun(A,$o)),Na),L)) ).

% set_drop_conv
tff(fact_6934_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg3: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary),Deg3)
    <=> ( ( Deg = Deg3 )
        & $let(
            n2: nat,
            n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m: nat,
              m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg),n2),
              ( ! [X5: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                 => vEBT_VEBT_valid(X5,n2) )
              & vEBT_VEBT_valid(Summary,m)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X9)
                  & ! [X5: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_6935_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xa,Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xa = vEBT_Leaf((Uu),(Uv))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m: nat,
                      m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                      ( ! [X5: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_VEBT_valid(X5,n2) )
                      & vEBT_VEBT_valid(Summary2,m)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                          & ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                          aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_6936_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                          ( ! [X5: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X5,n2) )
                          & vEBT_VEBT_valid(Summary2,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                              & ! [X5: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_6937_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X5,n2) )
                            & vEBT_VEBT_valid(Summary2,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_6938_option__Inf__Sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(option(A)))] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(set(option(A)),option(A),complete_Inf_Inf(option(A)),aa(set(set(option(A))),set(option(A)),image(set(option(A)),option(A),complete_Sup_Sup(option(A))),A5))),aa(set(option(A)),option(A),complete_Sup_Sup(option(A)),aa(set(set(option(A))),set(option(A)),image(set(option(A)),option(A),complete_Inf_Inf(option(A))),aa(fun(set(option(A)),$o),set(set(option(A))),collect(set(option(A))),aTP_Lamp_abv(set(set(option(A))),fun(set(option(A)),$o),A5))))) ) ).

% option_Inf_Sup
tff(fact_6939_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_abw(set(set(A)),fun(set(A),$o),A5))))) ) ).

% Inf_Sup_le
tff(fact_6940_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_abx(set(set(A)),fun(set(A),$o),A5))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A5))) ) ).

% Sup_Inf_le
tff(fact_6941_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A5: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A5))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aby(set(set(A)),fun(set(A),$o),A5))))) ) ).

% finite_Inf_Sup
tff(fact_6942_Inf__filter__def,axiom,
    ! [A: $tType,S2: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S2) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),$o),set(filter(A)),collect(filter(A)),aTP_Lamp_abz(set(filter(A)),fun(filter(A),$o),S2))) ).

% Inf_filter_def
tff(fact_6943_Pow__Compl,axiom,
    ! [A: $tType,A5: set(A)] : pow(A,aa(set(A),set(A),uminus_uminus(set(A)),A5)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aca(set(A),fun(set(A),$o),A5)) ).

% Pow_Compl
tff(fact_6944_Sup__real__def,axiom,
    ! [X4: set(real)] : aa(set(real),real,complete_Sup_Sup(real),X4) = ord_Least(real,aTP_Lamp_acb(set(real),fun(real,$o),X4)) ).

% Sup_real_def
tff(fact_6945_Sup__int__def,axiom,
    ! [X4: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X4) = the(int,aTP_Lamp_acc(set(int),fun(int,$o),X4)) ).

% Sup_int_def
tff(fact_6946_Union__maximal__sets,axiom,
    ! [A: $tType,F16: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),F16)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_acd(set(set(A)),fun(set(A),$o),F16))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F16) ) ) ).

% Union_maximal_sets
tff(fact_6947_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xa,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xa),Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xa = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu),(Uv))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary2) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Deg2),n2),
                            ( ! [X5: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X5,n2) )
                            & vEBT_VEBT_valid(Summary2,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X9)
                                & ! [X5: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary2),n2),m)),Mima) ) ) ) ) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg2,TreeList2,Summary2)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_6948_set__bits__int__def,axiom,
    ! [F3: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F3) = $ite(
        ? [N5: nat] :
        ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),M5)
         => ( aa(nat,$o,F3,M5) = aa(nat,$o,F3,N5) ) ),
        $let(
          n2: nat,
          n2:= ord_Least(nat,aTP_Lamp_ace(fun(nat,$o),fun(nat,$o),F3)),
          aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F3),upt(zero_zero(nat),aa(nat,nat,suc,n2))))) ),
        zero_zero(int) ) ).

% set_bits_int_def
tff(fact_6949_set__bits__int__unfold_H,axiom,
    ! [F3: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F3) = $ite(
        ? [N5: nat] :
        ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
         => ~ aa(nat,$o,F3,N11) ),
        groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F3),upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_acf(fun(nat,$o),fun(nat,$o),F3))))),
        $ite(
          ? [N5: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
           => aa(nat,$o,F3,N11) ),
          $let(
            n2: nat,
            n2:= ord_Least(nat,aTP_Lamp_acg(fun(nat,$o),fun(nat,$o),F3)),
            aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list($o),list($o),aa(list($o),fun(list($o),list($o)),append($o),aa(list(nat),list($o),map(nat,$o,F3),upt(zero_zero(nat),n2))),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),$true),nil($o))))) ),
          zero_zero(int) ) ) ).

% set_bits_int_unfold'
tff(fact_6950_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_ach(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_6951_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_aci(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).

% min_ext_def
tff(fact_6952_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Na: nat] : lexn(A,R2,Na) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_acj(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),Na))) ).

% lexn_conv
tff(fact_6953_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_6954_brk__rel__def,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : brk_rel(A,B,R3) = aa(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B))),aa(set(product_prod(product_prod($o,A),product_prod($o,B))),fun(set(product_prod(product_prod($o,A),product_prod($o,B))),set(product_prod(product_prod($o,A),product_prod($o,B)))),sup_sup(set(product_prod(product_prod($o,A),product_prod($o,B)))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_ack(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),R3))),aa(fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),set(product_prod(product_prod($o,A),product_prod($o,B))),collect(product_prod(product_prod($o,A),product_prod($o,B))),aTP_Lamp_acl(product_prod(product_prod($o,A),product_prod($o,B)),$o))) ).

% brk_rel_def
tff(fact_6955_bin__last__set__bits,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
     => ( ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F3))
      <=> aa(nat,$o,F3,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_6956_wf__set__bits__int__Suc,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_io(fun(nat,$o),fun(nat,$o),F3))
    <=> bit_wf_set_bits_int(F3) ) ).

% wf_set_bits_int_Suc
tff(fact_6957_wf__set__bits__int__simps,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
    <=> ? [N5: nat] :
          ( ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
             => ~ aa(nat,$o,F3,N11) )
          | ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
             => aa(nat,$o,F3,N11) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_6958_zeros,axiom,
    ! [Na: nat,F3: fun(nat,$o)] :
      ( ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
         => ~ aa(nat,$o,F3,N4) )
     => bit_wf_set_bits_int(F3) ) ).

% zeros
tff(fact_6959_wf__set__bits__int_Osimps,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
    <=> ( ? [N5: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
           => ~ aa(nat,$o,F3,N11) )
        | ? [N5: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),N11)
           => aa(nat,$o,F3,N11) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_6960_wf__set__bits__int_Ocases,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
     => ( ! [N: nat] :
            ~ ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
               => ~ aa(nat,$o,F3,N3) )
       => ~ ! [N: nat] :
              ~ ! [N3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N3)
                 => aa(nat,$o,F3,N3) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_6961_ones,axiom,
    ! [Na: nat,F3: fun(nat,$o)] :
      ( ! [N4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
         => aa(nat,$o,F3,N4) )
     => bit_wf_set_bits_int(F3) ) ).

% ones
tff(fact_6962_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P2: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image(product_prod(B,B),product_prod(A,A),pairself(B,A,F3)),aa(fun(product_prod(B,B),$o),set(product_prod(B,B)),collect(product_prod(B,B)),aa(fun(B,fun(B,$o)),fun(product_prod(B,B),$o),product_case_prod(B,B,$o),P2))) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_acm(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F3),P2)) ).

% pairself_image_eq
tff(fact_6963_map__to__set__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A,V3: B] : map_to_set(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V3))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),map_to_set(A,B,M)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_acn(A,fun(product_prod(A,B),$o),K)))) ).

% map_to_set_upd
tff(fact_6964_map__to__set__empty,axiom,
    ! [B: $tType,A: $tType] : map_to_set(A,B,aTP_Lamp_cp(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% map_to_set_empty
tff(fact_6965_pairself_Oelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xa),Xaa) = Y )
     => ~ ! [A4: B,B4: B] :
            ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4) )
           => ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xa,A4)),aa(B,A,Xa,B4)) ) ) ) ).

% pairself.elims
tff(fact_6966_pairself_Osimps,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A2: B,B2: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F3),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F3,A2)),aa(B,A,F3,B2)) ).

% pairself.simps
tff(fact_6967_map__to__set__inverse,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : set_to_map(A,B,map_to_set(A,B,M)) = M ).

% map_to_set_inverse
tff(fact_6968_map__to__set__empty__iff_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( bot_bot(set(product_prod(A,B))) = map_to_set(A,B,M) )
    <=> ! [X5: A] : aa(A,option(B),M,X5) = none(B) ) ).

% map_to_set_empty_iff(2)
tff(fact_6969_map__to__set__empty__iff_I1_J,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B))] :
      ( ( map_to_set(A,B,M) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X5: A] : aa(A,option(B),M,X5) = none(B) ) ).

% map_to_set_empty_iff(1)
tff(fact_6970_map__to__set__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : map_to_set(A,B,M) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_aco(fun(A,option(B)),fun(A,fun(B,$o)),M))) ).

% map_to_set_def
tff(fact_6971_pairself_Opelims,axiom,
    ! [A: $tType,B: $tType,Xa: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xa),Xaa) = Y )
     => ( aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),Xa),Xaa))
       => ~ ! [A4: B,B4: B] :
              ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4) )
             => ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xa,A4)),aa(B,A,Xa,B4)) )
               => ~ aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),Xa),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B4))) ) ) ) ) ).

% pairself.pelims
tff(fact_6972_int__set__bits__unfold__BIT,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
     => ( bit_bi4170147762399595738t_bits(int,F3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,F3,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,F3),suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_6973_bin__rest__set__bits,axiom,
    ! [F3: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,F3)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,aa(fun(nat,nat),fun(nat,$o),comp(nat,$o,nat,F3),suc)) ) ) ).

% bin_rest_set_bits
tff(fact_6974_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,A),X4: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image(B,A,F3),X4) = top_top(set(A)) )
     => ( ! [X: B] :
            ( member(B,X,X4)
           => ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F3),X) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F3),X) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_6975_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F3)
    <=> ! [Y5: A,X5: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y5)),aa(A,fun(B,B),F3,X5)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X5)),aa(A,fun(B,B),F3,Y5)) ) ).

% comp_fun_commute_def
tff(fact_6976_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),Y: A,Xa: A] :
      ( finite6289374366891150609ommute(A,B,F3)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xa)),aa(A,fun(B,B),F3,Y)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_6977_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B))] :
      ( ! [Y3: A,X: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y3)),aa(A,fun(B,B),F3,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X)),aa(A,fun(B,B),F3,Y3))
     => finite6289374366891150609ommute(A,B,F3) ) ).

% comp_fun_commute.intro
tff(fact_6978_comp__fun__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X: A] :
          ( member(A,X,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X)),aa(A,fun(B,B),F3,X)) = aa(A,fun(B,B),F3,X) ) )
     => finite4980608107308702382axioms(A,B,S2,F3) ) ).

% comp_fun_idem_on_axioms.intro
tff(fact_6979_comp__fun__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4980608107308702382axioms(A,B,S2,F3)
    <=> ! [X5: A] :
          ( member(A,X5,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X5)),aa(A,fun(B,B),F3,X5)) = aa(A,fun(B,B),F3,X5) ) ) ) ).

% comp_fun_idem_on_axioms_def
tff(fact_6980_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( ! [X: A,Y3: A] :
          ( member(A,X,S2)
         => ( member(A,Y3,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y3)),aa(A,fun(B,B),F3,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X)),aa(A,fun(B,B),F3,Y3)) ) ) )
     => finite4664212375090638736ute_on(A,B,S2,F3) ) ).

% comp_fun_commute_on.intro
tff(fact_6981_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,Y: A,G3: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xa,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Y)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Xa)),G3)) = aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Xa)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F3,Y)),G3)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_6982_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A,Y: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( member(A,Xa,S2)
       => ( member(A,Y,S2)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y)),aa(A,fun(B,B),F3,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xa)),aa(A,fun(B,B),F3,Y)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_6983_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
    <=> ! [X5: A,Y5: A] :
          ( member(A,X5,S2)
         => ( member(A,Y5,S2)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y5)),aa(A,fun(B,B),F3,X5)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X5)),aa(A,fun(B,B),F3,Y5)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_6984_fun__comp__eq__conv,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(C,B),G3: fun(A,C),Fg: fun(A,B)] :
      ( ( aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G3) = Fg )
    <=> ! [X5: A] : aa(C,B,F3,aa(A,C,G3,X5)) = aa(A,B,Fg,X5) ) ).

% fun_comp_eq_conv
tff(fact_6985_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,Xa: fun(A,B),Y: fun(A,B),F3: fun(B,C)] :
      ( ( Xa = Y )
     => ( aa(fun(A,B),fun(A,C),comp(B,C,A,F3),Xa) = aa(fun(A,B),fun(A,C),comp(B,C,A,F3),Y) ) ) ).

% comp_cong_right
tff(fact_6986_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xa: fun(A,B),Y: fun(A,B),F3: fun(C,A)] :
      ( ( Xa = Y )
     => ( aa(fun(C,A),fun(C,B),comp(A,B,C,Xa),F3) = aa(fun(C,A),fun(C,B),comp(A,B,C,Y),F3) ) ) ).

% comp_cong_left
tff(fact_6987_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Na: nat,F3: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Na),comp(B,B,A,F3)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Na),F3)) ).

% comp_funpow
tff(fact_6988_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xa: A] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( member(A,Xa,S2)
       => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Xa)),aa(A,fun(B,B),F3,Xa)) = aa(A,fun(B,B),F3,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_6989_funpow__Suc__right,axiom,
    ! [A: $tType,Na: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Na)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3)),F3) ).

% funpow_Suc_right
tff(fact_6990_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Na: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Na)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3)) ).

% funpow.simps(2)
tff(fact_6991_funpow__add,axiom,
    ! [A: $tType,M: nat,Na: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3)) ).

% funpow_add
tff(fact_6992_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H2: fun(B,A),G3: fun(C,B),A5: set(C)] :
          ( ( aa(B,A,H2,zero_zero(B)) = zero_zero(A) )
         => ( ! [X: B,Y3: B] : aa(B,A,H2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H2,X)),aa(B,A,H2,Y3))
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),comp(B,A,C,H2),G3)),A5) = aa(B,A,H2,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),A5)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_6993_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_6994_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_6995_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6996_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_6997_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A5: set(A),H2: fun(A,B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A5)
               => ( member(A,Y3,A5)
                 => ( ( X != Y3 )
                   => ( ( aa(A,B,H2,X) = aa(A,B,H2,Y3) )
                     => ( aa(B,C,G3,aa(A,B,H2,X)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G3),aa(set(A),set(B),image(A,B,H2),A5)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G3),H2)),A5) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_6998_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A5: set(A),H2: fun(A,B),G3: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A5)
               => ( member(A,Y3,A5)
                 => ( ( X != Y3 )
                   => ( ( aa(A,B,H2,X) = aa(A,B,H2,Y3) )
                     => ( aa(B,C,G3,aa(A,B,H2,X)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G3),aa(set(A),set(B),image(A,B,H2),A5)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G3),H2)),A5) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6999_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G3: fun(C,B),F3: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G3,aa(A,C,F3,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G3),aa(set(A),set(C),image(A,C,F3),I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,G3),F3)),I5)) ) ) ) ).

% sum_image_le
tff(fact_7000_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_7001_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7002_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_7003_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G3,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_7004_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_7005_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_7006_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_su(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_7007_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_su(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7008_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_su(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7009_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G3: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aTP_Lamp_su(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_7010_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7011_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G3: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G3),set_or1337092689740270186AtMost(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G3),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_7012_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_7013_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: B,Xa: A,Sa: set(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),topolo174197925503356063within(A,Xa,Sa))
        <=> ! [X9: fun(nat,A)] :
              ( ! [I3: nat] : member(A,aa(nat,A,X9,I3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Sa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
             => ( filterlim(nat,A,X9,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
               => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),comp(A,B,nat,F3),X9),topolo7230453075368039082e_nhds(B,A2),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_7014_less__eq__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C12: $o,C23: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less_eq(char),char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))),char2((C0),(C12),(C23),(C32),(C42),(C52),(C62),(C72)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr($o,nat,aTP_Lamp_acp($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_acp($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C12)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C23)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o))))))))),zero_zero(nat))) ) ).

% less_eq_char_simp
tff(fact_7015_less__char__simp,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C12: $o,C23: $o,C32: $o,C42: $o,C52: $o,C62: $o,C72: $o] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less(char),char2((B0),(B1),(B22),(B32),(B42),(B52),(B62),(B72))),char2((C0),(C12),(C23),(C32),(C42),(C52),(C62),(C72)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),foldr($o,nat,aTP_Lamp_acp($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_acp($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C12)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C23)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o))))))))),zero_zero(nat))) ) ).

% less_char_simp
tff(fact_7016_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(A,C),G3: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_acq(C,set(B))),F3) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image(D,B,G3)),aTP_Lamp_acr(A,set(D))) ).

% empty_natural
tff(fact_7017_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(B,B)),G3: fun(C,A)] :
      ( finite6289374366891150609ommute(A,B,F3)
     => finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ).

% comp_fun_commute.comp_comp_fun_commute
tff(fact_7018_card_Ocomp__fun__commute__on,axiom,
    aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) = aa(fun(nat,nat),fun(nat,nat),comp(nat,nat,nat,suc),suc) ).

% card.comp_fun_commute_on
tff(fact_7019_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X3: list(A)] : aa(list(A),set(B),aa(fun(list(A),list(B)),fun(list(A),set(B)),comp(list(B),set(B),list(A),set2(B)),map(A,B,F3)),X3) = aa(set(A),set(B),image(A,B,F3),aa(list(A),set(A),set2(A),X3)) ).

% set_oo_map_alt
tff(fact_7020_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(set(A)),G3: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,B5)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [A15: set(A)] :
                ( member(set(A),A15,B5)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B5)
                   => ( ( A15 != A24 )
                     => ! [X: A] :
                          ( member(A,X,A15)
                         => ( member(A,X,A24)
                           => ( aa(A,B,G3,X) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G3),B5) ) ) ) ) ).

% prod.Union_comp
tff(fact_7021_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G3: fun(B,A),A5: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G3),A5) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G3),zero_zero(A),A5) ) ).

% sum.eq_fold
tff(fact_7022_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R3: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G3),top_top(set(C)))),S2)
       => finite4664212375090638736ute_on(C,B,R3,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_7023_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G3: fun(C,A),R3: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G3),top_top(set(C)))),S2)
       => finite673082921795544331dem_on(C,B,R3,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G3)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_7024_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S2: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S2)
    <=> ~ aa(set(nat),$o,finite_finite2(nat),aa(set(int),set(nat),image(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),S2)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_7025_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C4: set(set(A)),G3: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,C4)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( member(set(A),X,C4)
               => ! [Xa5: set(A)] :
                    ( member(set(A),Xa5,C4)
                   => ( ( X != Xa5 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G3),C4) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_7026_sup__SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),B5),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F3),B5,A5) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_7027_inf__INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),B5: B,F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),B5),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F3),B5,A5) ) ) ) ).

% inf_INF_fold_inf
tff(fact_7028_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(set(A)),G3: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,B5)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [A15: set(A)] :
                ( member(set(A),A15,B5)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B5)
                   => ( ( A15 != A24 )
                     => ! [X: A] :
                          ( member(A,X,A15)
                         => ( member(A,X,A24)
                           => ( aa(A,B,G3,X) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B5)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G3),B5) ) ) ) ) ).

% sum.Union_comp
tff(fact_7029_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A5)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F3),bot_bot(B),A5) ) ) ) ).

% SUP_fold_sup
tff(fact_7030_INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A5)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F3),top_top(B),A5) ) ) ) ).

% INF_fold_inf
tff(fact_7031_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C4: set(set(A)),G3: fun(A,B)] :
          ( ! [X: set(A)] :
              ( member(set(A),X,C4)
             => aa(set(A),$o,finite_finite2(A),X) )
         => ( ! [X: set(A)] :
                ( member(set(A),X,C4)
               => ! [Xa5: set(A)] :
                    ( member(set(A),Xa5,C4)
                   => ( ( X != Xa5 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),Xa5) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G3),C4) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_7032_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F3: fun(A,real),F4: filter(A)] :
      ( eventually(A,aTP_Lamp_aas(fun(A,real),fun(A,$o),F3),F4)
     => ( filterlim(A,real,F3,at_top(real),F4)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F4) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_7033_Code__Numeral_Onegative__def,axiom,
    code_negative = aa(fun(num,code_integer),fun(num,code_integer),comp(code_integer,code_integer,num,uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_7034_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G3: fun(B,A),Y6: set(B),X4: set(A),F4: filter(B),F3: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G3),Y6)),X4)
     => ( eventually(B,aTP_Lamp_acs(set(B),fun(B,$o),Y6),F4)
       => ( map_filter_on(A,C,X4,F3,map_filter_on(B,A,Y6,G3,F4)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F3),G3),F4) ) ) ) ).

% map_filter_on_comp
tff(fact_7035_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F4: filter(A)] :
          ( topolo6773858410816713723filter(A,F4)
        <=> ! [E4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E4)
             => ? [P7: fun(A,$o)] :
                  ( eventually(A,P7,F4)
                  & ! [X5: A,Y5: A] :
                      ( ( aa(A,$o,P7,X5)
                        & aa(A,$o,P7,Y5) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X5,Y5)),E4) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_7036_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_act(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_7037_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(C,B),Xa: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xa),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),aa(C,B,F3,Y)) ).

% apsnd_conv
tff(fact_7038_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F3: fun(C,B),G3: fun(D,C),Xa: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G3),Xa)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),comp(C,B,D,F3),G3)),Xa) ).

% apsnd_compose
tff(fact_7039_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_7040_nhds__imp__cauchy__filter,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [F4: filter(A),Xa: A] :
          ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),topolo7230453075368039082e_nhds(A,Xa))
         => topolo6773858410816713723filter(A,F4) ) ) ).

% nhds_imp_cauchy_filter
tff(fact_7041_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_7042_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_acu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_acv(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_7043_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))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),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_7044_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_acw(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_7045_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Na: nat,Xa: word(A),A2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
         => ( ( Xa = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),Xa)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
               => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_7046_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7047_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_7048_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_7049_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_numeral_numeral
tff(fact_7050_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: num,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(num,word(A),numeral_numeral(word(A)),W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W2)),Na) ) ) ) ).

% bit_numeral_word_iff
tff(fact_7051_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_lt2p
tff(fact_7052_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_bounded
tff(fact_7053_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_lt2p
tff(fact_7054_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% exp_eq_zero_iff
tff(fact_7055_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_Suc_numeral
tff(fact_7056_scast__sbintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W2: num] : aa(word(B),word(A),ring_1_signed(B,word(A)),aa(num,word(B),numeral_numeral(word(B)),W2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(num,int,numeral_numeral(int),W2))) ) ).

% scast_sbintr
tff(fact_7057_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_7058_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_7059_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_7060_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_7061_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_7062_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_7063_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: num,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W2))),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W2))),Na) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_7064_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ) ) ) ).

% unat_power_lower
tff(fact_7065_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_Suc_minus_numeral
tff(fact_7066_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% word_less_sub_le
tff(fact_7067_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2))),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))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_numeral_minus_1
tff(fact_7068_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),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))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_minus_numeral_minus_1
tff(fact_7069_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),M)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_iff
tff(fact_7070_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M) )
            <=> ( Na = M ) ) ) ) ) ).

% two_power_eq
tff(fact_7071_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Na: nat,W2: word(B)] :
          aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se2584673776208193580ke_bit(word(B)),Na),W2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B))),aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),aa(word(B),A,ring_1_signed(B,A),W2)),aa(word(B),A,ring_1_signed(B,A),W2)) ) ).

% signed_take_bit_eq
tff(fact_7072_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_iff
tff(fact_7073_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_7074_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = Xa ) ) ) ).

% ucast_ucast_id
tff(fact_7075_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( Na = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_7076_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),N)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),B2),N) ) )
         => ( A2 = B2 ) ) ) ).

% bit_word_eqI
tff(fact_7077_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_7078_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),M: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) )
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),M) ) ) ).

% test_bit_conj_lt
tff(fact_7079_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( Xa = Y )
        <=> ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),N5)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N5) ) ) ) ) ).

% word_eq_iff
tff(fact_7080_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Xa)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xa),Na) ) ) ) ).

% test_bit_wi
tff(fact_7081_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Na) ) ) ) ).

% bit_word_of_int_iff
tff(fact_7082_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_7083_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na) ) ) ) ).

% bit_uint_iff
tff(fact_7084_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),Na) ) ) ) ).

% bit_ucast_iff
tff(fact_7085_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_7086_bit__word__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B)))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),Na) ) ) ) ).

% bit_word_ucast_iff
tff(fact_7087_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),Na) ) ) ) ).

% test_bit_bin
tff(fact_7088_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W2)),Na)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_7089_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W2: word(B),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,Na,W2)),M)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_7090_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),M)),aa(nat,nat,aa(nat,fun(nat,nat),bit_se2584673776208193580ke_bit(nat),type_len0_len_of(A,type2(A))),Na)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_7091_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B),C3: word(A),D2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( word_cat(A,B,C,A2,B2) = word_cat(A,B,C,C3,D2) )
          <=> ( ( A2 = C3 )
              & ( B2 = D2 ) ) ) ) ) ).

% word_cat_inj
tff(fact_7092_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(A),Y: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_7093_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: int] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),W2)) = aa(int,word(A),ring_1_of_int(word(A)),W2) ) ) ) ).

% wi_bintr
tff(fact_7094_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),K)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_eq_iff
tff(fact_7095_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),Na),W2) = W2 ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_7096_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),ring_1_signed(A,word(B)),Xa) = aa(word(A),word(B),ring_1_signed(A,word(B)),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_7097_up__scast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),ring_1_signed(B,word(A)),Xa) = aa(word(B),word(A),ring_1_signed(B,word(A)),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(B),nat,size_size(word(B)),Xa)),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_scast_inj
tff(fact_7098_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Xa),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_7099_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_7100_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_7101_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),Na),aa(word(A),int,semiring_1_unsigned(A,int),W2)) = aa(word(A),int,semiring_1_unsigned(A,int),W2) ) ) ) ).

% bintr_uint
tff(fact_7102_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) ) ) ) ).

% unat_ucast_up_simp
tff(fact_7103_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( Xa = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
           => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_7104_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_7105_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( Xa = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_7106_ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(C)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( Xa = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_7107_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_7108_up__ucast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( Xa = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_7109_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B2: word(B),A2: word(A),D2: word(B),C3: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B2),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = aa(int,word(C),ring_1_of_int(word(C)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D2),aa(word(A),int,semiring_1_unsigned(A,int),C3))) )
          <=> ( ( B2 = D2 )
              & ( A2 = C3 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_7110_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(nat,$o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),P2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,P2,Na) ) ) ) ).

% bit_set_bits_word_iff
tff(fact_7111_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,$o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),F3)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F3,Na) ) ) ) ).

% word_test_bit_set_bits
tff(fact_7112_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W2)),Na)
          <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ) ).

% bin_nth_sint
tff(fact_7113_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xa)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),minus_minus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_7114_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [V3: word(B),W2: word(C),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,V3,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(C,type2(C))),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),W2),Na),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),V3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_7115_signed__scast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W2: word(C)] : aa(word(B),A,ring_1_signed(B,A),aa(word(C),word(B),ring_1_signed(C,word(B)),W2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(C),A,ring_1_signed(C,A),W2)) ) ).

% signed_scast_eq
tff(fact_7116_signed__take__bit__decr__length__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [K: A,L: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),K) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),L) )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),type_len0_len_of(B,type2(B))),K) = aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),type_len0_len_of(B,type2(B))),L) ) ) ) ).

% signed_take_bit_decr_length_iff
tff(fact_7117_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),V3: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W2 = V3 ) ) ) ).

% size_0_same
tff(fact_7118_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W2: word(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(word(B),A,semiring_1_unsigned(B,A),W2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% unsigned_less
tff(fact_7119_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),type_len0_len_of(A,type2(A)))
       => ( aa(num,word(A),numeral_numeral(word(A)),bit0(one2)) != zero_zero(word(A)) ) ) ) ).

% not_degenerate_imp_2_neq_0
tff(fact_7120_bit__word__scast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),ring_1_signed(B,word(A)),W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),Na)
              | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),Na)
                & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ) ).

% bit_word_scast_iff
tff(fact_7121_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( Xa = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_7122_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xa) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_7123_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_7124_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: nat] :
              ( ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),N) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nat_cases
tff(fact_7125_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) != zero_zero(word(A)) ) ) ) ).

% More_Word.power_not_zero
tff(fact_7126_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_7127_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_7128_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_7129_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),M)
        <=> ( ( M = Na )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_7130_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),Na),W2) = W2 )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% take_bit_word_eq_self_iff
tff(fact_7131_sint__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(word(A),int,ring_1_signed(A,int),W2) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(A),int,semiring_1_unsigned(A,int),W2)) ) ).

% sint_uint
tff(fact_7132_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)) = Xa ) ) ) ).

% ucast_ucast_len
tff(fact_7133_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono
tff(fact_7134_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W2)),Na)
        <=> $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% nth_sint
tff(fact_7135_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,ring_1_signed(A,int),W2)),Na)
        <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))) )
            | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na) ) ) ) ).

% bit_sint_iff
tff(fact_7136_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% less_Suc_unat_less_bound
tff(fact_7137_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_7138_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% lt2p_lem
tff(fact_7139_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_le_mono
tff(fact_7140_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% two_power_increasing
tff(fact_7141_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_7142_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),Xa)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_7143_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = Xa ) ) ) ).

% unat_of_nat_len
tff(fact_7144_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xa) = Na )
          <=> ( Xa = aa(nat,word(A),semiring_1_of_nat(word(A)),Na) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_7145_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P2,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ! [N5: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N5) = Xa )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(nat,$o,P2,N5) ) ) ) ).

% unat_split
tff(fact_7146_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: nat,A2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R2) = A2 )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),A2) = R2 ) ) ) ) ).

% of_nat_inverse
tff(fact_7147_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(nat,$o),Xa: word(A)] :
          ( aa(nat,$o,P2,aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))
        <=> ~ ? [N5: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N5) = Xa )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(nat,$o,P2,N5) ) ) ) ).

% unat_split_asm
tff(fact_7148_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( Xa = zero_zero(word(A)) )
              | ( Xa = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_7149_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na))),M)
        <=> ( ( M = Na )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_7150_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% word_1_le_power
tff(fact_7151_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = aa(nat,word(B),semiring_1_of_nat(word(B)),Xa) ) ) ) ).

% ucast_of_nat_small
tff(fact_7152_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(B)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(B),int,semiring_1_unsigned(B,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_sub_lt2p
tff(fact_7153_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_7154_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W2)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_7155_scast__1_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(word(B),word(A),ring_1_signed(B,word(A)),one_one(word(B))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),one_one(int))) ) ) ).

% scast_1'
tff(fact_7156_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ucast_less
tff(fact_7157_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: int] : aa(word(B),A,ring_1_signed(B,A),aa(int,word(B),ring_1_of_int(word(B)),Na)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),Na)) ) ).

% signed_of_int
tff(fact_7158_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_7159_signed__ucast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W2: word(C)] : aa(word(B),A,ring_1_signed(B,A),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(C),A,semiring_1_unsigned(C,A),W2)) ) ).

% signed_ucast_eq
tff(fact_7160_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( ( Xa != aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A)))) ) ) ).

% upper_trivial
tff(fact_7161_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Q3: word(A),M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),Q3)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Q3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Q3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ) ).

% word_power_less_diff
tff(fact_7162_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_7163_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_7164_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_7165_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Na) = zero_zero(word(A)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( Na = zero_zero(nat) ) ) ) ) ).

% More_Word.of_nat_0
tff(fact_7166_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,word(A),semiring_1_of_nat(word(A)),K) != zero_zero(word(A)) ) ) ) ) ).

% Word.of_nat_neq_0
tff(fact_7167_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xa)
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_7168_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(4)
tff(fact_7169_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_range'
tff(fact_7170_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_lt
tff(fact_7171_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xa)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_7172_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ~ ! [N: int] :
              ( ( Xa = aa(int,word(A),ring_1_of_int(word(A)),N) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N)
               => ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_int_cases
tff(fact_7173_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Y: int] :
          ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
           => ( ( aa(int,word(A),ring_1_of_int(word(A)),Xa) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( Xa = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_7174_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_7175_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B2: word(A),F3: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),F3)),B2)
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),F3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_7176_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_m2p_not_non_neg
tff(fact_7177_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),zero_zero(int)) ) ).

% uint_m2p_neg
tff(fact_7178_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F3: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F3)),Na)
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),F3),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_7179_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,F3: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),F3)),Na)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),F3),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_7180_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na) ) ) ) ).

% uint_power_lower
tff(fact_7181_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P3),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_7182_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% nth_bounded
tff(fact_7183_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,P3: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N11)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P3),N11) ) )
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),P3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% upper_bits_unset_is_l2p
tff(fact_7184_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_7185_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_le
tff(fact_7186_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% word_2p_mult_inc
tff(fact_7187_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Na)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Na),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_less
tff(fact_7188_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ).

% word_power_less_1
tff(fact_7189_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_7190_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A2 = zero_zero(word(A)) )
             => ( aa(word(A),int,ring_1_signed(A,int),A2) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A2 = one_one(word(A)) )
               => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),type_len0_len_of(A,type2(A)))
               => ( aa(word(A),int,ring_1_signed(A,int),one_one(word(A))) != one_one(int) ) ) ) ) ) ).

% sint_1_cases
tff(fact_7191_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P2) = groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),aa(list(nat),list($o),map(nat,$o,P2),upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_7192_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_7193_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Na: nat] : aa(word(B),A,ring_1_signed(B,A),aa(nat,word(B),semiring_1_of_nat(word(B)),Na)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(nat,int,semiring_1_of_nat(int),Na))) ) ).

% signed_of_nat
tff(fact_7194_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_7195_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = aa(fun(word(B),$o),set(word(B)),collect(word(B)),aTP_Lamp_acx(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_7196_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% unat_plus_if'
tff(fact_7197_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),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))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(5)
tff(fact_7198_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if'
tff(fact_7199_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add_nat
tff(fact_7200_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_7201_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_7202_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_7203_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_7204_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sint_less
tff(fact_7205_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_7206_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_7207_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_less_mono1
tff(fact_7208_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),aa(word(A),int,ring_1_signed(A,int),Xa)) ) ).

% sint_ge
tff(fact_7209_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),K),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% unat_less_power
tff(fact_7210_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P2,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ! [I3: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(int,$o,P2,I3) ) ) ) ).

% uint_split
tff(fact_7211_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(int,$o),Xa: word(A)] :
          ( aa(int,$o,P2,aa(word(A),int,semiring_1_unsigned(A,int),Xa))
        <=> ~ ? [I3: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I3) = Xa )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(int,$o,P2,I3) ) ) ) ).

% uint_split_asm
tff(fact_7212_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: int,A2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R2) = A2 )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),int,semiring_1_unsigned(A,int),A2) = R2 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_7213_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_7214_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt'
tff(fact_7215_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt''
tff(fact_7216_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A2) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(word(A)) )
            | ( A2 = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ).

% double_eq_zero_iff
tff(fact_7217_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Y))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add
tff(fact_7218_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add'
tff(fact_7219_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y)
         => ? [Z3: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xa),Z3) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xa)),aa(word(A),int,semiring_1_unsigned(A,int),Z3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_le_exists'
tff(fact_7220_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: nat,Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),P3)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_7221_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_plus_if'
tff(fact_7222_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans
tff(fact_7223_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Na),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans2
tff(fact_7224_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_sub_if'
tff(fact_7225_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_two_pow_divI
tff(fact_7226_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ( ( Xa != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) != zero_zero(word(A)) ) ) ) ) ) ).

% word_power_nonzero
tff(fact_7227_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint'
tff(fact_7228_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xa))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xa))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint''
tff(fact_7229_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),dvd_dvd(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)))
        <=> ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$o,aa(word(A),fun(word(A),$o),dvd_dvd(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_7230_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Na) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),Na) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_7231_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
         => ( ! [X: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X) = zero_zero(A) )
               => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),C3),X) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ).

% CHAR_eqI
tff(fact_7232_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_word_eq_word_of_nat
tff(fact_7233_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),aa(word(A),int,ring_1_signed(A,int),W2)) ) ).

% sint_greater_eq
tff(fact_7234_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),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),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa)) = aa(nat,int,semiring_1_of_nat(int),Xa) ) ) ) ).

% int_eq_sint
tff(fact_7235_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_less_mono1
tff(fact_7236_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_7237_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ! [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),type_len0_len_of(A,type2(A)))
               => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P3),N3) ) ) ) ) ).

% le2p_bits_unset
tff(fact_7238_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P3: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N3)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P3),N3) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_7239_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Na: nat,Xa: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),Na)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
                 => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),Y)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ) ) ) ) ).

% word_add_offset_less
tff(fact_7240_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ws: list(word(A)),Na: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_7241_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xa)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y),Xa))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_7242_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: nat,A2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),A2))),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),B2))) ) ) ) ).

% sint_of_nat_le
tff(fact_7243_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),semiring_1_of_nat(word(A)),Xa))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_7244_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_max_plus_1
tff(fact_7245_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
           => ( aa(word(A),int,ring_1_signed(A,int),aa(int,word(A),ring_1_of_int(word(A)),Xa)) = Xa ) ) ) ) ).

% sint_of_int_eq
tff(fact_7246_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_7247_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),K)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_7248_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
            & ( aa(nat,A,semiring_1_of_nat(A),N5) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7249_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
           => ( ! [X: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),C3)
                   => ( aa(nat,A,semiring_1_of_nat(A),X) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7250_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N5: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N5)
           => ( aa(nat,A,semiring_1_of_nat(A),N5) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7251_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),Y)),Y) = Xa ) ) ) ) ).

% Word.word_div_mult
tff(fact_7252_sint__int__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,ring_1_signed(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_min
tff(fact_7253_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),K)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans_ofnat
tff(fact_7254_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: fun(word(A),$o),A2: word(A)] :
          ( aa(word(A),$o,P2,zero_zero(word(A)))
         => ( ! [A4: word(A)] :
                ( aa(word(A),$o,P2,A4)
               => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),A4)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P2,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4)) ) ) )
           => ( ! [A4: word(A)] :
                  ( aa(word(A),$o,P2,A4)
                 => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P2,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4))) ) )
             => aa(word(A),$o,P2,A2) ) ) ) ) ).

% word_bit_induct
tff(fact_7255_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),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),type_len0_len_of(A,type2(A))),Sz)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),aa(nat,word(A),semiring_1_of_nat(word(A)),K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)),K) ) ) ) ).

% unat_mult_power_lem
tff(fact_7256_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: $o] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa($o,word(A),zero_neq_one_of_bool(word(A)),(B2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A2 ) ) ) ).

% bit_word_half_eq
tff(fact_7257_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( aa(fun(word(A),$o),set(word(A)),collect(word(A)),aa(nat,fun(word(A),$o),aTP_Lamp_acy(nat,fun(nat,fun(word(A),$o)),Na),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_acz(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_7258_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Us2),Sz)
           => ( 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),Sz),Us2)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us2)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_7259_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),dvd_dvd(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),K)) ).

% bit_cut_integer_def
tff(fact_7260_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_7261_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na) ) ) ).

% decr_length_less_iff
tff(fact_7262_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),type_len0_len_of(A,type2(A))) ) ).

% len_gt_0
tff(fact_7263_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% length_not_greater_eq_2_iff
tff(fact_7264_len__num0,axiom,
    ! [Uu2: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu2) = zero_zero(nat) ).

% len_num0
tff(fact_7265_len__not__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( type_len0_len_of(A,type2(A)) != zero_zero(nat) ) ) ).

% len_not_eq_0
tff(fact_7266_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% two_less_eq_exp_length
tff(fact_7267_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P2: fun(A,$o),F3: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P2,word_int_case(A,B,F3,Xa))
        <=> ~ ? [N5: int] :
                ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),N5) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N5)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),N5),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P2,aa(int,A,F3,N5)) ) ) ) ).

% word_int_split_asm
tff(fact_7268_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P2: fun(A,$o),F3: fun(int,A),Xa: word(B)] :
          ( aa(A,$o,P2,word_int_case(A,B,F3,Xa))
        <=> ! [I3: int] :
              ( ( ( Xa = aa(int,word(B),ring_1_of_int(word(B)),I3) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))) )
             => aa(A,$o,P2,aa(int,A,F3,I3)) ) ) ) ).

% word_int_split
tff(fact_7269_word__reverse__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : word_reverse(A,W2) = groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),rev($o,aa(list(nat),list($o),map(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2)),upt(zero_zero(nat),type_len0_len_of(A,type2(A)))))) ) ).

% word_reverse_def
tff(fact_7270_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2))) ) ).

% smod_word_min
tff(fact_7271_bit__word__reverse__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_reverse(A,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))) ) ) ) ).

% bit_word_reverse_iff
tff(fact_7272_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% smod_word_max
tff(fact_7273_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),B2)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(3)
tff(fact_7274_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),aa(int,int,uminus_uminus(int),B2)) ) ) ).

% smod_int_compares(5)
tff(fact_7275_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_7276_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(2)
tff(fact_7277_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_7278_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(6)
tff(fact_7279_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_7280_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2)) ) ) ).

% smod_int_compares(8)
tff(fact_7281_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),A2),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) ) ) ) ).

% smod_mod_positive
tff(fact_7282_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs2: list($o),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(list($o),nat,size_size(list($o)),Bs2)))
            & aa(nat,$o,nth($o,Bs2),Na) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_7283_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_roti(A,K,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Na)),K)),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_word_roti_iff
tff(fact_7284_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = M )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),Na) ) ) ).

% min_eq_arg(1)
tff(fact_7285_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = Na )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),M) ) ) ).

% min_eq_arg(2)
tff(fact_7286_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = Na ) ) ) ).

% min_arg_le(1)
tff(fact_7287_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na))
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = M ) ) ) ).

% min_arg_le(2)
tff(fact_7288_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% min.bounded_iff
tff(fact_7289_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_7290_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_7291_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_7292_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_7293_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_7294_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_7295_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min_simps(1)
tff(fact_7296_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2) ) ) ).

% min_less_self_conv(2)
tff(fact_7297_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),A2) ) ) ).

% min_less_self_conv(1)
tff(fact_7298_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na)),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = Na ) ) ) ).

% min_arg_not_ge(2)
tff(fact_7299_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na)),M)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_7300_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_7301_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% min_bot
tff(fact_7302_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)) ).

% min_Suc_Suc
tff(fact_7303_min__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7304_min__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% min_0L
tff(fact_7305_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_7306_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_7307_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V3)) ) ).

% min_number_of(1)
tff(fact_7308_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_7309_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_7310_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),B2),aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_7311_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A2),B2)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,A2)),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_7312_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) ) ).

% min_number_of(4)
tff(fact_7313_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V3)) ) ).

% min_number_of(3)
tff(fact_7314_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V3: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V3))) ) ).

% min_number_of(2)
tff(fact_7315_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I: nat,Xa: A,J: nat,Y: B] : zip(A,B,replicate(A,I,Xa),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I),J),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)) ).

% zip_replicate
tff(fact_7316_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ).

% Min_insert
tff(fact_7317_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_Suc_numeral
tff(fact_7318_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,Na))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_7319_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_7320_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_7321_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_7322_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ).

% nat_mult_min_left
tff(fact_7323_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_min_right
tff(fact_7324_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: 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),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_7325_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% min_add_distrib_right
tff(fact_7326_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ).

% Min.in_idem
tff(fact_7327_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_7328_min__diff,axiom,
    ! [M: nat,I: nat,Na: 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),Na),I)) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),I) ).

% min_diff
tff(fact_7329_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Y: 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),Xa),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),minus_minus(A),Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_7330_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2),X3,Xa2) ) ).

% min_def_raw
tff(fact_7331_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D2)) ) ) ) ).

% min.mono
tff(fact_7332_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_7333_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_7334_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3) ) ) ) ).

% min.boundedE
tff(fact_7335_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3)) ) ) ) ).

% min.boundedI
tff(fact_7336_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.order_iff
tff(fact_7337_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2) ) ).

% min.cobounded1
tff(fact_7338_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2) ) ).

% min.cobounded2
tff(fact_7339_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_7340_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_7341_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI1
tff(fact_7342_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.coboundedI2
tff(fact_7343_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_7344_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7345_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y) = Xa ) ) ) ).

% min_absorb1
tff(fact_7346_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_7347_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C3: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI2
tff(fact_7348_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C3: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C3) ) ) ).

% min.strict_coboundedI1
tff(fact_7349_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
        <=> ( ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_7350_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A2),C3) ) ) ) ).

% min.strict_boundedE
tff(fact_7351_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_7352_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% max_mult_distrib_left
tff(fact_7353_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Y))) ) ).

% min_mult_distrib_left
tff(fact_7354_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% max_mult_distrib_right
tff(fact_7355_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P3))) ) ).

% min_mult_distrib_right
tff(fact_7356_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3))) ) ).

% max_divide_distrib_right
tff(fact_7357_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Y: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Y)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P3))) ) ).

% min_divide_distrib_right
tff(fact_7358_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),S2)) = $ite(S2 = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,complete_Inf_Inf(A),S2))) ) ) ) ).

% Inf_insert_finite
tff(fact_7359_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H2: fun(A,A),N6: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H2,X)),aa(A,A,H2,Y3))
         => ( aa(set(A),$o,finite_finite2(A),N6)
           => ( ( N6 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic643756798350308766er_Min(A),N6)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image(A,A,H2),N6)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_7360_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B5)),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) = aa(set(A),A,lattic643756798350308766er_Min(A),A5) ) ) ) ) ) ).

% Min.subset
tff(fact_7361_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A5)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_7362_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A5),A5) ) ) ) ) ).

% Min.closed
tff(fact_7363_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A5)),aa(set(A),A,lattic643756798350308766er_Min(A),B5)) ) ) ) ) ) ) ).

% Min.union
tff(fact_7364_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,A,ord_min(A),Xa,A5) ) ) ) ).

% Min.eq_fold
tff(fact_7365_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_7366_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_7367_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_ada(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Min.eq_fold'
tff(fact_7368_bit__slice__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,M,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),type_len0_len_of(B,type2(B)))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M))) ) ) ) ).

% bit_slice_iff
tff(fact_7369_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_adb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys)) ).

% set_zip
tff(fact_7370_bit__slice1__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice1(B,A,M,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),M))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),M))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_slice1_iff
tff(fact_7371_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Na),W2) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))) ) ) ) ).

% signed_drop_bit_beyond
tff(fact_7372_int__of__integer__min,axiom,
    ! [K: code_integer,L: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),ord_min(code_integer),K),L)) = aa(int,int,aa(int,fun(int,int),ord_min(int),aa(code_integer,int,code_int_of_integer,K)),aa(code_integer,int,code_int_of_integer,L)) ).

% int_of_integer_min
tff(fact_7373_signed__drop__bit__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),zero_zero(nat)),W2) = W2 ) ).

% signed_drop_bit_0
tff(fact_7374_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Na),one_one(word(A))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( Na = zero_zero(nat) ) )) ) ).

% signed_drop_bit_of_1
tff(fact_7375_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),M),W2)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_7376_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_7377_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_7378_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_7379_signed_Olength__insort,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [F3: fun(A,word(B)),Xa: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),insort_key(word(B),A,word_sle(B),F3,Xa,Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% signed.length_insort
tff(fact_7380_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,B2: $o] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa($o,A,zero_neq_one_of_bool(A),(B2))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (B2) )) ) ).

% drop_bit_of_bool
tff(fact_7381_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_7382_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_7383_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_7384_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),bit1(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_7385_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_7386_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_sle_no
tff(fact_7387_signed_OMax__const,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [A5: set(A),C3: word(B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(word(B)),word(B),lattices_Max(word(B),word_sle(B)),aa(set(A),set(word(B)),image(A,word(B),aTP_Lamp_adc(word(B),fun(A,word(B)),C3)),A5)) = C3 ) ) ) ) ).

% signed.Max_const
tff(fact_7388_signed_OMin__const,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [A5: set(A),C3: word(B)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(word(B)),word(B),lattices_Min(word(B),word_sle(B)),aa(set(A),set(word(B)),image(A,word(B),aTP_Lamp_adc(word(B),fun(A,word(B)),C3)),A5)) = C3 ) ) ) ) ).

% signed.Min_const
tff(fact_7389_drop__bit__Suc__minus__bit0,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_7390_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_7391_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Na)),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_7392_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_7393_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),Na)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_7394_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_7395_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_7396_drop__bit__Suc__minus__bit1,axiom,
    ! [Na: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit1(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Na),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_7397_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_7398_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_7399_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_7400_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_Suc_numeral
tff(fact_7401_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,word(A)),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F3,aa(nat,nat,suc,N))),aa(nat,word(A),F3,N))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F3,N2)),aa(nat,word(A),F3,Na)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_7402_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,word(A)),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F3,N)),aa(nat,word(A),F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),F3,Na)),aa(nat,word(A),F3,N2)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_7403_signed_OMax__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M7: set(word(A)),N6: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),M7),N6)
         => ( ( M7 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),N6)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Max(word(A),word_sle(A)),M7)),aa(set(word(A)),word(A),lattices_Max(word(A),word_sle(A)),N6)) ) ) ) ) ).

% signed.Max_mono
tff(fact_7404_signed_OMin__antimono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M7: set(word(A)),N6: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),M7),N6)
         => ( ( M7 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),N6)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),N6)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),M7)) ) ) ) ) ).

% signed.Min_antimono
tff(fact_7405_signed_OMax_Osubset__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A5: set(word(A)),B5: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),A5),B5)
         => ( ( A5 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),B5)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Max(word(A),word_sle(A)),A5)),aa(set(word(A)),word(A),lattices_Max(word(A),word_sle(A)),B5)) ) ) ) ) ).

% signed.Max.subset_imp
tff(fact_7406_signed_OMin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A5: set(word(A)),B5: set(word(A))] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),A5),B5)
         => ( ( A5 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,finite_finite2(word(A)),B5)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),B5)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A5)) ) ) ) ) ).

% signed.Min.subset_imp
tff(fact_7407_signed_OMin_Osubset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A5: set(word(A)),B5: set(word(A))] :
          ( aa(set(word(A)),$o,finite_finite2(word(A)),A5)
         => ( ( B5 != bot_bot(set(word(A))) )
           => ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),B5),A5)
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),min(word(A),word_sle(A)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),B5)),aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A5)) = aa(set(word(A)),word(A),lattices_Min(word(A),word_sle(A)),A5) ) ) ) ) ) ).

% signed.Min.subset
tff(fact_7408_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2)) ) ) ).

% word_sle_eq
tff(fact_7409_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = A2 )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_7410_signed_Ofinite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [S2: set(A),P2: fun(set(A),$o),F3: fun(A,word(B))] :
          ( aa(set(A),$o,finite_finite2(A),S2)
         => ( aa(set(A),$o,P2,bot_bot(set(A)))
           => ( ! [X: A,S4: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S4)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S4)
                       => aa(word(B),$o,aa(word(B),fun(word(B),$o),word_sle(B),aa(A,word(B),F3,Y4)),aa(A,word(B),F3,X)) )
                   => ( aa(set(A),$o,P2,S4)
                     => aa(set(A),$o,P2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),S4)) ) ) )
             => aa(set(A),$o,P2,S2) ) ) ) ) ).

% signed.finite_ranking_induct
tff(fact_7411_signed_Osorted01,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(word(A)),nat,size_size(list(word(A))),Xs)),one_one(nat))
         => sorted_wrt(word(A),word_sle(A),Xs) ) ) ).

% signed.sorted01
tff(fact_7412_signed_Osorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono_less
tff(fact_7413_signed_Osorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),aa(nat,nat,suc,I3))) ) ) ) ).

% signed.sorted_iff_nth_Suc
tff(fact_7414_signed_Osorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),rev(word(A),Xs))
        <=> ! [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),aa(nat,nat,suc,I3))),aa(nat,word(A),nth(word(A),Xs),I3)) ) ) ) ).

% signed.sorted_rev_iff_nth_Suc
tff(fact_7415_signed_Osorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),I: nat,J: nat] :
          ( sorted_wrt(word(A),word_sle(A),rev(word(A),Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),J)),aa(nat,word(A),nth(word(A),Xs),I)) ) ) ) ) ).

% signed.sorted_rev_nth_mono
tff(fact_7416_signed_Osorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),rev(word(A),Xs))
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),J3)),aa(nat,word(A),nth(word(A),Xs),I3)) ) ) ) ) ).

% signed.sorted_rev_iff_nth_mono
tff(fact_7417_signed_Osorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A))] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I3)),aa(nat,word(A),nth(word(A),Xs),J3)) ) ) ) ) ).

% signed.sorted_iff_nth_mono
tff(fact_7418_signed_Osorted__nth__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(word(A)),I: nat,J: nat] :
          ( sorted_wrt(word(A),word_sle(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(word(A)),nat,size_size(list(word(A))),Xs))
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),aa(nat,word(A),nth(word(A),Xs),I)),aa(nat,word(A),nth(word(A),Xs),J)) ) ) ) ) ).

% signed.sorted_nth_mono
tff(fact_7419_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_7420_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Na),A2) = $ite(Na = zero_zero(nat),A2,aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_7421_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),zero_zero(word(A))),Xa) ) ) ).

% word_0_sle_from_less
tff(fact_7422_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,A2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(3)
tff(fact_7423_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_7424_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Na = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_7425_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),W2) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_7426_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Na)),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(12)
tff(fact_7427_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(10)
tff(fact_7428_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),Na)),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(11)
tff(fact_7429_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),aa(num,word(A),numeral_numeral(word(A)),Na))) ) ) ).

% extra_sle_sless_unfolds(8)
tff(fact_7430_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),one_one(word(A))),zero_zero(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(9)
tff(fact_7431_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),zero_zero(word(A))),one_one(word(A)))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),zero_zero(word(A)))),aa(word(A),int,ring_1_signed(A,int),one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(7)
tff(fact_7432_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_Suc_numeral
tff(fact_7433_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% sless_word_numeral_numeral
tff(fact_7434_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,zero_zero(nat))),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,zero_zero(nat))),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_numeral_bit0_1
tff(fact_7435_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_Suc_minus_numeral
tff(fact_7436_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_7437_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A2))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_7438_drop__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa),Xb)) ).

% drop_bit_integer.abs_eq
tff(fact_7439_drop__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% drop_bit_integer.rep_eq
tff(fact_7440_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),A2)),aa(word(A),int,ring_1_signed(A,int),B2)) ) ) ).

% word_sless_alt
tff(fact_7441_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,word(A)),Na: nat,N2: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F3,N)),aa(nat,word(A),F3,aa(nat,nat,suc,N)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F3,Na)),aa(nat,word(A),F3,N2)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_7442_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,word(A)),Na: nat,M: nat] :
          ( ! [N: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F3,N)),aa(nat,word(A),F3,aa(nat,nat,suc,N)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),aa(nat,word(A),F3,Na)),aa(nat,word(A),F3,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_7443_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat] : word_succ(A,aa(nat,word(A),semiring_1_of_nat(word(A)),A2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,A2)) ) ).

% Abs_fnat_hom_Suc
tff(fact_7444_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Na),W2)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),Na),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W2)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_7445_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,ring_1_signed(A,int),Xa)),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(word(A),int,ring_1_signed(A,int),Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_7446_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))) ) ).

% word_arith_nat_Suc
tff(fact_7447_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),W2) = zero_zero(word(A)) )
        <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ).

% drop_bit_eq_zero_iff_not_bit_last
tff(fact_7448_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
                & bit_se6407376104438227557le_bit(A,type2(A),Na) ))),
            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),Na),M))) ) ).

% drop_bit_exp_eq
tff(fact_7449_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% bit_minus_2_iff
tff(fact_7450_possible__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( bit_se6407376104438227557le_bit(word(A),type2(word(A)),M)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ).

% possible_bit_word
tff(fact_7451_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_7452_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7453_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Na: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Na)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7454_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% fold_possible_bit
tff(fact_7455_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7456_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7457_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),Na)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)))
            & ( Na != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ).

% bit_double_iff
tff(fact_7458_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(word(B),A,ring_1_signed(B,A),W2)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),Na)) ) ) ) ).

% bit_signed_iff
tff(fact_7459_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = $let(
                q: word(A),
                q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xa)),Y)),
                $let(
                  r2: word(A),
                  r2:= aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),r2),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),r2),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r2)) ) ) ) ) ) ).

% div_half_word
tff(fact_7460_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => ( 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),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_7461_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_7462_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_7463_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,bit_se2239418461657761734s_mask(A),Na) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_7464_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( aa(nat,A,bit_se2239418461657761734s_mask(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_7465_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_7466_Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% Word.mask_Suc_0
tff(fact_7467_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_7468_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Na),W2) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_7469_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_7470_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),A2))
        <=> ( ( Na != zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_7471_less__eq__mask__iff__take__bit__eq__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
        <=> ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se2584673776208193580ke_bit(word(A)),Na),W2) = W2 ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
tff(fact_7472_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,suc,zero_zero(nat))) = one_one(word(A)) ) ) ).

% More_Word.mask_Suc_0
tff(fact_7473_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,I: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),I)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(word(A),nat,size_size(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) ) ) ) ).

% nth_mask
tff(fact_7474_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Na: nat,W2: word(B)] : aa(word(B),A,ring_1_signed(B,A),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4730199178511100633sh_bit(word(B)),Na),W2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),aa(word(B),A,ring_1_signed(B,A),W2))) ) ).

% signed_push_bit_eq
tff(fact_7475_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
        <=> ! [X5: nat] :
              ( member(nat,X5,set_or7035219750837199246ssThan(nat,Na,aa(word(A),nat,size_size(word(A)),W2)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),X5) ) ) ) ).

% le_mask_high_bits
tff(fact_7476_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_7477_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,bit_se2239418461657761734s_mask(A),M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_iff
tff(fact_7478_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)),Na)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),type_len0_len_of(A,type2(A))),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_7479_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7480_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_7481_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),Na),A2) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_add(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_7482_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,Na)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)),aa(nat,A,bit_se2239418461657761734s_mask(A),Na)) ) ).

% mask_Suc_exp
tff(fact_7483_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] : aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_7484_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,W2: word(B)] :
          slice1(B,A,Na,W2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(B,type2(B))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),Na)),W2)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),type_len0_len_of(B,type2(B)))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W2))) ) ).

% slice1_def
tff(fact_7485_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A2) = $let(
            l: A,
            l:= aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),aa(nat,nat,suc,Na)),A2),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_7486_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% mask_lt_2pn
tff(fact_7487_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(nat,A,bit_se2239418461657761734s_mask(A),aa(nat,nat,suc,Na)) = 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))),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) ) ).

% mask_Suc_double
tff(fact_7488_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% le_mask_iff_lt_2n
tff(fact_7489_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),M: nat,Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
           => ( ( W2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Na),W2) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_7490_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_7491_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xa),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),Y)) = $ite(
                aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),one_one(word(A)))),Y),
                $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),zero_zero(word(A))),Xa),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),Y))),
                $let(
                  q: word(A),
                  q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xa)),Y)),
                  $let(
                    r2: word(A),
                    r2:= aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),r2),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),r2),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r2)) ) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_7492_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,$o),Na: nat,W2: word(A)] :
          code_T2661198915054445665ts_aux(A,F3,aa(nat,nat,suc,Na),W2) = code_T2661198915054445665ts_aux(A,F3,Na,
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),W2)),
              $ite(aa(nat,$o,F3,Na),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_7493_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_7494_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_7495_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_7496_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,$o),W2: word(A)] : code_T2661198915054445665ts_aux(A,F3,zero_zero(nat),W2) = W2 ) ).

% set_bits_aux_0
tff(fact_7497_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Na),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))),Na) ).

% push_bit_of_Suc_0
tff(fact_7498_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),M),K)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_7499_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_7500_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(nat,int,bit_se2239418461657761734s_mask(int),Na)) ).

% mask_nonnegative_int
tff(fact_7501_less__eq__mask,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) ).

% less_eq_mask
tff(fact_7502_mask__integer_Orep__eq,axiom,
    ! [Xa: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),Xa)) = aa(nat,int,bit_se2239418461657761734s_mask(int),Xa) ).

% mask_integer.rep_eq
tff(fact_7503_push__bit__integer_Orep__eq,axiom,
    ! [Xa: nat,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Xa),Xaa)) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% push_bit_integer.rep_eq
tff(fact_7504_mask__integer_Oabs__eq,axiom,
    ! [Xa: nat] : aa(nat,code_integer,bit_se2239418461657761734s_mask(code_integer),Xa) = aa(int,code_integer,code_integer_of_int,aa(nat,int,bit_se2239418461657761734s_mask(int),Xa)) ).

% mask_integer.abs_eq
tff(fact_7505_push__bit__integer_Oabs__eq,axiom,
    ! [Xa: nat,Xb: int] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Xa),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Xa),Xb)) ).

% push_bit_integer.abs_eq
tff(fact_7506_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) ) ).

% less_mask
tff(fact_7507_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),Q3)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_7508_Suc__mask__eq__exp,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% Suc_mask_eq_exp
tff(fact_7509_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2239418461657761734s_mask(nat),Na)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% mask_nat_less_exp
tff(fact_7510_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,$o),Na: nat,W2: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),code_T2661198915054445665ts_aux(A,F3,Na,W2)),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),aa(nat,$o,F3,M),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na))) ) ) ) ).

% bit_set_bit_aux
tff(fact_7511_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F3: fun(nat,$o),Na: nat,W2: word(A)] :
          code_T2661198915054445665ts_aux(A,F3,Na,W2) = $ite(
            Na = zero_zero(nat),
            W2,
            $let(
              n3: nat,
              n3:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),
              code_T2661198915054445665ts_aux(A,F3,n3,
                aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),W2)),
                  $ite(aa(nat,$o,F3,n3),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_7512_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),Na)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(word(A),nat,size_size(word(A)),W2)),Na))) ) ) ) ).

% and_mask_arith'
tff(fact_7513_word__of__int__via__signed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: fun(int,word(A)),Arbitrary2: fun(int,word(A))] :
          ( ( Mask = aa(nat,int,bit_se2239418461657761734s_mask(int),type_len0_len_of(A,type2(A))) )
         => ( ( Shift = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),type_len0_len_of(A,type2(A))),one_one(int)) )
           => ( ( Index = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)) )
             => ( ( Overflow = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) )
               => ( ( Least = aa(int,int,uminus_uminus(int),Overflow) )
                 => ( aa(int,word(A),ring_1_of_int(word(A)),I) = $let(
                        i: int,
                        i:= aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask),
                        $ite(
                          aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),i),Index),
                          $ite(
                            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)) ),
                            aa(int,word(A),Arbitrary1,i),
                            aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),minus_minus(int),i),Shift)) ),
                          $ite(
                            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),i),Least)
                            | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_7514_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_7515_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_7516_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xa) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_7517_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_7518_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_7519_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_7520_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_7521_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_7522_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W2 )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ).

% and_mask_eq_iff_le_mask
tff(fact_7523_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = Xa ) ) ) ).

% le_mask_imp_and_mask
tff(fact_7524_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ) ).

% mask_twice2
tff(fact_7525_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_7526_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya2)),Z) ) ) ).

% AND_upper1''
tff(fact_7527_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_7528_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,Y: word(A)] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xa)),Na)
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),Na) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y))),Na) ) ) ).

% word_unat_and_lt
tff(fact_7529_word__and__le1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),A2)),A2) ) ).

% word_and_le1
tff(fact_7530_word__and__le2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),Y)),A2) ) ).

% word_and_le2
tff(fact_7531_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_7532_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya2)),Z) ) ) ).

% AND_upper1'
tff(fact_7533_AND__upper2,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Y) ) ).

% AND_upper2
tff(fact_7534_AND__upper1,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)),Xa) ) ).

% AND_upper1
tff(fact_7535_AND__lower,axiom,
    ! [Xa: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Y)) ) ).

% AND_lower
tff(fact_7536_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xa) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xa = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_7537_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Na),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7538_word__and__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Na),one_one(word(A))) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Na),zero_zero(nat)),one_one(word(A)),zero_zero(word(A))) ) ).

% word_and_1
tff(fact_7539_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xa: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_min(word(A)),Xa),Y) ) ).

% bit_twiddle_min
tff(fact_7540_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),
              $ite(aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),ord_max(word(A)),Xa),Y) ) ).

% bit_twiddle_max
tff(fact_7541_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Na: nat,Xa: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xa),aa(nat,word(B),bit_se2239418461657761734s_mask(word(B)),Na))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xa) ) ) ) ).

% ucast_mask_drop
tff(fact_7542_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_7543_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = Xa ) ) ) ).

% less_mask_eq
tff(fact_7544_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% and_mask_lt_2p
tff(fact_7545_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W2 )
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,semiring_1_unsigned(A,int),W2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% mask_eq_iff
tff(fact_7546_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Na: nat,V3: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),V3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),V3) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_7547_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) = W2 )
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_7548_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),Xa))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less_size
tff(fact_7549_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),minus_minus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)),one_one(word(A)))) ) ).

% word_and_mask_le_2pm1
tff(fact_7550_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xa),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_7551_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_7552_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ).

% and_mask_less'
tff(fact_7553_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( ( W2 = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% eq_mask_less
tff(fact_7554_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(word(A),nat,size_size(word(A)),W2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% word_unat_mask_lt
tff(fact_7555_drop__bit__int__code_I2_J,axiom,
    ! [Na: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Na)),zero_zero(int)) = zero_zero(int) ).

% drop_bit_int_code(2)
tff(fact_7556_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,W2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na)) ) ) ) ).

% neg_mask_is_div'
tff(fact_7557_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_7558_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_7559_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_7560_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_7561_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_7562_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_7563_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W2),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_7564_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_7565_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit1(Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_7566_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_7567_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_7568_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = Y )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_7569_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% mask_lower_twice
tff(fact_7570_and__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),Xa),Xaa)) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(code_integer,int,code_int_of_integer,Xa)),aa(code_integer,int,code_int_of_integer,Xaa)) ).

% and_integer.rep_eq
tff(fact_7571_and__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),aa(int,code_integer,code_integer_of_int,Xa)),aa(int,code_integer,code_integer_of_int,Xb)) = aa(int,code_integer,code_integer_of_int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Xb)) ).

% and_integer.abs_eq
tff(fact_7572_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Na)
        <=> ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_7573_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(nat,fun(A,A),bit_se2584673776208193580ke_bit(A),M),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,bit_se2239418461657761734s_mask(A),Na))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7574_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_7575_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_7576_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),M))) ) ) ) ).

% mask_subsume
tff(fact_7577_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_7578_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),Xa))
         => ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),Na)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),Na)
            <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na)
                <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Na) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),Na)
            <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),Na) ) ) ) ) ).

% word_ops_nth_size
tff(fact_7579_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Na)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_7580_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_7581_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat,Xa: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Na)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),bit_se2239418461657761734s_mask(word(A)),Na))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_7582_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) ) ) ).

% msb0
tff(fact_7583_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = $ite(
        ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(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(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),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),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_7584_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),zero_zero(nat)),I) = I ).

% drop_bit_int_code(1)
tff(fact_7585_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N2: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Na))),N2)
          <=> ( N2 != Na ) ) ) ) ).

% complement_nth_w2p
tff(fact_7586_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),zero_zero(nat)),I) = I ).

% push_bit_int_code(1)
tff(fact_7587_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_7588_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xa),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xa),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xa),Y)),zero_zero(nat))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xa)),zero_zero(nat))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xa),zero_zero(nat)) ) ) ) ).

% word_ops_lsb
tff(fact_7589_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_7590_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_7591_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_7592_not__integer_Oabs__eq,axiom,
    ! [Xa: int] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(int,code_integer,code_integer_of_int,Xa)) = aa(int,code_integer,code_integer_of_int,aa(int,int,bit_ri4277139882892585799ns_not(int),Xa)) ).

% not_integer.abs_eq
tff(fact_7593_not__integer_Orep__eq,axiom,
    ! [Xa: code_integer] : aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),Xa)) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(code_integer,int,code_int_of_integer,Xa)) ).

% not_integer.rep_eq
tff(fact_7594_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_rotl(A,M,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),type_len0_len_of(A,type2(A)))))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotl_iff
tff(fact_7595_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),M: nat,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),W2),M)),Na)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),M)),Na)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )) ) ) ).

% bit_sshiftr_iff
tff(fact_7596_sshiftr__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),W2),zero_zero(nat)) = W2 ) ).

% sshiftr_of_0
tff(fact_7597_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(num,word(A),numeral_numeral(word(A)),M)),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_Suc
tff(fact_7598_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Na: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))),aa(nat,nat,suc,Na)) = aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),aa(nat,nat,suc,Na)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_Suc
tff(fact_7599_sshiftr__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Na: nat] :
          aa(nat,word(A),aa(word(A),fun(nat,word(A)),bit_Sh8784991116023147202shiftr(A),one_one(word(A))),Na) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( Na = zero_zero(nat) ) )) ) ).

% sshiftr_1
tff(fact_7600_shiftl__Suc__0,axiom,
    ! [Na: nat] : bit_Sh4282982442137083160shiftl(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% shiftl_Suc_0
tff(fact_7601_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W2: word(A)] :
          word_roti(A,I,W2) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),word_rotr(A,aa(int,nat,nat2,I),W2),word_rotl(A,aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I)),W2)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_7602_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),Na) = zero_zero(A) ) ).

% shiftl_0
tff(fact_7603_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083160shiftl(A,A2,zero_zero(nat)) = A2 ) ).

% shiftl_of_0
tff(fact_7604_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: nat] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_Suc
tff(fact_7605_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: nat] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_Suc
tff(fact_7606_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Na: nat] : bit_Sh4282982442137083160shiftl(A,A2,aa(nat,nat,suc,Na)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ).

% shiftl_of_Suc
tff(fact_7607_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W2: word(A),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_rotr(A,M,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_7608_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,M: nat,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083160shiftl(A,A2,M)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)) ) ) ) ).

% bit_shiftl_iff
tff(fact_7609_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Na: nat] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_Suc
tff(fact_7610_bit__revcast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W2: word(B),Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),revcast(B,A,W2)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_revcast_iff
tff(fact_7611_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),Na) = zero_zero(A) ) ).

% shiftr_0
tff(fact_7612_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_Sh4282982442137083166shiftr(A,A2,zero_zero(nat)) = A2 ) ).

% shiftr_of_0
tff(fact_7613_shiftr__Suc__0,axiom,
    ! [Na: nat] : bit_Sh4282982442137083166shiftr(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa($o,nat,zero_neq_one_of_bool(nat),Na = zero_zero(nat)) ).

% shiftr_Suc_0
tff(fact_7614_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: nat] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,Na)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Na)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_Suc
tff(fact_7615_shiftr__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : bit_Sh4282982442137083166shiftr(A,one_one(A),Na) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% shiftr_1
tff(fact_7616_signed_Orsp,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => aa(fun(int,A),$o,aa(fun(int,A),fun(fun(int,A),$o),bNF_rel_fun(int,int,A,A,aTP_Lamp_ade(int,fun(int,$o)),fequal(A)),aa(fun(int,int),fun(int,A),comp(int,A,int,ring_1_of_int(A)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))))),aa(fun(int,int),fun(int,A),comp(int,A,int,ring_1_of_int(A)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% signed.rsp
tff(fact_7617_inj__on__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(nat,word(A),semiring_1_of_nat(word(A)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_nat
tff(fact_7618_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] : inj_on(A,B,F3,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_7619_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_7620_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_adf(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7621_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F3,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_7622_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: A,A5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5))
    <=> ( inj_on(A,B,F3,A5)
        & ~ member(B,aa(A,B,F3,A2),aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_7623_not__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),bit_ri4277139882892585799ns_not(int)),bit_ri4277139882892585799ns_not(int)) ).

% not_integer.rsp
tff(fact_7624_xor__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344971392196577ns_xor(int)),bit_se5824344971392196577ns_xor(int)) ).

% xor_integer.rsp
tff(fact_7625_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),Na: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_7626_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( aa(A,B,F3,X) != aa(A,B,F3,Y3) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7627_less__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less(int)),ord_less(int)) ).

% less_integer.rsp
tff(fact_7628_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A5: set(A),F3: fun(A,B)] :
          ( ! [X: A,Y3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y3)
             => ( member(A,X,A5)
               => ( member(A,Y3,A5)
                 => ( aa(A,B,F3,X) != aa(A,B,F3,Y3) ) ) ) )
         => ( ! [X: A,Y3: A] :
                ( member(A,X,A5)
               => ( member(A,Y3,A5)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y3)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),X) ) ) )
           => inj_on(A,B,F3,A5) ) ) ) ).

% linorder_inj_onI
tff(fact_7629_less__eq__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less_eq(int)),ord_less_eq(int)) ).

% less_eq_integer.rsp
tff(fact_7630_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A5: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
       => inj_on(A,B,F3,A5) ) ) ).

% subset_inj_on
tff(fact_7631_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
       => inj_on(A,B,F3,B5) ) ) ).

% inj_on_subset
tff(fact_7632_minus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),minus_minus(int)),minus_minus(int)) ).

% minus_integer.rsp
tff(fact_7633_dup_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),aTP_Lamp_adg(int,int)),aTP_Lamp_adg(int,int)) ).

% dup.rsp
tff(fact_7634_bit__integer_Orsp,axiom,
    aa(fun(int,fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(int,fun(nat,$o)),$o),bNF_rel_fun(int,int,fun(nat,$o),fun(nat,$o),fequal(int),fequal(fun(nat,$o))),bit_se5641148757651400278ts_bit(int)),bit_se5641148757651400278ts_bit(int)) ).

% bit_integer.rsp
tff(fact_7635_abs__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),abs_abs(int)),abs_abs(int)) ).

% abs_integer.rsp
tff(fact_7636_sgn__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),sgn_sgn(int)),sgn_sgn(int)) ).

% sgn_integer.rsp
tff(fact_7637_natural__of__integer_Orsp,axiom,
    aa(fun(int,nat),$o,aa(fun(int,nat),fun(fun(int,nat),$o),bNF_rel_fun(int,int,nat,nat,fequal(int),fequal(nat)),nat2),nat2) ).

% natural_of_integer.rsp
tff(fact_7638_divide__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),divide_divide(int)),divide_divide(int)) ).

% divide_integer.rsp
tff(fact_7639_modulo__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),modulo_modulo(int)),modulo_modulo(int)) ).

% modulo_integer.rsp
tff(fact_7640_uminus__integer_Orsp,axiom,
    aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int)),uminus_uminus(int)),uminus_uminus(int)) ).

% uminus_integer.rsp
tff(fact_7641_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
             => ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R3,R3),uminus_uminus(A)),uminus_uminus(B))
               => aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R3),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7642_plus__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),plus_plus(int)),plus_plus(int)) ).

% plus_integer.rsp
tff(fact_7643_times__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),times_times(int)),times_times(int)) ).

% times_integer.rsp
tff(fact_7644_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A5: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A5) ) ) ).

% inj_on_mult
tff(fact_7645_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
             => aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R3),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7646_or__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se1065995026697491101ons_or(int)),bit_se1065995026697491101ons_or(int)) ).

% or_integer.rsp
tff(fact_7647_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_7648_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),Xa: B,B5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( member(B,Xa,aa(set(A),set(B),image(A,B,F3),A5))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => member(A,the_inv_into(A,B,A5,F3,Xa),B5) ) ) ) ).

% the_inv_into_into
tff(fact_7649_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A5),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,F3),B5)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_7650_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G3: fun(A,B),A5: set(A),F3: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G3,A5)
     => ( finite_fold(B,C,F3,Z,aa(set(A),set(B),image(A,B,G3),A5)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F3),G3),Z,A5) ) ) ).

% fold_image
tff(fact_7651_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A),D5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(B,A,F3,D5)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_adh(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A5),F3),D5))) ) ) ).

% finite_inverse_image_gen
tff(fact_7652_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( inj_on(B,A,F3,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_adi(set(A),fun(fun(B,A),fun(B,$o)),S2),F3))) ) ) ).

% finite_Collect
tff(fact_7653_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A5: set(A),F3: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(B,A,F3,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_adi(set(A),fun(fun(B,A),fun(B,$o)),A5),F3))) ) ) ).

% finite_inverse_image
tff(fact_7654_range__inj__infinite,axiom,
    ! [A: $tType,F3: fun(nat,A)] :
      ( inj_on(nat,A,F3,top_top(set(nat)))
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image(nat,A,F3),top_top(set(nat)))) ) ).

% range_inj_infinite
tff(fact_7655_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( inj_on(A,A,F3,top_top(set(A)))
       => ( aa(set(A),set(A),image(A,A,F3),top_top(set(A))) = top_top(set(A)) ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_7656_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),set(A),image(A,A,F3),top_top(set(A))) = top_top(set(A)) )
       => inj_on(A,A,F3,top_top(set(A))) ) ) ).

% finite_UNIV_surj_inj
tff(fact_7657_inj__on__image__Fpow,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => inj_on(set(A),set(B),image(A,B,F3),finite_Fpow(A,A5)) ) ).

% inj_on_image_Fpow
tff(fact_7658_finite__imageD,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,F3),A5))
     => ( inj_on(B,A,F3,A5)
       => aa(set(B),$o,finite_finite2(B),A5) ) ) ).

% finite_imageD
tff(fact_7659_finite__image__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,F3),A5))
      <=> aa(set(A),$o,finite_finite2(A),A5) ) ) ).

% finite_image_iff
tff(fact_7660_card__image,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) ) ) ).

% card_image
tff(fact_7661_inj__on__iff__eq__card,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( inj_on(A,B,F3,A5)
      <=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_7662_eq__card__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F3),A5)) = aa(set(A),nat,finite_card(A),A5) )
       => inj_on(A,B,F3,A5) ) ) ).

% eq_card_imp_inj_on
tff(fact_7663_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),A10: set(B)] :
      ( ( A5 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A5)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A5)),A10) )
      <=> ? [G8: fun(B,A)] : aa(set(B),set(A),image(B,A,G8),A10) = A5 ) ) ).

% inj_on_iff_surj
tff(fact_7664_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),C4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
         => ( ( aa(set(A),set(B),image(A,B,F3),A5) = aa(set(A),set(B),image(A,B,F3),B5) )
          <=> ( A5 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_7665_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A2: A,A5: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( member(A,A2,B5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( member(B,aa(A,B,F3,A2),aa(set(A),set(B),image(A,B,F3),A5))
          <=> member(A,A2,A5) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_7666_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(B,A),T3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(B),set(A),image(B,A,F3),T3))
    <=> ? [U7: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U7),T3)
          & inj_on(B,A,F3,U7)
          & ( S2 = aa(set(B),set(A),image(B,A,F3),U7) ) ) ) ).

% subset_image_inj
tff(fact_7667_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B5: set(A),A5: set(A)] :
      ( inj_on(A,B,F3,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A5),B5)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,F3),B5)) ) ) ).

% inj_on_strict_subset
tff(fact_7668_endo__inj__surj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,F3),A5)),A5)
       => ( inj_on(A,A,F3,A5)
         => ( aa(set(A),set(A),image(A,A,F3),A5) = A5 ) ) ) ) ).

% endo_inj_surj
tff(fact_7669_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),B5)
       => ( aa(set(B),$o,finite_finite2(B),B5)
         => aa(set(A),$o,finite_finite2(A),A5) ) ) ) ).

% inj_on_finite
tff(fact_7670_finite__surj__inj,axiom,
    ! [A: $tType,A5: set(A),F3: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(set(A),set(A),image(A,A,F3),A5))
       => inj_on(A,A,F3,A5) ) ) ).

% finite_surj_inj
tff(fact_7671_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,F3),B5))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ) ).

% inj_image_subset_iff
tff(fact_7672_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A5: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F3),A5))),aa(set(B),nat,finite_card(B),A5))
     => ~ inj_on(B,A,F3,A5) ) ).

% pigeonhole
tff(fact_7673_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C4: set(A),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)),C4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C4)
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,F3),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_7674_inj__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S2: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),S2)
         => inj_on(nat,A,infini527867602293511546merate(A,S2),top_top(set(nat))) ) ) ).

% inj_enumerate
tff(fact_7675_less__word_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_adj(int,fun(int,$o)),fequal($o))),aTP_Lamp_adk(int,fun(int,$o))),aTP_Lamp_adk(int,fun(int,$o))) ).

% less_word.rsp
tff(fact_7676_less__eq__word_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_adj(int,fun(int,$o)),fequal($o))),aTP_Lamp_adl(int,fun(int,$o))),aTP_Lamp_adl(int,fun(int,$o))) ).

% less_eq_word.rsp
tff(fact_7677_and__integer_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),fequal(int),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5824344872417868541ns_and(int)),bit_se5824344872417868541ns_and(int)) ).

% and_integer.rsp
tff(fact_7678_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A5: fun(A,set(B)),F3: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( member(A,I2,I5)
         => ( member(A,J2,I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A5,I2)),aa(A,set(B),A5,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A5,J2)),aa(A,set(B),A5,I2)) ) ) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F3,aa(A,set(B),A5,I2)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_7679_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [N: nat,F2: fun(nat,A)] :
          ( ( A5 = aa(set(nat),set(A),image(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N))) )
          & inj_on(nat,A,F2,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7680_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F3: fun(B,C),A5: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F3,aa(A,set(B),A5,I2)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),A5),I5))) ) ) ).

% inj_on_INTER
tff(fact_7681_word__sle_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_adj(int,fun(int,$o)),fequal($o))),aTP_Lamp_adm(int,fun(int,$o))),aTP_Lamp_adm(int,fun(int,$o))) ).

% word_sle.rsp
tff(fact_7682_word__sless_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,$o,$o,aTP_Lamp_adj(int,fun(int,$o)),fequal($o))),aTP_Lamp_adn(int,fun(int,$o))),aTP_Lamp_adn(int,fun(int,$o))) ).

% word_sless.rsp
tff(fact_7683_signed__modulo__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_adj(int,fun(int,$o)))),aTP_Lamp_ado(int,fun(int,int))),aTP_Lamp_ado(int,fun(int,int))) ).

% signed_modulo_word.rsp
tff(fact_7684_signed__divide__word_Orsp,axiom,
    aa(fun(int,fun(int,int)),$o,aa(fun(int,fun(int,int)),fun(fun(int,fun(int,int)),$o),bNF_rel_fun(int,int,fun(int,int),fun(int,int),aTP_Lamp_adj(int,fun(int,$o)),bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_adj(int,fun(int,$o)))),aTP_Lamp_adp(int,fun(int,int))),aTP_Lamp_adp(int,fun(int,int))) ).

% signed_divide_word.rsp
tff(fact_7685_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S2: set(A),T3: set(B),F3: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),S2)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(B),nat,finite_card(B),T3) )
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),S2)),T3)
           => ( ! [X5: B] :
                  ( member(B,X5,T3)
                 => ? [Xa3: A] :
                      ( member(A,Xa3,S2)
                      & ( aa(A,B,F3,Xa3) = X5 ) ) )
            <=> inj_on(A,B,F3,S2) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_7686_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B),G3: fun(B,A)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),B5)
       => ( inj_on(B,A,G3,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G3),B5)),A5)
           => ( aa(set(A),$o,finite_finite2(A),A5)
             => ( aa(set(B),$o,finite_finite2(B),B5)
               => ( aa(set(A),nat,finite_card(A),A5) = aa(set(B),nat,finite_card(B),B5) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_7687_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X: nat] :
            ( member(nat,X,I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_7688_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),uminus_uminus(set(A)),A5))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,F3),A5))) ) ).

% inj_image_Compl_subset
tff(fact_7689_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
    <=> ? [F7: fun(nat,A)] :
          ( inj_on(nat,A,F7,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F7),top_top(set(nat)))),S2) ) ) ).

% infinite_iff_countable_subset
tff(fact_7690_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S2)
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F2),top_top(set(nat)))),S2) ) ) ).

% infinite_countable_subset
tff(fact_7691_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),G3: fun(A,B),B5: set(A)] :
      ( inj_on(A,B,F3,A5)
     => ( inj_on(A,B,G3,B5)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),aa(set(A),set(B),image(A,B,G3),B5)) = 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_adq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F3),A5),G3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_7692_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),C4: set(A),A5: set(C),B5: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F3,C4)
     => ( ! [X: C] :
            ( member(C,X,A5)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B5,X)),C4) )
       => ( member(C,J,A5)
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B5),A5))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_adr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B5)),A5)) ) ) ) ) ).

% image_INT
tff(fact_7693_inj__on__funpow__least,axiom,
    ! [A: $tType,Na: nat,F3: fun(A,A),Sa: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F3),Sa) = Sa )
     => ( ! [M3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M3),F3),Sa) != Sa ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_ads(fun(A,A),fun(A,fun(nat,A)),F3),Sa),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ) ).

% inj_on_funpow_least
tff(fact_7694_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A5)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F7),A5)),B5) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_7695_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(B)] :
      ( inj_on(A,B,F3,A5)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A5)),B5)
       => ( aa(set(B),$o,finite_finite2(B),B5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5)) ) ) ) ).

% card_inj_on_le
tff(fact_7696_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A5: set(A),B5: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ( aa(set(B),$o,finite_finite2(B),B5)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A5)),aa(set(B),nat,finite_card(B),B5))
         => ? [F2: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F2),A5)),B5)
              & inj_on(A,B,F2,A5) ) ) ) ) ).

% card_le_inj
tff(fact_7697_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A5: set(A),B5: set(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5))
    <=> ( inj_on(A,B,F3,A5)
        & inj_on(A,B,F3,B5)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),B5))),aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B5),A5))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_7698_bit__word_Orsp,axiom,
    aa(fun(int,fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(int,fun(nat,$o)),$o),bNF_rel_fun(int,int,fun(nat,$o),fun(nat,$o),aTP_Lamp_adj(int,fun(int,$o)),fequal(fun(nat,$o))),aTP_Lamp_adt(int,fun(nat,$o))),aTP_Lamp_adt(int,fun(nat,$o))) ).

% bit_word.rsp
tff(fact_7699_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xs: list(A),Ys: list(A)] :
          ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Xs))
             => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Ys))
               => ( distinct(B,aa(list(A),list(B),map(A,B,F3),Ys))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7700_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),Xa: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_adu(fun(A,A),fun(A,fun(A,$o)),P3),Xa)))
       => ~ ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),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),P3),Xa) != Xa ) ) ) ) ).

% funpow_inj_finite
tff(fact_7701_the__signed__int_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),fequal(int)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% the_signed_int.rsp
tff(fact_7702_signed__cast_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(int,int),$o,aa(fun(int,int),fun(fun(int,int),$o),bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_ade(int,fun(int,$o))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% signed_cast.rsp
tff(fact_7703_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R3,bNF_rel_fun(A,B,A,B,R3,R3)),plus_plus(A)),plus_plus(B))
             => aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R3),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7704_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A5: set(A),F3: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A5)
     => ( inj_on(A,B,F3,A5)
       => ( aa(list(B),list(A),map(B,A,inv_on(A,B,F3,A5)),aa(list(A),list(B),map(A,B,F3),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_7705_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A5: set(A),Xa: A] :
      ( inj_on(A,B,F3,A5)
     => ( member(A,Xa,A5)
       => ( aa(B,A,inv_on(A,B,F3,A5),aa(A,B,F3,Xa)) = Xa ) ) ) ).

% inv_on_f_f
tff(fact_7706_xor__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344971392196577ns_xor(nat)),bit_se5824344971392196577ns_xor(nat)) ).

% xor_natural.rsp
tff(fact_7707_drop__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4197421643247451524op_bit(int)),bit_se4197421643247451524op_bit(int)) ).

% drop_bit_integer.rsp
tff(fact_7708_drop__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4197421643247451524op_bit(nat)),bit_se4197421643247451524op_bit(nat)) ).

% drop_bit_natural.rsp
tff(fact_7709_take__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2584673776208193580ke_bit(int)),bit_se2584673776208193580ke_bit(int)) ).

% take_bit_integer.rsp
tff(fact_7710_take__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2584673776208193580ke_bit(nat)),bit_se2584673776208193580ke_bit(nat)) ).

% take_bit_natural.rsp
tff(fact_7711_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_adw(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_7712_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_7713_less__eq__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less_eq(nat)),ord_less_eq(nat)) ).

% less_eq_natural.rsp
tff(fact_7714_inj__on__diff__nat,axiom,
    ! [N6: set(nat),K: nat] :
      ( ! [N: nat] :
          ( member(nat,N,N6)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N) )
     => inj_on(nat,nat,aTP_Lamp_do(nat,fun(nat,nat),K),N6) ) ).

% inj_on_diff_nat
tff(fact_7715_minus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),minus_minus(nat)),minus_minus(nat)) ).

% minus_natural.rsp
tff(fact_7716_sub_Orsp,axiom,
    aa(fun(num,fun(num,int)),$o,aa(fun(num,fun(num,int)),fun(fun(num,fun(num,int)),$o),bNF_rel_fun(num,num,fun(num,int),fun(num,int),fequal(num),bNF_rel_fun(num,num,int,int,fequal(num),fequal(int))),aTP_Lamp_adx(num,fun(num,int))),aTP_Lamp_adx(num,fun(num,int))) ).

% sub.rsp
tff(fact_7717_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R3: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R3,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R3,one_one(A)),one_one(B))
           => aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R3),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_7718_bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),fequal(fun(nat,$o))),bit_se5641148757651400278ts_bit(nat)),bit_se5641148757651400278ts_bit(nat)) ).

% bit_natural.rsp
tff(fact_7719_unset__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se2638667681897837118et_bit(int)),bit_se2638667681897837118et_bit(int)) ).

% unset_bit_integer.rsp
tff(fact_7720_flip__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se8732182000553998342ip_bit(int)),bit_se8732182000553998342ip_bit(int)) ).

% flip_bit_integer.rsp
tff(fact_7721_inj__Suc,axiom,
    ! [N6: set(nat)] : inj_on(nat,nat,suc,N6) ).

% inj_Suc
tff(fact_7722_Suc_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),suc),suc) ).

% Suc.rsp
tff(fact_7723_VEBTi_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S2: fun(A,fun(B,$o))] : aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBTi,B))),$o,aa(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A))),fun(fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(fun($o,fun($o,B)),fun(vEBT_VEBTi,B))),$o),bNF_rel_fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(fun($o,fun($o,A)),fun(vEBT_VEBTi,A)),fun(fun($o,fun($o,B)),fun(vEBT_VEBTi,B)),bNF_rel_fun(option(product_prod(nat,nat)),option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B))),fequal(option(product_prod(nat,nat))),bNF_rel_fun(nat,nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)),fequal(nat),bNF_rel_fun(array(vEBT_VEBTi),array(vEBT_VEBTi),fun(vEBT_VEBTi,A),fun(vEBT_VEBTi,B),fequal(array(vEBT_VEBTi)),bNF_rel_fun(vEBT_VEBTi,vEBT_VEBTi,A,B,fequal(vEBT_VEBTi),S2)))),bNF_rel_fun(fun($o,fun($o,A)),fun($o,fun($o,B)),fun(vEBT_VEBTi,A),fun(vEBT_VEBTi,B),bNF_rel_fun($o,$o,fun($o,A),fun($o,B),fequal($o),bNF_rel_fun($o,$o,A,B,fequal($o),S2)),bNF_rel_fun(vEBT_VEBTi,vEBT_VEBTi,A,B,fequal(vEBT_VEBTi),S2))),vEBT_case_VEBTi(A)),vEBT_case_VEBTi(B)) ).

% VEBTi.case_transfer
tff(fact_7724_divide__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),divide_divide(nat)),divide_divide(nat)) ).

% divide_natural.rsp
tff(fact_7725_modulo__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),modulo_modulo(nat)),modulo_modulo(nat)) ).

% modulo_natural.rsp
tff(fact_7726_Code__Numeral_Oset__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se5668285175392031749et_bit(int)),bit_se5668285175392031749et_bit(int)) ).

% Code_Numeral.set_bit_integer.rsp
tff(fact_7727_plus__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),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_7728_times__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),times_times(nat)),times_times(nat)) ).

% times_natural.rsp
tff(fact_7729_or__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se1065995026697491101ons_or(nat)),bit_se1065995026697491101ons_or(nat)) ).

% or_natural.rsp
tff(fact_7730_integer__of__natural_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),semiring_1_of_nat(int)),semiring_1_of_nat(int)) ).

% integer_of_natural.rsp
tff(fact_7731_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_ady(A,fun(B,product_prod(B,A)))),A5) ).

% swap_inj_on
tff(fact_7732_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X4: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_adz(fun(A,B),fun(A,product_prod(A,B)),F3),X4) ).

% inj_on_convol_ident
tff(fact_7733_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C3: fun(A,B),S2: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_aea(fun(A,B),fun(A,product_prod(B,A)),C3),S2) ).

% inj_Pair(2)
tff(fact_7734_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C3: fun(A,B),S2: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_adz(fun(A,B),fun(A,product_prod(A,B)),C3),S2) ).

% inj_Pair(1)
tff(fact_7735_inj__singleton,axiom,
    ! [A: $tType,A5: set(A)] : inj_on(A,set(A),aTP_Lamp_ki(A,set(A)),A5) ).

% inj_singleton
tff(fact_7736_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_7737_mask__integer_Orsp,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,int),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,int,int,fequal(nat),fequal(int)),bit_se2239418461657761734s_mask(int)),bit_se2239418461657761734s_mask(int)) ).

% mask_integer.rsp
tff(fact_7738_mask__natural_Orsp,axiom,
    aa(fun(nat,nat),$o,aa(fun(nat,nat),fun(fun(nat,nat),$o),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat)),bit_se2239418461657761734s_mask(nat)),bit_se2239418461657761734s_mask(nat)) ).

% mask_natural.rsp
tff(fact_7739_push__bit__integer_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,fequal(int),fequal(int))),bit_se4730199178511100633sh_bit(int)),bit_se4730199178511100633sh_bit(int)) ).

% push_bit_integer.rsp
tff(fact_7740_push__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se4730199178511100633sh_bit(nat)),bit_se4730199178511100633sh_bit(nat)) ).

% push_bit_natural.rsp
tff(fact_7741_and__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5824344872417868541ns_and(nat)),bit_se5824344872417868541ns_and(nat)) ).

% and_natural.rsp
tff(fact_7742_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F3: fun(B,A),A5: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F3),A5))
     => ( aa(B,A,F3,aa(A,B,inv_on(B,A,F3,A5),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_7743_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F3: fun(B,A),A5: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F3),A5))
     => member(B,aa(A,B,inv_on(B,A,F3,A5),Y),A5) ) ).

% inv_on_f_range
tff(fact_7744_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ? [F2: fun(A,nat),N: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F2),A5) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N)) )
          & inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7745_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A5)
     => ~ ! [F2: fun(A,nat)] :
            ( ? [N: nat] : aa(set(A),set(nat),image(A,nat,F2),A5) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),N))
           => ~ inj_on(A,nat,F2,A5) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_7746_summable__reindex,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G3,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X))
         => summable(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G3)) ) ) ) ).

% summable_reindex
tff(fact_7747_is__up_Orsp,axiom,
    aa(fun(fun(int,int),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(int,int),$o),$o),bNF_rel_fun(fun(int,int),fun(int,int),$o,$o,bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_ade(int,fun(int,$o))),fequal($o)),aTP_Lamp_aeb(fun(int,int),$o)),aTP_Lamp_aeb(fun(int,int),$o)) ).

% is_up.rsp
tff(fact_7748_is__down_Orsp,axiom,
    aa(fun(fun(int,int),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(int,int),$o),$o),bNF_rel_fun(fun(int,int),fun(int,int),$o,$o,bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_ade(int,fun(int,$o))),fequal($o)),aTP_Lamp_aec(fun(int,int),$o)),aTP_Lamp_aec(fun(int,int),$o)) ).

% is_down.rsp
tff(fact_7749_suminf__reindex__mono,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G3,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G3))),suminf(real,F3)) ) ) ) ).

% suminf_reindex_mono
tff(fact_7750_suminf__reindex,axiom,
    ! [F3: fun(nat,real),G3: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G3,top_top(set(nat)))
       => ( ! [X: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X))
         => ( ! [X: nat] :
                ( ~ member(nat,X,aa(set(nat),set(nat),image(nat,nat,G3),top_top(set(nat))))
               => ( aa(nat,real,F3,X) = zero_zero(real) ) )
           => ( suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G3)) = suminf(real,F3) ) ) ) ) ) ).

% suminf_reindex
tff(fact_7751_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => inj_on(real,real,aTP_Lamp_mi(nat,fun(real,real),Na),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7752_signed__drop__bit_Orsp,axiom,
    aa(fun(nat,fun(int,int)),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(int,int)),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(int,int),fequal(nat),bNF_rel_fun(int,int,int,int,aTP_Lamp_adj(int,fun(int,$o)),aTP_Lamp_adj(int,fun(int,$o)))),aTP_Lamp_aed(nat,fun(int,int))),aTP_Lamp_aed(nat,fun(int,int))) ).

% signed_drop_bit.rsp
tff(fact_7753_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P2: fun(set(A),$o)] :
      ( ? [T11: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T11),aa(set(B),set(A),image(B,A,F3),S2))
          & aa(set(A),$o,P2,T11) )
    <=> ? [T11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T11),S2)
          & inj_on(B,A,F3,T11)
          & aa(set(A),$o,P2,aa(set(B),set(A),image(B,A,F3),T11)) ) ) ).

% ex_subset_image_inj
tff(fact_7754_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P2: fun(set(A),$o)] :
      ( ! [T11: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T11),aa(set(B),set(A),image(B,A,F3),S2))
         => aa(set(A),$o,P2,T11) )
    <=> ! [T11: set(B)] :
          ( ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T11),S2)
            & inj_on(B,A,F3,T11) )
         => aa(set(A),$o,P2,aa(set(B),set(A),image(B,A,F3),T11)) ) ) ).

% all_subset_image_inj
tff(fact_7755_unset__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se2638667681897837118et_bit(nat)),bit_se2638667681897837118et_bit(nat)) ).

% unset_bit_natural.rsp
tff(fact_7756_flip__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se8732182000553998342ip_bit(nat)),bit_se8732182000553998342ip_bit(nat)) ).

% flip_bit_natural.rsp
tff(fact_7757_set__bit__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,nat)),$o,aa(fun(nat,fun(nat,nat)),fun(fun(nat,fun(nat,nat)),$o),bNF_rel_fun(nat,nat,fun(nat,nat),fun(nat,nat),fequal(nat),bNF_rel_fun(nat,nat,nat,nat,fequal(nat),fequal(nat))),bit_se5668285175392031749et_bit(nat)),bit_se5668285175392031749et_bit(nat)) ).

% set_bit_natural.rsp
tff(fact_7758_signed__drop__bit_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(nat,fun(word(A),word(A))),$o,aa(fun(nat,fun(int,int)),fun(fun(nat,fun(word(A),word(A))),$o),bNF_rel_fun(nat,nat,fun(int,int),fun(word(A),word(A)),fequal(nat),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_aed(nat,fun(int,int))),signed_drop_bit(A)) ) ).

% signed_drop_bit.transfer
tff(fact_7759_signed_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & ring_1(B) )
     => aa(fun(word(A),B),$o,aa(fun(int,B),fun(fun(word(A),B),$o),bNF_rel_fun(int,word(A),B,B,pcr_word(A),fequal(B)),aa(fun(int,int),fun(int,B),comp(int,B,int,ring_1_of_int(B)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,B)) ) ).

% signed.transfer
tff(fact_7760_less__eq__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_adl(int,fun(int,$o))),ord_less_eq(word(A))) ) ).

% less_eq_word.transfer
tff(fact_7761_less__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_adk(int,fun(int,$o))),ord_less(word(A))) ) ).

% less_word.transfer
tff(fact_7762_bit__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(word(A),fun(nat,$o)),$o),bNF_rel_fun(int,word(A),fun(nat,$o),fun(nat,$o),pcr_word(A),fequal(fun(nat,$o))),aTP_Lamp_adt(int,fun(nat,$o))),bit_se5641148757651400278ts_bit(word(A))) ) ).

% bit_word.transfer
tff(fact_7763_word__test__bit__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,$o)),$o,aa(fun(int,fun(nat,$o)),fun(fun(word(A),fun(nat,$o)),$o),bNF_rel_fun(int,word(A),fun(nat,$o),fun(nat,$o),pcr_word(A),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),aTP_Lamp_adt(int,fun(nat,$o))),bit_se5641148757651400278ts_bit(word(A))) ) ).

% word_test_bit_transfer
tff(fact_7764_word__sle_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_adm(int,fun(int,$o))),word_sle(A)) ) ).

% word_sle.transfer
tff(fact_7765_word__sless_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(word(A),fun(word(A),$o)),$o),bNF_rel_fun(int,word(A),fun(int,$o),fun(word(A),$o),pcr_word(A),bNF_rel_fun(int,word(A),$o,$o,pcr_word(A),fequal($o))),aTP_Lamp_adn(int,fun(int,$o))),word_sless(A)) ) ).

% word_sless.transfer
tff(fact_7766_signed__divide__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_adp(int,fun(int,int))),signed7115095781618012415divide(word(A))) ) ).

% signed_divide_word.transfer
tff(fact_7767_sshiftr__transfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(nat,word(A))),$o,aa(fun(int,fun(nat,int)),fun(fun(word(A),fun(nat,word(A))),$o),bNF_rel_fun(int,word(A),fun(nat,int),fun(nat,word(A)),pcr_word(A),bNF_rel_fun(nat,nat,int,word(A),fequal(nat),pcr_word(A))),aTP_Lamp_aee(int,fun(nat,int))),bit_Sh8784991116023147202shiftr(A)) ) ).

% sshiftr_transfer
tff(fact_7768_signed__cast_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => aa(fun(word(A),word(B)),$o,aa(fun(int,int),fun(fun(word(A),word(B)),$o),bNF_rel_fun(int,word(A),int,word(B),pcr_word(A),pcr_word(B)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),signed_cast(A,B)) ) ).

% signed_cast.transfer
tff(fact_7769_the__signed__int_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),int),$o,aa(fun(int,int),fun(fun(word(A),int),$o),bNF_rel_fun(int,word(A),int,int,pcr_word(A),fequal(int)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),the_signed_int(A)) ) ).

% the_signed_int.transfer
tff(fact_7770_signed__modulo__word_Otransfer,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(fun(word(A),fun(word(A),word(A))),$o,aa(fun(int,fun(int,int)),fun(fun(word(A),fun(word(A),word(A))),$o),bNF_rel_fun(int,word(A),fun(int,int),fun(word(A),word(A)),pcr_word(A),bNF_rel_fun(int,word(A),int,word(A),pcr_word(A),pcr_word(A))),aTP_Lamp_ado(int,fun(int,int))),signed6721504322012087516modulo(word(A))) ) ).

% signed_modulo_word.transfer
tff(fact_7771_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G3: fun(A,B),C4: set(A),B5: set(A),Xa: A] :
      ( inj_on(A,B,G3,C4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_aef(fun(A,B),fun(set(A),fun(A,fun(B,A))),G3),C4),Xa),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_7772_refl__ge__eq,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o))] :
      ( ! [X: A] : aa(A,$o,aa(A,fun(A,$o),R3,X),X)
     => aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R3) ) ).

% refl_ge_eq
tff(fact_7773_ge__eq__refl,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xa: A] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R3)
     => aa(A,$o,aa(A,fun(A,$o),R3,Xa),Xa) ) ).

% ge_eq_refl
tff(fact_7774_Ball__Collect,axiom,
    ! [A: $tType,A5: set(A),P2: fun(A,$o)] :
      ( ! [X5: A] :
          ( member(A,X5,A5)
         => aa(A,$o,P2,X5) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),aa(fun(A,$o),set(A),collect(A),P2)) ) ).

% Ball_Collect
tff(fact_7775_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A16: set(B),B13: set(A),F22: fun(C,D),B23: set(C),A25: set(D)] :
      ( ( aa(set(B),set(A),image(B,A,F1),A16) = B13 )
     => ( inj_on(C,D,F22,B23)
       => ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image(C,D,F22),B23)),A25)
         => ( ( ( B23 = bot_bot(set(C)) )
             => ( A25 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B23,B13) = aa(set(fun(D,B)),set(fun(C,A)),image(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F22)),bNF_Wellorder_Func(D,B,A25,A16)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_7776_lexord__take__index__conv,axiom,
    ! [A: $tType,Xa: list(A),Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Y))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xa),Y) = Xa ) )
        | ? [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Y)))
            & ( take(A,I3,Xa) = take(A,I3,Y) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth(A,Xa),I3)),aa(nat,A,nth(A,Y),I3)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_7777_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,Xa: list(A),B2: A,Y: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xa)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)),lexord(A,R2))
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
        | ( ( A2 = B2 )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2)) ) ) ) ).

% lexord_cons_cons
tff(fact_7778_lexord__linear,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: list(A),Y: list(A)] :
      ( ! [A4: A,B4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B4),R2)
          | ( A4 = B4 )
          | member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A4),R2) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xa),Y),lexord(A,R2))
        | ( Xa = Y )
        | member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y),Xa),lexord(A,R2)) ) ) ).

% lexord_linear
tff(fact_7779_lexord__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lexord(A,R2)) ) ).

% lexord_irreflexive
tff(fact_7780_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R2: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X: A,Y3: A,Z3: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),R2)
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y3),Z3),R2)
             => member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3),R2) ) ) )
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexord(A,R2))
       => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R2))
         => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs),lexord(A,R2)) ) ) ) ).

% lexord_partial_trans
tff(fact_7781_lexord__append__leftD,axiom,
    ! [A: $tType,Xa: list(A),U: list(A),V3: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xa),V3)),lexord(A,R2))
     => ( ! [A4: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),A4),R2)
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V3),lexord(A,R2)) ) ) ).

% lexord_append_leftD
tff(fact_7782_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R2: set(product_prod(A,A)),U: list(A),Xa: list(A),Y: list(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2),R2)
     => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xa))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R2)) ) ).

% lexord_append_left_rightI
tff(fact_7783_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),lexord(A,R2))
    <=> ( ? [X5: A] :
            ( member(A,X5,aa(list(A),set(A),set2(A),Xs))
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5),R2) )
        | member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lexord(A,R2)) ) ) ).

% lexord_same_pref_iff
tff(fact_7784_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V3: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),W2),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V3)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W2),Z)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_7785_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G3: fun(A,B),A25: set(A),A16: set(B),F1: fun(B,C),B13: set(C),F22: fun(D,A),B23: set(D)] :
      ( member(fun(A,B),G3,bNF_Wellorder_Func(A,B,A25,A16))
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,F1),A16)),B13)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(D),set(A),image(D,A,F22),B23)),A25)
         => member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B23,F1,F22),G3),bNF_Wellorder_Func(D,C,B23,B13)) ) ) ) ).

% Func_map
tff(fact_7786_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B5: set(A),A5: set(B)] :
      ( ( B5 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A5,B5) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_7787_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A5: set(A),B5: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A5,B5) = bot_bot(set(fun(A,B))) )
    <=> ( ( A5 != bot_bot(set(A)) )
        & ( B5 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_7788_lexord__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexord(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aeg(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lexord_def
tff(fact_7789_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A2)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A2) ) ) ).

% msb_big
tff(fact_7790_integer__of__num_I3_J,axiom,
    ! [Na: num] :
      aa(num,code_integer,code_integer_of_num,bit1(Na)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k)),one_one(code_integer)) ) ).

% integer_of_num(3)
tff(fact_7791_word__sless__msb__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),Xa),Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xa) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xa)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Xa),Y) ) ) ) ) ).

% word_sless_msb_less
tff(fact_7792_integer__of__num__def,axiom,
    code_integer_of_num = numeral_numeral(code_integer) ).

% integer_of_num_def
tff(fact_7793_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(word(A),int,ring_1_signed(A,int),W2)),zero_zero(int)) ) ) ).

% word_msb_sint
tff(fact_7794_word__sle__msb__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),Xa),Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xa) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xa)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xa),Y) ) ) ) ) ).

% word_sle_msb_le
tff(fact_7795_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W2)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ).

% msb_word_iff_bit
tff(fact_7796_integer__of__num__triv_I1_J,axiom,
    aa(num,code_integer,code_integer_of_num,one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_7797_integer__of__num_I2_J,axiom,
    ! [Na: num] :
      aa(num,code_integer,code_integer_of_num,bit0(Na)) = $let(
        k: code_integer,
        k:= aa(num,code_integer,code_integer_of_num,Na),
        aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),k),k) ) ).

% integer_of_num(2)
tff(fact_7798_integer__of__num__triv_I2_J,axiom,
    aa(num,code_integer,code_integer_of_num,bit0(one2)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% integer_of_num_triv(2)
tff(fact_7799_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V3: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),V3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),one_one(nat))))
         => ~ most_s684356279273892711sb_msb(word(A),V3) ) ) ).

% not_msb_from_less
tff(fact_7800_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P2: fun(A,$o),F3: fun(A,B),A5: B,B5: set(B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,topolo174197925503356063within(B,A5,B5)))
        <=> ? [S9: set(B)] :
              ( topolo1002775350975398744n_open(B,S9)
              & member(B,A5,S9)
              & ! [X5: A] :
                  ( member(B,aa(A,B,F3,X5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S9),B5)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A5),bot_bot(set(B)))))
                 => aa(A,$o,P2,X5) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_7801_take__hd__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,Na,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,drop(A,Na,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Na),Xs) ) ) ).

% take_hd_drop
tff(fact_7802_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( hd(nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_7803_hd__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( hd(A,replicate(A,Na,Xa)) = Xa ) ) ).

% hd_replicate
tff(fact_7804_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_7805_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( hd(A,Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_7806_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F13: filter(B),F24: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),filtercomap(A,B,F3,F13)),filtercomap(A,B,F3,F24))),filtercomap(A,B,F3,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F13),F24))) ).

% filtercomap_sup
tff(fact_7807_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,F4: filter(A),F6: filter(A),F3: fun(B,A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F4),F6)
     => aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),filtercomap(B,A,F3,F4)),filtercomap(B,A,F3,F6)) ) ).

% filtercomap_mono
tff(fact_7808_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),F4: filter(B),G5: filter(A)] :
      ( filterlim(A,B,F3,F4,G5)
    <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),filtercomap(A,B,F3,F4)) ) ).

% filterlim_iff_le_filtercomap
tff(fact_7809_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( hd(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),hd(A,Xs)),hd(B,Ys)) ) ) ) ).

% hd_zip
tff(fact_7810_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N8: B] :
            ! [X5: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N8),aa(A,B,F3,X5))
             => aa(A,$o,P2,X5) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_7811_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_top(B)))
        <=> ? [N8: B] :
            ! [X5: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N8),aa(A,B,F3,X5))
             => aa(A,$o,P2,X5) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_7812_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N8: B] :
            ! [X5: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,X5)),N8)
             => aa(A,$o,P2,X5) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_7813_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P2: fun(A,$o),F3: fun(A,B)] :
          ( eventually(A,P2,filtercomap(A,B,F3,at_bot(B)))
        <=> ? [N8: B] :
            ! [X5: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F3,X5)),N8)
             => aa(A,$o,P2,X5) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_7814_sorted__hd__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),hd(A,Xs)),X3) ) ) ) ) ).

% sorted_hd_min
tff(fact_7815_hd__drop__conv__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( hd(A,drop(A,Na,Xs)) = aa(nat,A,nth(A,Xs),Na) ) ) ).

% hd_drop_conv_nth
tff(fact_7816_hd__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))
     => ( hd(A,butlast(A,Xs)) = hd(A,Xs) ) ) ).

% hd_butlast
tff(fact_7817_sorted__hd__last,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),L)
         => ( ( L != nil(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),hd(A,L)),last(A,L)) ) ) ) ).

% sorted_hd_last
tff(fact_7818_hd__last__singletonI,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( hd(A,Xs) = last(A,Xs) )
       => ( distinct(A,Xs)
         => ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),hd(A,Xs)),nil(A)) ) ) ) ) ).

% hd_last_singletonI
tff(fact_7819_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(A,C),F4: fun(B,filter(C)),B5: set(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_aeh(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F3),F4)),B5))),filtercomap(A,C,F3,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image(B,filter(C),F4),B5)))) ).

% filtercomap_SUP
tff(fact_7820_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hd(A,slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_7821_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A2: A,Xs: list(A),F3: fun(A,B)] :
          ( member(A,A2,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),aa(list(A),list(B),map(A,B,F3),Xs))
           => ( ( hd(A,filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_aei(A,fun(fun(A,B),fun(A,$o)),A2),F3),Xs)) = A2 )
             => ( linorder_insort_key(A,B,F3,A2,remove1(A,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7822_test__bit__rcat,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Sw: nat,Wl: list(word(A)),Rc: word(B),Na: nat] :
          ( ( Sw = aa(word(A),nat,size_size(word(A)),hd(word(A),Wl)) )
         => ( ( Rc = word_rcat(A,B,Wl) )
           => ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),Rc),Na)
            <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(B),nat,size_size(word(B)),Rc))
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Sw)),aa(list(word(A)),nat,size_size(list(word(A))),Wl))
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),rev(word(A),Wl)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Na),Sw))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Na),Sw)) ) ) ) ) ) ).

% test_bit_rcat
tff(fact_7823_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_7824_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_7825_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xa: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),Xa)) = aa(product_prod(A,C),A,product_fst(A,C),Xa) ).

% fst_apsnd
tff(fact_7826_img__fst,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S2)
     => member(A,A2,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S2)) ) ).

% img_fst
tff(fact_7827_fst__divmod__nat,axiom,
    ! [M: nat,Na: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Na) ).

% fst_divmod_nat
tff(fact_7828_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_7829_sorted__wrt__map__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_aej(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L)) ) ) ).

% sorted_wrt_map_linord
tff(fact_7830_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ep(B,fun(A,product_prod(A,B))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_7831_sorted__wrt__map__rev__linord,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [L: list(product_prod(A,B))] :
          ( sorted_wrt(product_prod(A,B),aTP_Lamp_aek(product_prod(A,B),fun(product_prod(A,B),$o)),L)
        <=> sorted_wrt(A,ord_less_eq(A),rev(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))) ) ) ).

% sorted_wrt_map_rev_linord
tff(fact_7832_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_div_numeral
tff(fact_7833_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_fst_mk_fst
tff(fact_7834_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(A,B)),B5: set(A),Xa: A,Y: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A5)),B5)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),A5)
       => member(A,Xa,B5) ) ) ).

% fst_image_mp
tff(fact_7835_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,Xa: A,S2: set(product_prod(A,B))] :
      ( member(A,Xa,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S2))
     => ~ ! [Y3: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y3),S2) ) ).

% in_fst_imageE
tff(fact_7836_inj__on__fst__map__to__set,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),map_to_set(A,B,M)) ).

% inj_on_fst_map_to_set
tff(fact_7837_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ej(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_7838_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,Na)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) ) ).

% fst_divmod
tff(fact_7839_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_cg(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_7840_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(A,C)] : aTP_Lamp_ael(fun(A,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_aem(fun(A,C),fun(A,fun(B,C)),F3)) ).

% fn_fst_conv
tff(fact_7841_fstE,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(A,B),A2: A,B2: B,P2: fun(A,$o)] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
     => ( aa(A,$o,P2,aa(product_prod(A,B),A,product_fst(A,B),Xa))
       => aa(A,$o,P2,A2) ) ) ).

% fstE
tff(fact_7842_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xa: A,Y: B,A2: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)) = A2 )
     => ( Xa = A2 ) ) ).

% fst_eqD
tff(fact_7843_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X2: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X2)) = X1 ).

% fst_conv
tff(fact_7844_distinct__map__fst__filterI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),P2: fun(product_prod(A,B),$o)] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),filter2(product_prod(A,B),P2,Xs))) ) ).

% distinct_map_fst_filterI
tff(fact_7845_fstI,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(A,B),Y: A,Z: B] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),Xa) = Y ) ) ).

% fstI
tff(fact_7846_sorted__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Na,Xs))) ).

% sorted_enumerate
tff(fact_7847_fst__foldl,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(A,fun(C,A)),G3: fun(A,fun(B,fun(C,B))),A2: A,B2: B,Xs: list(C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(list(C),product_prod(A,B),aa(product_prod(A,B),fun(list(C),product_prod(A,B)),foldl(product_prod(A,B),C,aa(fun(A,fun(B,fun(C,product_prod(A,B)))),fun(product_prod(A,B),fun(C,product_prod(A,B))),product_case_prod(A,B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_aen(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),F3),G3))),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),Xs)) = aa(list(C),A,aa(A,fun(list(C),A),foldl(A,C,F3),A2),Xs) ).

% fst_foldl
tff(fact_7848_set__to__map__inverse,axiom,
    ! [B: $tType,A: $tType,S2: set(product_prod(A,B))] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S2)
     => ( map_to_set(A,B,set_to_map(A,B,S2)) = S2 ) ) ).

% set_to_map_inverse
tff(fact_7849_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))
     => ( 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))
       => ( 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_7850_distinct__map__fstD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(A,B)),Xa: A,Y: B,Z: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Z),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xs))
         => ( Y = Z ) ) ) ) ).

% distinct_map_fstD
tff(fact_7851_set__to__map__simp,axiom,
    ! [B: $tType,A: $tType,S2: set(product_prod(A,B)),K: A,V3: B] :
      ( inj_on(product_prod(A,B),A,product_fst(A,B),S2)
     => ( ( aa(A,option(B),set_to_map(A,B,S2),K) = aa(B,option(B),some(B),V3) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V3),S2) ) ) ).

% set_to_map_simp
tff(fact_7852_map__of__distinct__upd4,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),Ys))),Xa,none(B)) ) ) ) ).

% map_of_distinct_upd4
tff(fact_7853_map__of__distinct__upd3,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B,Y7: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y7)),Ys))),Xa,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd3
tff(fact_7854_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L) ).

% fst_divmod_integer
tff(fact_7855_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_abs(K,L)) = 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)) ).

% fst_divmod_abs
tff(fact_7856_map__of__rev__distinct,axiom,
    ! [B: $tType,A: $tType,M: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),M))
     => ( map_of(A,B,rev(product_prod(A,B),M)) = map_of(A,B,M) ) ) ).

% map_of_rev_distinct
tff(fact_7857_map__of__is__SomeI,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),Xa: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys))
       => ( aa(A,option(B),map_of(A,B,Xys),Xa) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_is_SomeI
tff(fact_7858_Some__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y: B,Xa: A] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(B,option(B),some(B),Y) = aa(A,option(B),map_of(A,B,Xys),Xa) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% Some_eq_map_of_iff
tff(fact_7859_map__of__eq__Some__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Xa: A,Y: B] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
     => ( ( aa(A,option(B),map_of(A,B,Xys),Xa) = aa(B,option(B),some(B),Y) )
      <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Xys)) ) ) ).

% map_of_eq_Some_iff
tff(fact_7860_map__of__None__filterD,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),Xa: B,P2: fun(product_prod(B,A),$o)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),Xa) = none(A) )
     => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P2,Xs)),Xa) = none(A) ) ) ).

% map_of_None_filterD
tff(fact_7861_map__of__SomeD,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y) )
     => member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs)) ) ).

% map_of_SomeD
tff(fact_7862_weak__map__of__SomeI,axiom,
    ! [A: $tType,B: $tType,K: A,Xa: B,L: list(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),Xa),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L))
     => ? [X: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X) ) ).

% weak_map_of_SomeI
tff(fact_7863_map__of__Cons__code_I2_J,axiom,
    ! [A: $tType,B: $tType,L: B,V3: A,Ps2: list(product_prod(B,A)),K: B] :
      aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),L),V3)),Ps2)),K) = $ite(L = K,aa(A,option(A),some(A),V3),aa(B,option(A),map_of(B,A,Ps2),K)) ).

% map_of_Cons_code(2)
tff(fact_7864_map__of__Some__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V3: A] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V3) )
     => ? [Ys5: list(product_prod(B,A)),Zs2: list(product_prod(B,A))] :
          ( ( Xs = aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(list(product_prod(B,A)),fun(list(product_prod(B,A)),list(product_prod(B,A))),append(product_prod(B,A)),Ys5),aa(list(product_prod(B,A)),list(product_prod(B,A)),aa(product_prod(B,A),fun(list(product_prod(B,A)),list(product_prod(B,A))),cons(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V3)),Zs2)) )
          & ( aa(B,option(A),map_of(B,A,Ys5),K) = none(A) ) ) ) ).

% map_of_Some_split
tff(fact_7865_map__of__map__to__set,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B)),M: fun(A,option(B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
     => ( ( map_of(A,B,L) = M )
      <=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) = map_to_set(A,B,M) ) ) ) ).

% map_of_map_to_set
tff(fact_7866_map__to__set__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),L))
     => ( map_to_set(A,B,map_of(A,B,L)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),L) ) ) ).

% map_to_set_map_of
tff(fact_7867_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_adz(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_7868_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),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_7869_map__of__Some__filter__not__in,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,V3: A,P2: fun(product_prod(B,A),$o)] :
      ( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),V3) )
     => ( ~ aa(product_prod(B,A),$o,P2,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V3))
       => ( distinct(B,aa(list(product_prod(B,A)),list(B),map(product_prod(B,A),B,product_fst(B,A)),Xs))
         => ( aa(B,option(A),map_of(B,A,filter2(product_prod(B,A),P2,Xs)),K) = none(A) ) ) ) ) ).

% map_of_Some_filter_not_in
tff(fact_7870_map__of__mapk__SomeI,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),T2: list(product_prod(A,C)),K: A,Xa: C] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(A,option(C),map_of(A,C,T2),K) = aa(C,option(C),some(C),Xa) )
       => ( 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_aeo(fun(A,B),fun(A,fun(C,product_prod(B,C))),F3))),T2)),aa(A,B,F3,K)) = aa(C,option(C),some(C),Xa) ) ) ) ).

% map_of_mapk_SomeI
tff(fact_7871_map__of__distinct__lookup,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( aa(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),Ys))),Xa) = aa(B,option(B),some(B),Y) ) ) ) ).

% map_of_distinct_lookup
tff(fact_7872_map__of__distinct__upd2,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)))
       => ( map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y)),Ys))) = fun_upd(A,option(B),map_of(A,B,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),Xs),Ys)),Xa,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_of_distinct_upd2
tff(fact_7873_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_7874_bezw_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      bezw(Xa,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Y)))))) ).

% bezw.simps
tff(fact_7875_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xa: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F3),Xa)) = aa(C,A,F3,aa(product_prod(B,C),C,product_snd(B,C),Xa)) ).

% snd_apsnd
tff(fact_7876_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),Xa: product_prod(A,C),G3: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F3),Xa) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G3),Xa) )
    <=> ( aa(C,B,F3,aa(product_prod(A,C),C,product_snd(A,C),Xa)) = aa(C,B,G3,aa(product_prod(A,C),C,product_snd(A,C),Xa)) ) ) ).

% apsnd_eq_conv
tff(fact_7877_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_7878_img__snd,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S2)
     => member(B,B2,aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S2)) ) ).

% img_snd
tff(fact_7879_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_7880_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(B,A)),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K)),L)) = L ).

% map_snd_mk_fst
tff(fact_7881_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F3)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),comp(B,C,product_prod(A,B),F3),product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_7882_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Na)) ) ).

% one_mod_numeral
tff(fact_7883_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_ady(A,fun(B,product_prod(B,A))),K)),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_snd_mk_snd
tff(fact_7884_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F3: fun(D,fun(B,C)),G3: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_aep(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F3),G3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_aeq(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F3),G3)) ).

% split_comp_eq
tff(fact_7885_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(A,fun(B,C)),X3: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F3),X3) = aa(B,C,aa(A,fun(B,C),F3,aa(product_prod(A,B),A,product_fst(A,B),X3)),aa(product_prod(A,B),B,product_snd(A,B),X3)) ).

% case_prod_beta'
tff(fact_7886_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa2: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X3),Xa2) = aa(B,C,aa(A,fun(B,C),X3,aa(product_prod(A,B),A,product_fst(A,B),Xa2)),aa(product_prod(A,B),B,product_snd(A,B),Xa2)) ).

% case_prod_unfold
tff(fact_7887_split__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) ).

% split_beta
tff(fact_7888_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,fun(C,A)),P3: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),P3) = aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),P3)),aa(product_prod(B,C),C,product_snd(B,C),P3)) ).

% case_prod_beta
tff(fact_7889_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,Xa: product_prod(A,B),A5: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),Xa,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A5)))
     => aa(B,$o,aa(A,fun(B,$o),A5,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_7890_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P2,aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_7891_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P2: fun(A,$o),F3: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F3),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P2,aa(C,A,aa(B,fun(C,A),F3,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_7892_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_7893_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_7894_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Xa: A,Y: B,A2: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P2,Xa),Y)
     => ( ( A2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Y) )
       => aa(B,$o,aa(A,fun(B,$o),P2,aa(product_prod(A,B),A,product_fst(A,B),A2)),aa(product_prod(A,B),B,product_snd(A,B),A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_7895_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_7896_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P3: product_prod(A,B),Q3: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P3) = aa(product_prod(A,B),A,product_fst(A,B),Q3) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P3) = aa(product_prod(A,B),B,product_snd(A,B),Q3) )
       => ( P3 = Q3 ) ) ) ).

% prod_eqI
tff(fact_7897_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,Sa: product_prod(A,B),T2: product_prod(A,B)] :
      ( ( Sa = T2 )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),Sa) = aa(product_prod(A,B),A,product_fst(A,B),T2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Sa) = aa(product_prod(A,B),B,product_snd(A,B),T2) ) ) ) ).

% prod_eq_iff
tff(fact_7898_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o))] :
      ( ? [A7: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),P2,A7),X_13)
    <=> ? [Z2: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P2,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2)) ) ).

% Ex_prod_contract
tff(fact_7899_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o))] :
      ( ! [A7: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),P2,A7),X_13)
    <=> ! [Z2: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P2,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2)) ) ).

% All_prod_contract
tff(fact_7900_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,C)] : aTP_Lamp_aer(fun(B,C),fun(product_prod(A,B),C),F3) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_aes(fun(B,C),fun(A,fun(B,C)),F3)) ).

% fn_snd_conv
tff(fact_7901_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,aa(fun(B,fun(A,A)),fun(product_prod(B,A),A),product_case_prod(B,A,A),aTP_Lamp_aet(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_7902_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A5: set(product_prod(B,A)),B5: set(A),Xa: B,Y: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),A5)),B5)
     => ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y),A5)
       => member(A,Y,B5) ) ) ).

% snd_image_mp
tff(fact_7903_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S2: set(product_prod(B,A))] :
      ( member(A,Y,aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),S2))
     => ~ ! [X: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),S2) ) ).

% in_snd_imageE
tff(fact_7904_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,Na)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) ) ).

% snd_divmod
tff(fact_7905_sndE,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),A2: A,B2: B,P2: fun(B,$o)] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
     => ( aa(B,$o,P2,aa(product_prod(A,B),B,product_snd(A,B),Xa))
       => aa(B,$o,P2,B2) ) ) ).

% sndE
tff(fact_7906_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xa: B,Y: A,A2: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_7907_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X2: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X2)) = X2 ).

% snd_conv
tff(fact_7908_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_7909_sndI,axiom,
    ! [A: $tType,B: $tType,Xa: product_prod(A,B),Y: A,Z: B] :
      ( ( Xa = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),Xa) = Z ) ) ).

% sndI
tff(fact_7910_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,fun(B,C)),S2: set(C)] : aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_aeu(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),F3),S2))) = aa(fun(A,$o),set(A),collect(A),aa(set(C),fun(A,$o),aTP_Lamp_aev(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),F3),S2)) ).

% fs_contract
tff(fact_7911_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F3: fun(D,fun(C,A)),G3: fun(B,D),Xa: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),comp(D,fun(C,A),B,F3),G3)),Xa) = aa(C,A,aa(D,fun(C,A),F3,aa(B,D,G3,aa(product_prod(B,C),B,product_fst(B,C),Xa))),aa(product_prod(B,C),C,product_snd(B,C),Xa)) ).

% case_prod_comp
tff(fact_7912_map__to__set__ran,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),map_to_set(B,A,M)) ).

% map_to_set_ran
tff(fact_7913_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_aew(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_7914_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o))] : the(product_prod(A,B),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2)) = the(product_prod(A,B),aTP_Lamp_aex(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P2)) ).

% The_case_prod
tff(fact_7915_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X4: set(A),A5: set(product_prod(A,B)),Y6: set(B),P2: fun(A,fun(B,$o)),Q2: fun(A,fun(B,$o))] :
      ( ( X4 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A5) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A5) )
       => ( ! [X: A] :
              ( member(A,X,X4)
             => ! [Xa5: B] :
                  ( member(B,Xa5,Y6)
                 => ( aa(B,$o,aa(A,fun(B,$o),P2,X),Xa5)
                   => aa(B,$o,aa(A,fun(B,$o),Q2,X),Xa5) ) ) )
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P2)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q2))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_7916_set__to__map__ran,axiom,
    ! [A: $tType,B: $tType,S2: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,set_to_map(B,A,S2))),aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),S2)) ).

% set_to_map_ran
tff(fact_7917_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_ep(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_7918_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ady(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_7919_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_ej(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_7920_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_aew(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_7921_rel__fun__Collect__case__prodD,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,A5: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o)),F3: fun(A,C),G3: fun(B,D),X4: set(product_prod(A,B)),Xa: product_prod(A,B)] :
      ( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A5,B5),F3),G3)
     => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),X4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A5)))
       => ( member(product_prod(A,B),Xa,X4)
         => aa(D,$o,aa(C,fun(D,$o),B5,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),comp(A,C,product_prod(A,B),F3),product_fst(A,B)),Xa)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),comp(B,D,product_prod(A,B),G3),product_snd(A,B)),Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_7922_ran__map__of,axiom,
    ! [A: $tType,B: $tType,Xs: list(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ran(B,A,map_of(B,A,Xs))),aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set2(product_prod(B,A)),Xs))) ).

% ran_map_of
tff(fact_7923_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R12: fun(A,fun(B,$o)),R23: fun(C,fun(D,$o)),P2: fun(A,fun(C,$o)),Q2: fun(B,fun(D,$o)),A2: product_prod(A,B),A5: set(product_prod(A,B)),B2: product_prod(C,D),B5: set(product_prod(C,D))] :
      ( aa(fun(B,fun(D,$o)),$o,aa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),$o),bNF_rel_fun(A,B,fun(C,$o),fun(D,$o),R12,bNF_rel_fun(C,D,$o,$o,R23,fequal($o))),P2),Q2)
     => ( member(product_prod(A,B),A2,A5)
       => ( member(product_prod(C,D),B2,B5)
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A5),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R12)))
           => ( aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),B5),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),R23)))
             => ( aa(C,$o,aa(A,fun(C,$o),P2,aa(product_prod(A,B),A,product_fst(A,B),A2)),aa(product_prod(C,D),C,product_fst(C,D),B2))
              <=> aa(D,$o,aa(B,fun(D,$o),Q2,aa(product_prod(A,B),B,product_snd(A,B),A2)),aa(product_prod(C,D),D,product_snd(C,D),B2)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_7924_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( member(product_prod(A,B),P3,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))
    <=> ? [N5: nat] :
          ( ( aa(nat,A,nth(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
          & ( aa(nat,B,nth(B,Ys),N5) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys)) ) ) ).

% in_set_zip
tff(fact_7925_set__to__map__insert,axiom,
    ! [B: $tType,A: $tType,Kv: product_prod(A,B),S2: set(product_prod(A,B))] :
      ( ~ member(A,aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S2))
     => ( set_to_map(A,B,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert(product_prod(A,B)),Kv),S2)) = fun_upd(A,option(B),set_to_map(A,B,S2),aa(product_prod(A,B),A,product_fst(A,B),Kv),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),Kv))) ) ) ).

% set_to_map_insert
tff(fact_7926_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(B,product_prod(A,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image(B,A,aa(fun(B,product_prod(A,C)),fun(B,A),comp(product_prod(A,C),A,B,product_fst(A,C)),F3)),top_top(set(B))))
     => ( aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image(B,C,aa(fun(B,product_prod(A,C)),fun(B,C),comp(product_prod(A,C),C,B,product_snd(A,C)),F3)),top_top(set(B))))
       => aa(set(product_prod(A,C)),$o,finite_finite2(product_prod(A,C)),aa(set(B),set(product_prod(A,C)),image(B,product_prod(A,C),F3),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_7927_Misc_Oran__distinct,axiom,
    ! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
      ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
     => ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Al)) ) ) ).

% Misc.ran_distinct
tff(fact_7928_bezw__non__0,axiom,
    ! [Y: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Y)
     => ( bezw(Xa,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Y))))) ) ) ).

% bezw_non_0
tff(fact_7929_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xa),Xaa)))))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_7930_fun_Oin__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R3: fun(B,fun(C,$o)),A2: fun(A,B),B2: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R3),A2),B2)
    <=> ? [Z2: fun(A,product_prod(B,C))] :
          ( member(fun(A,product_prod(B,C)),Z2,aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_aey(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R3)))
          & ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),Z2) = A2 )
          & ( aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),Z2) = B2 ) ) ) ).

% fun.in_rel
tff(fact_7931_snd__divmod__nat,axiom,
    ! [M: nat,Na: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Na) ).

% snd_divmod_nat
tff(fact_7932_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L) ).

% snd_divmod_integer
tff(fact_7933_mod__pure,axiom,
    ! [B2: $o,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(pure_assn((B2))),H2)
    <=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) )
        & (B2) ) ) ).

% mod_pure
tff(fact_7934_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% snd_divmod_abs
tff(fact_7935_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),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_7936_mod__emp,axiom,
    ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(one_one(assn)),H2)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2) = bot_bot(set(nat)) ) ) ).

% mod_emp
tff(fact_7937_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P2: fun(nat,$o),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_aez(fun(nat,$o),fun(product_prod(A,nat),$o),P2),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_afa(fun(nat,$o),fun(product_prod(A,nat),$o),P2),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_7938_mod__star__trueE_H,axiom,
    ! [P2: assn,H2: product_prod(heap_ext(product_unit),set(nat))] :
      ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),top_top(assn))),H2)
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] :
            ( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H5) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H2) )
           => ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H5)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H2))
             => ~ aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),H5) ) ) ) ).

% mod_star_trueE'
tff(fact_7939_nths__shift__lemma,axiom,
    ! [A: $tType,A5: set(nat),Xs: list(A),I: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afb(set(nat),fun(product_prod(A,nat),$o),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),$o),aTP_Lamp_afc(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A5),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7940_fun_Orel__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType,R3: fun(A,fun(B,$o)),Ra2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R3),Ra2)
     => aa(fun(fun(C,A),fun(fun(C,B),$o)),$o,aa(fun(fun(C,A),fun(fun(C,B),$o)),fun(fun(fun(C,A),fun(fun(C,B),$o)),$o),ord_less_eq(fun(fun(C,A),fun(fun(C,B),$o))),bNF_rel_fun(C,C,A,B,fequal(C),R3)),bNF_rel_fun(C,C,A,B,fequal(C),Ra2)) ) ).

% fun.rel_mono
tff(fact_7941_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),Na: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P3,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Na,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(product_prod(nat,A),nat,product_fst(nat,A),P3))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),Na)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7942_one__mod__minus__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na)))) ).

% one_mod_minus_numeral
tff(fact_7943_minus__one__mod__numeral,axiom,
    ! [Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Na))) ).

% minus_one_mod_numeral
tff(fact_7944_numeral__mod__minus__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na)))) ).

% numeral_mod_minus_numeral
tff(fact_7945_minus__numeral__mod__numeral,axiom,
    ! [M: num,Na: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)) = adjust_mod(aa(num,int,numeral_numeral(int),Na),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Na))) ).

% minus_numeral_mod_numeral
tff(fact_7946_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      adjust_mod(L,R2) = $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),L),R2)) ).

% Divides.adjust_mod_def
tff(fact_7947_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G3: fun(B,nat),P3: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G3,P3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(product_prod(A,B),A,product_fst(A,B),P3))),aa(B,nat,G3,aa(product_prod(A,B),B,product_snd(A,B),P3)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_7948_fun__mono,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,C4: fun(A,fun(B,$o)),A5: fun(A,fun(B,$o)),B5: fun(C,fun(D,$o)),D5: fun(C,fun(D,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),C4),A5)
     => ( aa(fun(C,fun(D,$o)),$o,aa(fun(C,fun(D,$o)),fun(fun(C,fun(D,$o)),$o),ord_less_eq(fun(C,fun(D,$o))),B5),D5)
       => aa(fun(fun(A,C),fun(fun(B,D),$o)),$o,aa(fun(fun(A,C),fun(fun(B,D),$o)),fun(fun(fun(A,C),fun(fun(B,D),$o)),$o),ord_less_eq(fun(fun(A,C),fun(fun(B,D),$o))),bNF_rel_fun(A,B,C,D,A5,B5)),bNF_rel_fun(A,B,C,D,C4,D5)) ) ) ).

% fun_mono
tff(fact_7949_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat)] : nths(A,Xs,A5) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_afb(set(nat),fun(product_prod(A,nat),$o),A5),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7950_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P2: fun(A,fun(B,$o)),Y: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),P2,Y),Xa)
     => aa(B,$o,aa(A,fun(B,$o),P2,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xa),Y))) ) ).

% exI_realizer
tff(fact_7951_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_7952_nths__singleton,axiom,
    ! [A: $tType,Xa: A,A5: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),A5) = $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),nil(A)) ).

% nths_singleton
tff(fact_7953_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I5) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_7954_drop__eq__nths,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : drop(A,Na,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),Na))) ).

% drop_eq_nths
tff(fact_7955_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_7956_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_7957_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_afd(list(A),fun(set(nat),fun(nat,$o)),Xs),I5))) ).

% length_nths
tff(fact_7958_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A5: set(nat),B5: set(nat)] : nths(A,nths(A,Xs,A5),B5) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_aff(set(nat),fun(set(nat),fun(nat,$o)),A5),B5))) ).

% nths_nths
tff(fact_7959_filter__eq__nths,axiom,
    ! [A: $tType,P2: fun(A,$o),Xs: list(A)] : filter2(A,P2,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_mu(fun(A,$o),fun(list(A),fun(nat,$o)),P2),Xs))) ).

% filter_eq_nths
tff(fact_7960_nths__Cons,axiom,
    ! [A: $tType,Xa: A,L: list(A),A5: set(nat)] :
      nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),L),A5) = aa(list(A),list(A),
        aa(list(A),fun(list(A),list(A)),append(A),
          $ite(member(nat,zero_zero(nat),A5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),nil(A))),
        nths(A,L,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_afg(set(nat),fun(nat,$o),A5)))) ).

% nths_Cons
tff(fact_7961_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_afh(list(A),fun(set(nat),fun(A,$o)),Xs),I5)) ).

% set_nths
tff(fact_7962_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P2: fun(A,$o),P3: A,Q2: fun(B,$o),Q3: B] :
      ( aa(A,$o,P2,P3)
     => ( aa(B,$o,Q2,Q3)
       => ( aa(A,$o,P2,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3)))
          & aa(B,$o,Q2,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P3),Q3))) ) ) ) ).

% conjI_realizer
tff(fact_7963_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),Xa: product_prod(C,A),X4: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,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)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),Xa),bot_bot(set(product_prod(C,A)))),S2)),X4) = 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_afi(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xa)),X4,S2) ) ) ).

% insert_relcomp_union_fold
tff(fact_7964_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs12)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_7965_trancl__Int__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Sa: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),Sa)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R2)),Sa),R2)),Sa)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),Sa) ) ) ).

% trancl_Int_subset
tff(fact_7966_relcomp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: set(product_prod(A,B)),R2: set(product_prod(A,B)),S: set(product_prod(B,C)),Sa: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R4),R2)
     => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S),Sa)
       => aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R4,S)),relcomp(A,B,C,R2,Sa)) ) ) ).

% relcomp_mono
tff(fact_7967_max__ext__compat,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S2)),R3)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R3),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))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R3)) ) ).

% max_ext_compat
tff(fact_7968_min__ext__compat,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S2)),R3)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R3),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))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R3)) ) ).

% min_ext_compat
tff(fact_7969_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R2: set(product_prod(A,C)),Sa: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),C3),relcomp(A,C,B,R2,Sa))
     => ~ ! [B4: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),B4),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3),Sa) ) ) ).

% relcompEpair
tff(fact_7970_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),Sa: set(product_prod(C,B))] :
      ( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,Sa))
     => ~ ! [X: A,Y3: C,Z3: B] :
            ( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z3) )
           => ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y3),R2)
             => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y3),Z3),Sa) ) ) ) ).

% relcompE
tff(fact_7971_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set(product_prod(A,B)),C3: C,Sa: set(product_prod(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),R2)
     => ( member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3),Sa)
       => member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A2),C3),relcomp(A,B,C,R2,Sa)) ) ) ).

% relcomp.relcompI
tff(fact_7972_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A12: A,A23: B,R2: set(product_prod(A,C)),Sa: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),A23),relcomp(A,C,B,R2,Sa))
    <=> ? [A7: A,B7: C,C7: B] :
          ( ( A12 = A7 )
          & ( A23 = C7 )
          & member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),B7),R2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B7),C7),Sa) ) ) ).

% relcomp.simps
tff(fact_7973_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A12: A,A23: B,R2: set(product_prod(A,C)),Sa: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A12),A23),relcomp(A,C,B,R2,Sa))
     => ~ ! [B4: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A12),B4),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),A23),Sa) ) ) ).

% relcomp.cases
tff(fact_7974_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),nil(A)) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23) ).

% mergesort_by_rel_split.simps(1)
tff(fact_7975_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X1: A,X2: A,Xs: list(A)] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs23)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_7976_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),Sa: set(product_prod(C,B))] : relcomp(A,C,B,R2,Sa) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_afj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),Sa))) ).

% relcomp_unfold
tff(fact_7977_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xa,Xaa) = Y )
     => ( ! [Xs1: list(A),Xs22: list(A)] :
            ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
           => ( ( Xaa = nil(A) )
             => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) ) ) )
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
             => ! [X: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
                 => ( Y != aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(A),Xs22: list(A)] :
                ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
               => ! [X15: A,X23: A,Xs2: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)) )
                   => ( Y != merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs2) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_7978_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xa: A] : merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs12)),Xs23) ).

% mergesort_by_rel_split.simps(2)
tff(fact_7979_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(A,B)),S2: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R3)
     => ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S2)
       => ( relcomp(A,B,C,R3,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_afl(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))),R3) ) ) ) ).

% relcomp_fold
tff(fact_7980_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_afn(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_7981_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S2: set(product_prod(A,B)),Xa: product_prod(C,A),R3: set(product_prod(C,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S2)
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert(product_prod(C,A)),Xa),R3),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_afi(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xa)),relcomp(C,A,B,R3,S2),S2) ) ) ).

% insert_relcomp_fold
tff(fact_7982_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A12: list(A),P2: fun(fun(A,fun(A,$o)),fun(list(A),$o))] :
      ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),A0),A12))
     => ( ! [R7: fun(A,fun(A,$o)),Xs2: list(A)] :
            ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R7),Xs2))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P2,R7),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2))) )
             => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
                 => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P2,R7),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2))) )
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P2,R7),Xs2) ) ) )
       => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P2,A0),A12) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_7983_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,Xa: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xa,Xaa) = Y )
     => ( aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xa),Xaa))
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
             => ( ( Xaa = nil(A) )
               => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
                 => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),nil(A))) ) ) )
         => ( ! [Xs1: list(A),Xs22: list(A)] :
                ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
               => ! [X: A] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
                   => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1)),Xs22) )
                     => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) ) )
           => ~ ! [Xs1: list(A),Xs22: list(A)] :
                  ( ( Xa = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
                 => ! [X15: A,X23: A,Xs2: list(A)] :
                      ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)) )
                     => ( ( Y = merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs22)),Xs2) )
                       => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs2)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_7984_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4081349890594273596sample(A,A_of_integer,Zero,I) = $ite(
            aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),I),zero_zero(code_integer)),
            undefined(A),
            $ite(
              I = zero_zero(code_integer),
              Zero,
              $ite(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) = zero_zero(code_integer),aa(product_prod(A,A),A,product_snd(A,A),aa(code_integer,product_prod(A,A),A_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))),aa(product_prod(A,A),A,product_fst(A,A),aa(code_integer,product_prod(A,A),A_of_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),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),one_one(code_integer))))) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_7985_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      normalize(P3) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
        $let(
          a3: int,
          a3:= aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a3)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a3: int,
            a3:= aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),aa(product_prod(int,int),int,product_snd(int,int),P3))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P3)),a3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P3)),a3)) ) ) ) ).

% normalize_def
tff(fact_7986_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & ( B2 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_7987_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),Na))
    <=> ( ( M != zero_zero(int) )
        | ( Na != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_7988_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,B2: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2) ) ).

% Gcd_2
tff(fact_7989_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2) ) ).

% gcd_le1_int
tff(fact_7990_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2) ) ).

% gcd_le2_int
tff(fact_7991_gcd__cases__int,axiom,
    ! [Xa: int,Y: int,P2: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
             => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xa)),Y)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),zero_zero(int))
               => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xa)),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P2,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_7992_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B5: set(A),A5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),semiring_gcd_Gcd_fin(A,B5)),semiring_gcd_Gcd_fin(A,A5)) = semiring_gcd_Gcd_fin(A,A5) ) ) ) ).

% Gcd_fin.subset
tff(fact_7993_gcd__ge__0__int,axiom,
    ! [Xa: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Y)) ).

% gcd_ge_0_int
tff(fact_7994_gcd__unique__int,axiom,
    ! [D2: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),A2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),B2)
        & ! [E4: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E4),A2)
              & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E4),B2) )
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E4),D2) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2) ) ) ).

% gcd_unique_int
tff(fact_7995_gcd__non__0__int,axiom,
    ! [Y: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Y)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xa),Y)) ) ) ).

% gcd_non_0_int
tff(fact_7996_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A5: set(A)] :
          ( member(A,A2,A5)
         => ( semiring_gcd_Gcd_fin(A,A5) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7997_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,A5: set(A)] : semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A5)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),semiring_gcd_Gcd_fin(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7998_Func__empty,axiom,
    ! [B: $tType,A: $tType,B5: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B5) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert(fun(A,B)),aTP_Lamp_afo(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_7999_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A5: set(A)] :
          semiring_gcd_Gcd_fin(A,A5) = $ite(aa(set(A),$o,finite_finite2(A),A5),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A5),one_one(A)) ) ).

% Gcd_fin.eq_fold
tff(fact_8000_mod__h__bot__normalize,axiom,
    ! [A: $tType,H2: heap_ext(product_unit),P2: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H2)
     => ( aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),bot_bot(set(nat))))
      <=> aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(P2),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_normalize
tff(fact_8001_integer__set__bit__code,axiom,
    ! [Xa: code_integer,Na: code_integer,B2: $o] :
      bits_integer_set_bit(Xa,Na,(B2)) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer)),
        aa($o,code_integer,aa(code_integer,fun($o,code_integer),aa(code_integer,fun(code_integer,fun($o,code_integer)),undefined(fun(code_integer,fun(code_integer,fun($o,code_integer)))),Xa),Na),(B2)),
        $ite((B2),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xa),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),code_nat_of_integer(Na)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),Xa),aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),code_nat_of_integer(Na)),one_one(code_integer))))) ) ).

% integer_set_bit_code
tff(fact_8002_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_8003_gcd__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A2) = A2 ).

% gcd_nat.left_neutral
tff(fact_8004_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_8005_gcd__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),zero_zero(nat)) = A2 ).

% gcd_nat.right_neutral
tff(fact_8006_gcd__0__nat,axiom,
    ! [Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),zero_zero(nat)) = Xa ).

% gcd_0_nat
tff(fact_8007_gcd__0__left__nat,axiom,
    ! [Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),Xa) = Xa ).

% gcd_0_left_nat
tff(fact_8008_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_8009_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na))
    <=> ( ( M != zero_zero(nat) )
        | ( Na != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_8010_Gcd__in,axiom,
    ! [A5: set(nat)] :
      ( ! [A4: nat,B4: nat] :
          ( member(nat,A4,A5)
         => ( member(nat,B4,A5)
           => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A4),B4),A5) ) )
     => ( ( A5 != bot_bot(set(nat)) )
       => member(nat,gcd_Gcd(nat,A5),A5) ) ) ).

% Gcd_in
tff(fact_8011_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X: nat,Y3: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X))
        & ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y3)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_8012_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_8013_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2) ) ).

% gcd_le1_nat
tff(fact_8014_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),Na)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_8015_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_8016_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X: nat,Y3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),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)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_8017_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_8018_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Y: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Y) = $ite(Y = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y))) ).

% gcd_nat.simps
tff(fact_8019_gcd__non__0__nat,axiom,
    ! [Y: nat,Xa: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_8020_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_oa(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_8021_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_afp(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_8022_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xa),Xaa))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_8023_uint32__shiftl__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      uint32_shiftl(Xa,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4730199178511100633sh_bit(uint32)),Xa),Na),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),code_nat_of_integer(Na)),Xa) ) ).

% uint32_shiftl_def
tff(fact_8024_uint32__shiftr__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      uint32_shiftr(Xa,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4197421643247451524op_bit(uint32)),Xa),Na),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),code_nat_of_integer(Na)),Xa) ) ).

% uint32_shiftr_def
tff(fact_8025_uint32__test__bit__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      ( uint32_test_bit(Xa,Na)
    <=> $ite(
          ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
          | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),Xa),Na),
          aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xa),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_def
tff(fact_8026_integer__set__bit__def,axiom,
    ! [Xa: code_integer,Na: code_integer,B2: $o] :
      bits_integer_set_bit(Xa,Na,(B2)) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer)),aa($o,code_integer,aa(code_integer,fun($o,code_integer),aa(code_integer,fun(code_integer,fun($o,code_integer)),undefined(fun(code_integer,fun(code_integer,fun($o,code_integer)))),Xa),Na),(B2)),aa($o,code_integer,aa(nat,fun($o,code_integer),aa(code_integer,fun(nat,fun($o,code_integer)),generi7602027413899671122et_bit(code_integer),Xa),code_nat_of_integer(Na)),(B2))) ).

% integer_set_bit_def
tff(fact_8027_uint32__shiftr__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      rep_uint32(uint32_shiftr(W2,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint32(aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4197421643247451524op_bit(uint32)),W2),Na)),
        aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se4197421643247451524op_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),code_nat_of_integer(Na)),rep_uint32(W2)) ) ).

% uint32_shiftr_code
tff(fact_8028_uint32_Oless__iff__word__of,axiom,
    ! [P3: uint32,Q3: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),P3),Q3)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(P3)),rep_uint32(Q3)) ) ).

% uint32.less_iff_word_of
tff(fact_8029_less__uint32_Orep__eq,axiom,
    ! [Xa: uint32,Xaa: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),Xa),Xaa)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(Xa)),rep_uint32(Xaa)) ) ).

% less_uint32.rep_eq
tff(fact_8030_uint32_Oless__eq__iff__word__of,axiom,
    ! [P3: uint32,Q3: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),P3),Q3)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(P3)),rep_uint32(Q3)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_8031_less__eq__uint32_Orep__eq,axiom,
    ! [Xa: uint32,Xaa: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),Xa),Xaa)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(Xa)),rep_uint32(Xaa)) ) ).

% less_eq_uint32.rep_eq
tff(fact_8032_uint32__test__bit__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      ( uint32_test_bit(W2,Na)
    <=> $ite(
          ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
          | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),W2),Na),
          aa(nat,$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,$o),bit_se5641148757651400278ts_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(W2)),code_nat_of_integer(Na)) ) ) ).

% uint32_test_bit_code
tff(fact_8033_uint32__shiftl__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      rep_uint32(uint32_shiftl(W2,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint32(aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4730199178511100633sh_bit(uint32)),W2),Na)),
        aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se4730199178511100633sh_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),code_nat_of_integer(Na)),rep_uint32(W2)) ) ).

% uint32_shiftl_code
tff(fact_8034_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A2: A,M: nat,B2: $o,Na: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,aa(nat,fun($o,A),aa(A,fun(nat,fun($o,A)),generi7602027413899671122et_bit(A),A2),M),(B2))),Na)
        <=> ( $ite(M = Na,(B2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Na))
            & ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ) ).

% bit_set_bit_iff_2n
tff(fact_8035_integer__shiftl__def,axiom,
    ! [Xa: code_integer,Na: code_integer] :
      bits_integer_shiftl(Xa,Na) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),undefined(fun(code_integer,fun(code_integer,code_integer))),Xa),Na),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),code_nat_of_integer(Na)),Xa)) ).

% integer_shiftl_def
tff(fact_8036_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat,Xa: $o,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),Na),(Xa))),M)
        <=> $ite(
              M = Na,
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
              & (Xa) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),M) ) ) ) ).

% test_bit_set_gen
tff(fact_8037_test__bit__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat,Xa: $o] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),Na),(Xa))),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(word(A),nat,size_size(word(A)),W2))
            & (Xa) ) ) ) ).

% test_bit_set
tff(fact_8038_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),M: nat,B2: $o,Na: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),M),(B2))),Na)
        <=> $ite(
              M = Na,
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),type_len0_len_of(A,type2(A)))
              & (B2) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na) ) ) ) ).

% bit_set_bit_word_iff
tff(fact_8039_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat,B2: $o] :
          ( ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),Na),(B2)) = W2 )
        <=> ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W2),Na)
            <=> (B2) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W2)),Na) ) ) ) ).

% word_set_nth_iff
tff(fact_8040_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: word(A),Na: nat,B2: $o] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xa)),Na)
         => ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),Xa),Na),(B2)) = Xa ) ) ) ).

% set_bit_beyond
tff(fact_8041_word__set__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W2),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),Na),$true)) ) ).

% word_set_ge
tff(fact_8042_word__clr__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W2: word(A),Na: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W2),Na),$false)),W2) ) ).

% word_clr_le
tff(fact_8043_less__eq__uint32_Orsp,axiom,
    aa(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),$o,aa(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),fun(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),$o),bNF_rel_fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),fequal(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bNF_rel_fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,$o,fequal(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),fequal($o))),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% less_eq_uint32.rsp
tff(fact_8044_less__uint32_Orsp,axiom,
    aa(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),$o,aa(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),fun(fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o)),$o),bNF_rel_fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),fequal(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bNF_rel_fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,$o,fequal(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),fequal($o))),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))),ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% less_uint32.rsp
tff(fact_8045_uint32__set__bit__code,axiom,
    ! [W2: uint32,Na: code_integer,B2: $o] :
      rep_uint32(uint32_set_bit(W2,Na,(B2))) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
        rep_uint32(aa($o,uint32,aa(code_integer,fun($o,uint32),aa(uint32,fun(code_integer,fun($o,uint32)),aa(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))),undefined(fun(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))))),generi7602027413899671122et_bit(uint32)),W2),Na),(B2))),
        aa($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,fun($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))),generi7602027413899671122et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint32(W2)),code_nat_of_integer(Na)),(B2)) ) ).

% uint32_set_bit_code
tff(fact_8046_uint32__set__bit__def,axiom,
    ! [Xa: uint32,Na: code_integer,B2: $o] :
      uint32_set_bit(Xa,Na,(B2)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit1(bit1(bit1(bit1(one2)))))),Na) ),
        aa($o,uint32,aa(code_integer,fun($o,uint32),aa(uint32,fun(code_integer,fun($o,uint32)),aa(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))),undefined(fun(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))))),generi7602027413899671122et_bit(uint32)),Xa),Na),(B2)),
        aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Xa),code_nat_of_integer(Na)),(B2)) ) ).

% uint32_set_bit_def
tff(fact_8047_integer__shiftr__def,axiom,
    ! [Xa: code_integer,Na: code_integer] :
      bits_integer_shiftr(Xa,Na) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),undefined(fun(code_integer,fun(code_integer,code_integer))),Xa),Na),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),code_nat_of_integer(Na)),Xa)) ).

% integer_shiftr_def
tff(fact_8048_uint32__divmod__code,axiom,
    ! [Xa: uint32,Y: uint32] :
      uint32_divmod(Xa,Y) = $ite(
        aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),Y),
        $ite(aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less(uint32),Xa),Y),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),zero_zero(uint32)),Xa),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),one_one(uint32)),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),minus_minus(uint32),Xa),Y))),
        $ite(
          Y = zero_zero(uint32),
          aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),div0_uint32(Xa)),mod0_uint32(Xa)),
          $let(
            q: uint32,
            q:= aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),one_one(nat)),uint32_sdiv(aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),one_one(nat)),Xa),Y)),
            $let(
              r2: uint32,
              r2:= aa(uint32,uint32,aa(uint32,fun(uint32,uint32),minus_minus(uint32),Xa),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),q),Y)),
              $ite(aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),Y),r2),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),q),one_one(uint32))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),minus_minus(uint32),r2),Y)),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),q),r2)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_8049_uint32_Oset__bits__aux__code,axiom,
    ! [F3: fun(nat,$o),Na: nat,W2: uint32] :
      set_bits_aux_uint32(F3,Na,W2) = $ite(
        Na = zero_zero(nat),
        W2,
        $let(
          n3: nat,
          n3:= aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),one_one(nat)),
          set_bits_aux_uint32(F3,n3,
            aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se1065995026697491101ons_or(uint32),aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),one_one(nat)),W2)),
              $ite(aa(nat,$o,F3,n3),one_one(uint32),zero_zero(uint32)))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_8050_shiftr__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),Na),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftr(Xa,aa(nat,code_integer,code_integer_of_nat,Na)),zero_zero(uint32)) ).

% shiftr_uint32_code
tff(fact_8051_nat__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : code_nat_of_integer(aa(nat,code_integer,code_integer_of_nat,Na)) = Na ).

% nat_of_integer_integer_of_nat
tff(fact_8052_integer__of__nat_Orep__eq,axiom,
    ! [Xa: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,Xa)) = aa(nat,int,semiring_1_of_nat(int),Xa) ).

% integer_of_nat.rep_eq
tff(fact_8053_int__of__integer__integer__of__nat,axiom,
    ! [Na: nat] : aa(code_integer,int,code_int_of_integer,aa(nat,code_integer,code_integer_of_nat,Na)) = aa(nat,int,semiring_1_of_nat(int),Na) ).

% int_of_integer_integer_of_nat
tff(fact_8054_integer__of__nat__0,axiom,
    aa(nat,code_integer,code_integer_of_nat,zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_8055_integer__of__nat__1,axiom,
    aa(nat,code_integer,code_integer_of_nat,one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_8056_integer__of__nat__less__0__conv,axiom,
    ! [Na: nat] : ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),aa(nat,code_integer,code_integer_of_nat,Na)),zero_zero(code_integer)) ).

% integer_of_nat_less_0_conv
tff(fact_8057_integer__of__nat__eq__of__nat,axiom,
    code_integer_of_nat = semiring_1_of_nat(code_integer) ).

% integer_of_nat_eq_of_nat
tff(fact_8058_integer__of__nat_Oabs__eq,axiom,
    ! [Xa: nat] : aa(nat,code_integer,code_integer_of_nat,Xa) = aa(int,code_integer,code_integer_of_int,aa(nat,int,semiring_1_of_nat(int),Xa)) ).

% integer_of_nat.abs_eq
tff(fact_8059_integer__of__nat__numeral,axiom,
    ! [Na: num] : aa(nat,code_integer,code_integer_of_nat,aa(num,nat,numeral_numeral(nat),Na)) = aa(num,code_integer,numeral_numeral(code_integer),Na) ).

% integer_of_nat_numeral
tff(fact_8060_test__bit__uint32__code,axiom,
    ! [Xa: uint32,Na: nat] :
      ( aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xa),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))))
        & uint32_test_bit(Xa,aa(nat,code_integer,code_integer_of_nat,Na)) ) ) ).

% test_bit_uint32_code
tff(fact_8061_set__bit__uint32__code,axiom,
    ! [Xa: uint32,Na: nat,B2: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Xa),Na),(B2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_set_bit(Xa,aa(nat,code_integer,code_integer_of_nat,Na),(B2)),Xa) ).

% set_bit_uint32_code
tff(fact_8062_shiftl__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),Na),Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftl(Xa,aa(nat,code_integer,code_integer_of_nat,Na)),zero_zero(uint32)) ).

% shiftl_uint32_code
tff(fact_8063_sshiftr__uint32__code,axiom,
    ! [Na: nat,Xa: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,Na),Xa) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),
        uint32_sshiftr(Xa,aa(nat,code_integer,code_integer_of_nat,Na)),
        $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xa),aa(num,nat,numeral_numeral(nat),bit1(bit1(bit1(bit1(one2)))))),aa(uint32,uint32,uminus_uminus(uint32),one_one(uint32)),zero_zero(uint32)) ) ).

% sshiftr_uint32_code
tff(fact_8064_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xa),Xaa) = Y )
     => ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),Xa),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xaa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))))) )
           => ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),Xa),Xaa)) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_8065_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o))] : aa(list(A),list(A),mergesort_by_rel(A,R3),nil(A)) = nil(A) ).

% mergesort_by_rel_simps(1)
tff(fact_8066_set__mergesort__by__rel,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R3),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_mergesort_by_rel
tff(fact_8067_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Ys: list(A)] : merges9089515139780605204_merge(A,R3,nil(A),Ys) = Ys ).

% mergesort_by_rel_merge_simps(3)
tff(fact_8068_sorted__wrt__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] :
      ( ! [X: A,Y3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y3)
          | aa(A,$o,aa(A,fun(A,$o),R3,Y3),X) )
     => ( ! [X: A,Y3: A,Z3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y3)
           => ( aa(A,$o,aa(A,fun(A,$o),R3,Y3),Z3)
             => aa(A,$o,aa(A,fun(A,$o),R3,X),Z3) ) )
       => ( sorted_wrt(A,R3,merges9089515139780605204_merge(A,R3,Xs,Ys))
        <=> ( sorted_wrt(A,R3,Xs)
            & sorted_wrt(A,R3,Ys) ) ) ) ) ).

% sorted_wrt_mergesort_by_rel_merge
tff(fact_8069_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xa: A] : aa(list(A),list(A),mergesort_by_rel(A,R3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ).

% mergesort_by_rel_simps(2)
tff(fact_8070_set__mergesort__by__rel__merge,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),merges9089515139780605204_merge(A,R3,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_mergesort_by_rel_merge
tff(fact_8071_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),X1: A,X2: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs))) = aa(product_prod(list(A),list(A)),list(A),aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aTP_Lamp_afq(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),R3)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_8072_sorted__wrt__mergesort__by__rel,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
      ( ! [X: A,Y3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y3)
          | aa(A,$o,aa(A,fun(A,$o),R3,Y3),X) )
     => ( ! [X: A,Y3: A,Z3: A] :
            ( aa(A,$o,aa(A,fun(A,$o),R3,X),Y3)
           => ( aa(A,$o,aa(A,fun(A,$o),R3,Y3),Z3)
             => aa(A,$o,aa(A,fun(A,$o),R3,X),Z3) ) )
       => sorted_wrt(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),Xs)) ) ) ).

% sorted_wrt_mergesort_by_rel
tff(fact_8073_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),V3: A,Va2: list(A)] : merges9089515139780605204_merge(A,R3,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) ).

% mergesort_by_rel_merge.simps(3)
tff(fact_8074_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xa,Xaa,Xb) = Y )
     => ( ! [X: A,Xs2: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) )
           => ! [Y3: A,Ys5: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
               => ( Y != $ite(aa(A,$o,aa(A,fun(A,$o),Xa,X),Y3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,Xa,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2),Ys5))) ) ) )
       => ( ( ( Xb = nil(A) )
           => ( Y != Xaa ) )
         => ~ ( ( Xaa = nil(A) )
             => ! [V: A,Va: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
                 => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
tff(fact_8075_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xa: A,Xs: list(A),Y: A,Ys: list(A)] :
      merges9089515139780605204_merge(A,R3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(A,$o,aa(A,fun(A,$o),R3,Xa),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),merges9089515139780605204_merge(A,R3,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xs),Ys))) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_8076_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] : merges9089515139780605204_merge(A,R3,Xs,nil(A)) = Xs ).

% mergesort_by_rel_merge_simps(2)
tff(fact_8077_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_jo(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_8078_sorted__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),mergesort_by_rel(A,ord_less_eq(A)),Xs)) ) ).

% sorted_mergesort_by_rel
tff(fact_8079_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xa),Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xaa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xa),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xaa))))) ) ) ).

% mergesort_by_rel.elims
tff(fact_8080_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
      aa(list(A),list(A),mergesort_by_rel(A,R3),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ).

% mergesort_by_rel.simps
tff(fact_8081_uint32__sshiftr__def,axiom,
    ! [Xa: uint32,Na: code_integer] :
      uint32_sshiftr(Xa,Na) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        aa(uint32,uint32,aa(code_integer,fun(uint32,uint32),aa(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)))),signed489701013188660438uint32),Na),Xa),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,code_nat_of_integer(Na)),Xa) ) ).

% uint32_sshiftr_def
tff(fact_8082_uint32__sshiftr__code,axiom,
    ! [W2: uint32,Na: code_integer] :
      rep_uint32(uint32_sshiftr(W2,Na)) = $ite(
        ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Na),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Na) ),
        rep_uint32(aa(uint32,uint32,aa(code_integer,fun(uint32,uint32),aa(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)))),signed489701013188660438uint32),Na),W2)),
        aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),code_nat_of_integer(Na)),rep_uint32(W2)) ) ).

% uint32_sshiftr_code
tff(fact_8083_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R3: fun(A,fun(A,$o)),Xs: list(A)] :
      ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R3),Xs))
     => ( aa(list(A),list(A),mergesort_by_rel(A,R3),Xs) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R3,aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R3),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs))))) ) ) ).

% mergesort_by_rel.psimps
tff(fact_8084_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,Xa: fun(A,fun(A,$o)),Xaa: list(A),Xb: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xa,Xaa,Xb) = Y )
     => ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xaa),Xb)))
       => ( ! [X: A,Xs2: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) )
             => ! [Y3: A,Ys5: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5) )
                 => ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),Xa,X),Y3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),merges9089515139780605204_merge(A,Xa,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),merges9089515139780605204_merge(A,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2),Ys5))) )
                   => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys5)))) ) ) )
         => ( ( ( Xb = nil(A) )
             => ( ( Y = Xaa )
               => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xaa),nil(A)))) ) )
           => ~ ( ( Xaa = nil(A) )
               => ! [V: A,Va: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) )
                     => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xa),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_8085_mergesort__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% mergesort_def
tff(fact_8086_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_jo(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_8087_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = Y )
         => ( ! [X: A] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
               => ( Y != X ) )
           => ( ! [X: A,Y3: A,Zs2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                 => ( Y != $let(
                        m: A,
                        m:= arg_min_list(A,B,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)),
                        $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xa,X)),aa(A,B,Xa,m)),X,m) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_8088_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_afr(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Sup_fin.eq_fold'
tff(fact_8089_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Sup_fin.singleton
tff(fact_8090_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_8091_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_8092_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = the2(A,none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_8093_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_8094_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = aa(set(A),A,complete_Sup_Sup(A),A5) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_8095_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = aa(set(A),A,lattic5882676163264333800up_fin(A),X4) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_8096_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_8097_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_8098_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xa)
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A9),Xa) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_8099_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xa) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_8100_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),Xa)
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Xa) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_8101_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),Xa: A,Y: A,Zs: list(A)] :
          arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = $let(
            m: A,
            m:= arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F3,Xa)),aa(A,B,F3,m)),Xa,m) ) ) ).

% arg_min_list.simps(2)
tff(fact_8102_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_8103_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H2: fun(A,A),N6: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H2,X)),aa(A,A,H2,Y3))
         => ( aa(set(A),$o,finite_finite2(A),N6)
           => ( ( N6 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic5882676163264333800up_fin(A),N6)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image(A,A,H2),N6)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_8104_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A5) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_8105_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A5),A5) ) ) ) ) ).

% Sup_fin.closed
tff(fact_8106_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_8107_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_8108_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,A,sup_sup(A),Xa,A5) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_8109_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),B5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_afs(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_8110_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aft(set(A),fun(A,fun(A,$o)),A5),Xa))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_8111_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_8112_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_8113_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Y: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = Y )
         => ( aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),Xa),Xaa))
           => ( ! [X: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
                 => ( ( Y = X )
                   => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) ) )
             => ( ! [X: A,Y3: A,Zs2: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)) )
                   => ( ( Y = $let(
                            m: A,
                            m:= arg_min_list(A,B,Xa,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)),
                            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xa,X)),aa(A,B,Xa,m)),X,m) ) )
                     => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),Xa),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Zs2)))) ) )
               => ~ ( ( Xaa = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),Xa),nil(A))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_8114_mset__size2elem,axiom,
    ! [A: $tType,P2: multiset(A),Q3: A,Q6: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),P2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q3,zero_zero(multiset(A)))),add_mset(A,Q6,zero_zero(multiset(A))))),P2)
       => ( P2 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Q3,zero_zero(multiset(A)))),add_mset(A,Q6,zero_zero(multiset(A)))) ) ) ) ).

% mset_size2elem
tff(fact_8115_size__add__mset,axiom,
    ! [A: $tType,A2: A,A5: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),add_mset(A,A2,A5)) = aa(nat,nat,suc,aa(multiset(A),nat,size_size(multiset(A)),A5)) ).

% size_add_mset
tff(fact_8116_mset__le__subtract__add__mset__right,axiom,
    ! [A: $tType,Xa: A,B5: multiset(A),X4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B5)),X4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X4),B5))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),X4) ) ) ).

% mset_le_subtract_add_mset_right
tff(fact_8117_mset__le__subtract__add__mset__left,axiom,
    ! [A: $tType,Xa: A,B5: multiset(A),X4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B5)),X4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),X4),add_mset(A,Xa,zero_zero(multiset(A)))))
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),X4) ) ) ).

% mset_le_subtract_add_mset_left
tff(fact_8118_mset__le__add__mset__decr__left2,axiom,
    ! [A: $tType,C3: A,A2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C3,A2)),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C3,zero_zero(multiset(A)))),B2) ) ).

% mset_le_add_mset_decr_left2
tff(fact_8119_mset__le__single__cases,axiom,
    ! [A: $tType,M7: multiset(A),A2: A] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),M7),add_mset(A,A2,zero_zero(multiset(A))))
     => ( ( M7 != zero_zero(multiset(A)) )
       => ( M7 = add_mset(A,A2,zero_zero(multiset(A))) ) ) ) ).

% mset_le_single_cases
tff(fact_8120_mset__le__add__mset,axiom,
    ! [A: $tType,Xa: A,B5: multiset(A),C4: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,B5)),C4)
     => ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,Xa,zero_zero(multiset(A)))),C4)
        & aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),B5),C4) ) ) ).

% mset_le_add_mset
tff(fact_8121_mset__union__2__elem,axiom,
    ! [A: $tType,A2: A,B2: A,C3: A,M7: multiset(A)] :
      ( ( add_mset(A,A2,add_mset(A,B2,zero_zero(multiset(A)))) = add_mset(A,C3,M7) )
     => ( ( ( add_mset(A,A2,zero_zero(multiset(A))) = M7 )
          & ( B2 = C3 ) )
        | ( ( A2 = C3 )
          & ( add_mset(A,B2,zero_zero(multiset(A))) = M7 ) ) ) ) ).

% mset_union_2_elem
tff(fact_8122_mset__unplusm__dist__cases2,axiom,
    ! [A: $tType,B5: multiset(A),C4: multiset(A),Sa: A,A5: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),A5) )
     => ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),add_mset(A,Sa,zero_zero(multiset(A))))) )
         => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),add_mset(A,Sa,zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),add_mset(A,Sa,zero_zero(multiset(A))))) )
           => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),add_mset(A,Sa,zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases2
tff(fact_8123_mset__unplusm__dist__cases,axiom,
    ! [A: $tType,Sa: A,A5: multiset(A),B5: multiset(A),C4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),A5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),C4) )
     => ( ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),add_mset(A,Sa,zero_zero(multiset(A))))) )
         => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),B5),add_mset(A,Sa,zero_zero(multiset(A))))),C4) ) )
       => ~ ( ( C4 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),add_mset(A,Sa,zero_zero(multiset(A))))) )
           => ( A5 != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B5),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C4),add_mset(A,Sa,zero_zero(multiset(A))))) ) ) ) ) ).

% mset_unplusm_dist_cases
tff(fact_8124_mset__single__cases2_H,axiom,
    ! [A: $tType,Sa: A,C3: multiset(A),R4: A,C8: multiset(A)] :
      ( ( add_mset(A,Sa,C3) = add_mset(A,R4,C8) )
     => ( ( ( Sa = R4 )
         => ( C3 != C8 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,Sa,zero_zero(multiset(A)))) )
             => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Cc),add_mset(A,R4,zero_zero(multiset(A)))) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases2'
tff(fact_8125_mset__single__cases2,axiom,
    ! [A: $tType,Sa: A,C3: multiset(A),R4: A,C8: multiset(A)] :
      ( ( add_mset(A,Sa,C3) = add_mset(A,R4,C8) )
     => ( ( ( Sa = R4 )
         => ( C3 != C8 ) )
       => ~ ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A))))),add_mset(A,Sa,zero_zero(multiset(A)))) )
           => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A))))),add_mset(A,R4,zero_zero(multiset(A)))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases2
tff(fact_8126_mset__single__cases_H,axiom,
    ! [A: $tType,Sa: A,C3: multiset(A),R4: A,C8: multiset(A)] :
      ( ( add_mset(A,Sa,C3) = add_mset(A,R4,C8) )
     => ( ( ( Sa = R4 )
         => ( C3 != C8 ) )
       => ~ ! [Cc: multiset(A)] :
              ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),Cc) )
             => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R4,zero_zero(multiset(A)))),Cc) )
               => ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A)))) = Cc )
                 => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A)))) != Cc ) ) ) ) ) ) ).

% mset_single_cases'
tff(fact_8127_mset__single__cases,axiom,
    ! [A: $tType,Sa: A,C3: multiset(A),R4: A,C8: multiset(A)] :
      ( ( add_mset(A,Sa,C3) = add_mset(A,R4,C8) )
     => ( ( ( Sa = R4 )
         => ( C3 != C8 ) )
       => ~ ( ( C8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,Sa,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A))))) )
           => ( ( C3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,R4,zero_zero(multiset(A)))),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A))))) )
             => ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C3),add_mset(A,R4,zero_zero(multiset(A)))) != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),C8),add_mset(A,Sa,zero_zero(multiset(A)))) ) ) ) ) ) ).

% mset_single_cases
tff(fact_8128_size__eq__Suc__imp__eq__union,axiom,
    ! [A: $tType,M7: multiset(A),Na: nat] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M7) = aa(nat,nat,suc,Na) )
     => ? [A4: A,N10: multiset(A)] : M7 = add_mset(A,A4,N10) ) ).

% size_eq_Suc_imp_eq_union
tff(fact_8129_mset__le__add__mset__decr__left1,axiom,
    ! [A: $tType,C3: A,A2: multiset(A),B2: multiset(A)] :
      ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),add_mset(A,C3,A2)),B2)
     => aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),subseteq_mset(A),A2),B2) ) ).

% mset_le_add_mset_decr_left1
tff(fact_8130_size__mset__SucE,axiom,
    ! [A: $tType,A5: multiset(A),Na: nat] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),A5) = aa(nat,nat,suc,Na) )
     => ~ ! [A4: A,B8: multiset(A)] :
            ( ( A5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),add_mset(A,A4,zero_zero(multiset(A)))),B8) )
           => ( aa(multiset(A),nat,size_size(multiset(A)),B8) != Na ) ) ) ).

% size_mset_SucE
tff(fact_8131_size__Diff1__le,axiom,
    ! [A: $tType,M7: multiset(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),minus_minus(multiset(A)),M7),add_mset(A,Xa,zero_zero(multiset(A)))))),aa(multiset(A),nat,size_size(multiset(A)),M7)) ).

% size_Diff1_le
tff(fact_8132_mset__size__le1__cases,axiom,
    ! [A: $tType,M7: multiset(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(multiset(A),nat,size_size(multiset(A)),M7)),aa(nat,nat,suc,zero_zero(nat)))
     => ( ( M7 != zero_zero(multiset(A)) )
       => ~ ! [M3: A] : M7 != add_mset(A,M3,zero_zero(multiset(A))) ) ) ).

% mset_size_le1_cases
tff(fact_8133_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = the2(A,finite_fold(A,option(A),aTP_Lamp_afu(A,fun(option(A),option(A))),none(A),A5)) ) ).

% Inf_fin.eq_fold'
tff(fact_8134_is__up_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => aa(fun(fun(word(A),word(B)),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(word(A),word(B)),$o),$o),bNF_rel_fun(fun(int,int),fun(word(A),word(B)),$o,$o,bNF_rel_fun(int,word(A),int,word(B),pcr_word(A),pcr_word(B)),fequal($o)),aTP_Lamp_aeb(fun(int,int),$o)),is_up(A,B)) ) ).

% is_up.transfer
tff(fact_8135_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Inf_fin.singleton
tff(fact_8136_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_8137_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_8138_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = the2(A,none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_8139_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_8140_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_8141_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = aa(set(A),A,complete_Inf_Inf(A),A5) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_8142_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X4) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_8143_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),A2: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,A2,A5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),A2) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_8144_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))
             => ! [A9: A] :
                  ( member(A,A9,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A9) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_8145_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A4) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_8146_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5))
            <=> ! [X5: A] :
                  ( member(A,X5,A5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X5) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_8147_is__up_Orep__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: fun(word(A),word(B))] :
          ( aa(fun(word(A),word(B)),$o,is_up(A,B),Xa)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))) ) ) ).

% is_up.rep_eq
tff(fact_8148_is__up,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [C3: fun(word(A),word(B))] :
          ( aa(fun(word(A),word(B)),$o,is_up(A,B),C3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))) ) ) ).

% is_up
tff(fact_8149_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_8150_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H2: fun(A,A),N6: set(A)] :
          ( ! [X: A,Y3: A] : aa(A,A,H2,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H2,X)),aa(A,A,H2,Y3))
         => ( aa(set(A),$o,finite_finite2(A),N6)
           => ( ( N6 != bot_bot(set(A)) )
             => ( aa(A,A,H2,aa(set(A),A,lattic7752659483105999362nf_fin(A),N6)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image(A,A,H2),N6)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_8151_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A5)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_8152_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( ! [X: A,Y3: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y3),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A5),A5) ) ) ) ) ).

% Inf_fin.closed
tff(fact_8153_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ~ member(A,Xa,A5)
           => ( ( A5 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_8154_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A5),B5)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_8155_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = finite_fold(A,A,inf_inf(A),Xa,A5) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_8156_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic5882676163264333800up_fin(A),A5)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_8157_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_afv(set(A),fun(A,fun(A,$o)),A5),Xa))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_8158_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A5: set(A),B5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( ( A5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B5)
             => ( ( B5 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A5)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B5)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_afw(set(A),fun(set(A),fun(A,$o)),A5),B5))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_8159_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( member(A,Xa,A5)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A5) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_8160_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A5: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A5)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),A5)) = $ite(aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_8161_is__down_Otransfer,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => aa(fun(fun(word(A),word(B)),$o),$o,aa(fun(fun(int,int),$o),fun(fun(fun(word(A),word(B)),$o),$o),bNF_rel_fun(fun(int,int),fun(word(A),word(B)),$o,$o,bNF_rel_fun(int,word(A),int,word(B),pcr_word(A),pcr_word(B)),fequal($o)),aTP_Lamp_aec(fun(int,int),$o)),is_down(A,B)) ) ).

% is_down.transfer
tff(fact_8162_map__of__distinct__upd,axiom,
    ! [A: $tType,B: $tType,Xa: A,Xs: list(product_prod(A,B)),Y: B] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)))
     => ( map_add(A,B,fun_upd(A,option(B),aTP_Lamp_cp(A,option(B)),Xa,aa(B,option(B),some(B),Y)),map_of(A,B,Xs)) = fun_upd(A,option(B),map_of(A,B,Xs),Xa,aa(B,option(B),some(B),Y)) ) ) ).

% map_of_distinct_upd
tff(fact_8163_map__add__first__le,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [M: fun(A,option(B)),M6: fun(A,option(B)),Na: fun(A,option(B))] :
          ( aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),M),M6)
         => aa(fun(A,option(B)),$o,aa(fun(A,option(B)),fun(fun(A,option(B)),$o),ord_less_eq(fun(A,option(B))),map_add(A,B,M,Na)),map_add(A,B,M6,Na)) ) ) ).

% map_add_first_le
tff(fact_8164_map__add__find__left,axiom,
    ! [A: $tType,B: $tType,G3: fun(B,option(A)),K: B,F3: fun(B,option(A))] :
      ( ( aa(B,option(A),G3,K) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K) = aa(B,option(A),F3,K) ) ) ).

% map_add_find_left
tff(fact_8165_map__add__left__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),K: B,G3: fun(B,option(A))] :
      ( ( aa(B,option(A),F3,K) = none(A) )
     => ( aa(B,option(A),map_add(B,A,F3,G3),K) = aa(B,option(A),G3,K) ) ) ).

% map_add_left_None
tff(fact_8166_is__down,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [C3: fun(word(A),word(B))] :
          ( aa(fun(word(A),word(B)),$o,is_down(A,B),C3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))) ) ) ).

% is_down
tff(fact_8167_is__down_Orep__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xa: fun(word(A),word(B))] :
          ( aa(fun(word(A),word(B)),$o,is_down(A,B),Xa)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))) ) ) ).

% is_down.rep_eq
tff(fact_8168_map__of__concat,axiom,
    ! [B: $tType,A: $tType,Xss: list(list(product_prod(A,B)))] : map_of(A,B,concat(product_prod(A,B),Xss)) = foldr(list(product_prod(A,B)),fun(A,option(B)),aTP_Lamp_afx(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Xss,aTP_Lamp_cp(A,option(B))) ).

% map_of_concat
tff(fact_8169_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed7115095781618012415divide(word(A)),word2(A,Xa)),word2(A,Xb)) = word2(A,aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_divide_word.abs_eq
tff(fact_8170_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),signed6721504322012087516modulo(word(A)),word2(A,Xa)),word2(A,Xb)) = word2(A,aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_modulo_word.abs_eq
tff(fact_8171_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),word2(A,Xa)),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_eq_word.abs_eq
tff(fact_8172_less__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),word2(A,Xa)),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xa)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_word.abs_eq
tff(fact_8173_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,X3: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word2(A,Xa)),X3)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xa),X3) ) ) ) ).

% bit_word.abs_eq
tff(fact_8174_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int] : aa(word(A),int,the_signed_int(A),word2(A,Xa)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa) ) ).

% the_signed_int.abs_eq
tff(fact_8175_signed__cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xa: int] : aa(word(B),word(A),signed_cast(B,A),word2(B,Xa)) = word2(A,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),Xa)) ) ).

% signed_cast.abs_eq
tff(fact_8176_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sle(A),word2(A,Xa)),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb)) ) ) ).

% word_sle.abs_eq
tff(fact_8177_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),word_sless(A),word2(A,Xa)),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xa)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb)) ) ) ).

% word_sless.abs_eq
tff(fact_8178_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Xa: int] : aa(word(B),A,ring_1_signed(B,A),word2(B,Xa)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),Xa)) ) ).

% signed.abs_eq
tff(fact_8179_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xa: nat,Xb: int] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),signed_drop_bit(A),Xa),word2(A,Xb)) = word2(A,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xa),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xb))) ) ).

% signed_drop_bit.abs_eq
tff(fact_8180_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,Sa: set(A)] : topolo174197925503356063within(A,Xa,Sa) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_afy(A,fun(set(A),fun(set(A),filter(A))),Xa),Sa)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_afz(A,fun(set(A),$o),Xa)))) ) ).

% at_within_eq
tff(fact_8181_list__collect__set__alt,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),L: list(B)] : list_collect_set(B,A,F3,L) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(list(B),fun(set(A),$o),aTP_Lamp_aga(fun(B,set(A)),fun(list(B),fun(set(A),$o)),F3),L))) ).

% list_collect_set_alt
tff(fact_8182_list__collect__set__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),A2: B] : list_collect_set(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),nil(B))) = aa(B,set(A),F3,A2) ).

% list_collect_set_simps(2)
tff(fact_8183_list__collect__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,set(A))] : list_collect_set(B,A,F3,nil(B)) = bot_bot(set(A)) ).

% list_collect_set_simps(1)
tff(fact_8184_principal__le__iff,axiom,
    ! [A: $tType,A5: set(A),B5: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),principal(A,A5)),principal(A,B5))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A5),B5) ) ).

% principal_le_iff
tff(fact_8185_ATP_Olambda__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_wy(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),exp(A,Uu2)),one_one(A))),Uu2) ) ).

% ATP.lambda_1
tff(fact_8186_ATP_Olambda__2,axiom,
    ! [A: $tType,Uu2: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_jw(set(set(A)),int),Uu2) = 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)),Uu2)),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)),Uu2)))) ).

% ATP.lambda_2
tff(fact_8187_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_kg(A,set(product_prod(A,A))),Uu2) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_3
tff(fact_8188_ATP_Olambda__4,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_rq(nat,real),Uu2) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_4
tff(fact_8189_ATP_Olambda__5,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_ns(real,$o),Uu2)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu2),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu2) = zero_zero(real) ) ) ) ).

% ATP.lambda_5
tff(fact_8190_ATP_Olambda__6,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xt(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_6
tff(fact_8191_ATP_Olambda__7,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_ii(nat,real),Uu2) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu2)) ).

% ATP.lambda_7
tff(fact_8192_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: nat] : aa(nat,fun(int,int),aTP_Lamp_aed(nat,fun(int,int)),Uu2) = aa(fun(int,int),fun(int,int),comp(int,int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uu2)),bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_8
tff(fact_8193_ATP_Olambda__9,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xs(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu2)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu2))) ) ).

% ATP.lambda_9
tff(fact_8194_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: nat] :
          ( aa(nat,$o,aTP_Lamp_jm(nat,$o),Uu2)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu2) = zero_zero(A) ) ) ) ).

% ATP.lambda_10
tff(fact_8195_ATP_Olambda__11,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu2: fun(int,int)] :
          ( aa(fun(int,int),$o,aTP_Lamp_aec(fun(int,int),$o),Uu2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))) ) ) ).

% ATP.lambda_11
tff(fact_8196_ATP_Olambda__12,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Uu2: fun(int,int)] :
          ( aa(fun(int,int),$o,aTP_Lamp_aeb(fun(int,int),$o),Uu2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))) ) ) ).

% ATP.lambda_12
tff(fact_8197_ATP_Olambda__13,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu2: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_acx(word(B),$o),Uu2)
        <=> aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less(word(B)),Uu2),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ATP.lambda_13
tff(fact_8198_ATP_Olambda__14,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_su(nat,nat),Uu2) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_14
tff(fact_8199_ATP_Olambda__15,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_adg(int,int),Uu2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu2),Uu2) ).

% ATP.lambda_15
tff(fact_8200_ATP_Olambda__16,axiom,
    ! [B: $tType,Uu2: B] : aa(B,product_prod(B,B),aTP_Lamp_aew(B,product_prod(B,B)),Uu2) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu2),Uu2) ).

% ATP.lambda_16
tff(fact_8201_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu2: A] : aa(A,product_prod(A,A),aTP_Lamp_ej(A,product_prod(A,A)),Uu2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2) ).

% ATP.lambda_17
tff(fact_8202_ATP_Olambda__18,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_eq(A,A),Uu2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),one_one(A)) ) ).

% ATP.lambda_18
tff(fact_8203_ATP_Olambda__19,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,list(A),aTP_Lamp_my(A,list(A)),Uu2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu2),nil(A)) ) ).

% ATP.lambda_19
tff(fact_8204_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu2: A] : aa(A,set(A),aTP_Lamp_ki(A,set(A)),Uu2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))) ).

% ATP.lambda_20
tff(fact_8205_ATP_Olambda__21,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xr(nat,A),Uu2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_21
tff(fact_8206_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_mw(list(B),fun(nat,nat)),Uu2) = 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)),Uu2)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_22
tff(fact_8207_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_jv(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu2) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),Uu2)) ).

% ATP.lambda_23
tff(fact_8208_ATP_Olambda__24,axiom,
    ! [B: $tType,Uu2: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_mx(list(B),$o),Uu2)
    <=> ( Uu2 != nil(B) ) ) ).

% ATP.lambda_24
tff(fact_8209_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu2: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_mm(list(A),$o),Uu2)
    <=> ( Uu2 != nil(A) ) ) ).

% ATP.lambda_25
tff(fact_8210_ATP_Olambda__26,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_pn(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu2) = 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_pm(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu2)) ).

% ATP.lambda_26
tff(fact_8211_ATP_Olambda__27,axiom,
    ! [Uu2: real] : aa(real,real,aTP_Lamp_uj(real,real),Uu2) = suminf(real,aTP_Lamp_oh(real,fun(nat,real),Uu2)) ).

% ATP.lambda_27
tff(fact_8212_ATP_Olambda__28,axiom,
    ! [Uu2: nat] : aa(nat,set(nat),aTP_Lamp_oe(nat,set(nat)),Uu2) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_nw(nat,fun(nat,$o),Uu2)) ).

% ATP.lambda_28
tff(fact_8213_ATP_Olambda__29,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_adw(fun(A,B),set(product_prod(A,B))),Uu2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_adv(fun(A,B),fun(A,fun(B,$o)),Uu2))) ).

% ATP.lambda_29
tff(fact_8214_ATP_Olambda__30,axiom,
    ! [Uu2: nat] : aa(nat,real,aTP_Lamp_xo(nat,real),Uu2) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu2))) ).

% ATP.lambda_30
tff(fact_8215_ATP_Olambda__31,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_xj(nat,A),Uu2) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu2)) ) ).

% ATP.lambda_31
tff(fact_8216_ATP_Olambda__32,axiom,
    ! [B: $tType,Uu2: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_mv(list(B),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu2)) ).

% ATP.lambda_32
tff(fact_8217_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu2: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_mz(list(A),fun(nat,nat)),Uu2) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Uu2)) ).

% ATP.lambda_33
tff(fact_8218_ATP_Olambda__34,axiom,
    ! [Uu2: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_qb(nat,fun(nat,product_prod(nat,nat))),Uu2) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu2)) ).

% ATP.lambda_34
tff(fact_8219_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_abj(product_prod(A,A),$o),Uu2)
        <=> ? [X5: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X5),Y5) ) ) ) ).

% ATP.lambda_35
tff(fact_8220_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_abi(product_prod(A,A),$o),Uu2)
        <=> ? [X5: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y5),X5) ) ) ) ).

% ATP.lambda_36
tff(fact_8221_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu2: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_abk(product_prod(A,A),$o),Uu2)
        <=> ? [X5: A,Y5: A] :
              ( ( Uu2 = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5) )
              & ( X5 != Y5 ) ) ) ) ).

% ATP.lambda_37
tff(fact_8222_ATP_Olambda__38,axiom,
    ! [A: $tType,B: $tType,Uu2: product_prod(product_prod($o,A),product_prod($o,B))] :
      ( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_acl(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu2)
    <=> ? [X5: A,Y5: B] : Uu2 = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$true),X5)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y5)) ) ).

% ATP.lambda_38
tff(fact_8223_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_sb(A,fun(nat,A),Uu2),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)),zero_zero(A)) ) ).

% ATP.lambda_39
tff(fact_8224_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_of(nat,fun(nat,A),Uu2),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu2,Uua)),zero_zero(A)) ) ).

% ATP.lambda_40
tff(fact_8225_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_rw(A,fun(nat,A),Uu2),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua))) ) ).

% ATP.lambda_41
tff(fact_8226_ATP_Olambda__42,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: product_prod(A,C),Uua: product_prod(C,B)] :
      aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_afm(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu2),Uua) = $ite(aa(product_prod(A,C),C,product_snd(A,C),Uu2) = 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),Uu2)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).

% ATP.lambda_42
tff(fact_8227_ATP_Olambda__43,axiom,
    ! [A: $tType,Uu2: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_ho(list(A),fun(nat,option(A)),Uu2),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),aa(list(A),nat,size_size(list(A)),Uu2)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu2),Uua)),none(A)) ).

% ATP.lambda_43
tff(fact_8228_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_og(nat,fun(nat,A),Uu2),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu2,Uua)),zero_zero(A)) ) ).

% ATP.lambda_44
tff(fact_8229_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_mh(fun(A,A),fun(nat,A),Uu2),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),Uu2),bot_bot(A)) ) ).

% ATP.lambda_45
tff(fact_8230_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu2: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pt(option(A),fun(A,$o),Uu2),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_ps(A,fun(A,$o),Uua),Uu2) ) ) ).

% ATP.lambda_46
tff(fact_8231_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu2: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_pr(option(A),fun(A,$o),Uu2),Uua)
        <=> case_option($o,A,$false,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_47
tff(fact_8232_ATP_Olambda__48,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_aex(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu2),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_48
tff(fact_8233_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_sq(list(list(A)),fun(nat,list(A)),Uu2),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sp(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua)),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu2))) ).

% ATP.lambda_49
tff(fact_8234_ATP_Olambda__50,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_os(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_or(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_50
tff(fact_8235_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_oq(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_op(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_51
tff(fact_8236_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_ado(int,fun(int,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),signed6721504322012087516modulo(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ).

% ATP.lambda_52
tff(fact_8237_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_adp(int,fun(int,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),signed7115095781618012415divide(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ).

% ATP.lambda_53
tff(fact_8238_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adm(int,fun(int,$o)),Uu2),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ) ).

% ATP.lambda_54
tff(fact_8239_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adn(int,fun(int,$o)),Uu2),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu2)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uua)) ) ) ).

% ATP.lambda_55
tff(fact_8240_ATP_Olambda__56,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_nv(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uu2),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_56
tff(fact_8241_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_hn(nat,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),binomial(Uu2,Uua))) ) ).

% ATP.lambda_57
tff(fact_8242_ATP_Olambda__58,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_rf(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_58
tff(fact_8243_ATP_Olambda__59,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_re(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu2 ) ) ) ).

% ATP.lambda_59
tff(fact_8244_ATP_Olambda__60,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_vr(code_integer,fun(code_integer,int)),Uu2),Uua) = $let(
        l2: int,
        l2:= 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,Uu2)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_60
tff(fact_8245_ATP_Olambda__61,axiom,
    ! [Uu2: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_pa(nat,fun(nat,a)),Uu2),Uua) = $let(
        m2: a,
        m2:= 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),Uu2)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_61
tff(fact_8246_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_jy(real,fun(int,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu2)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu2),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_62
tff(fact_8247_ATP_Olambda__63,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_kw(rat,fun(int,$o),Uu2),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu2)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu2),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_63
tff(fact_8248_ATP_Olambda__64,axiom,
    ! [A: $tType,Uu2: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_dl(set(A),fun(list(A),$o),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu2)
        & distinct(A,Uua) ) ) ).

% ATP.lambda_64
tff(fact_8249_ATP_Olambda__65,axiom,
    ! [A: $tType,Uu2: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_co(set(A),fun(list(A),$o),Uu2),Uua)
    <=> ( ( aa(list(A),set(A),set2(A),Uua) = Uu2 )
        & distinct(A,Uua) ) ) ).

% ATP.lambda_65
tff(fact_8250_ATP_Olambda__66,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_oh(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_66
tff(fact_8251_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adl(int,fun(int,$o)),Uu2),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu2)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ) ).

% ATP.lambda_67
tff(fact_8252_ATP_Olambda__68,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adk(int,fun(int,$o)),Uu2),Uua)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu2)),aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua)) ) ) ).

% ATP.lambda_68
tff(fact_8253_ATP_Olambda__69,axiom,
    ! [B: $tType] :
      ( type_len(B)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ade(int,fun(int,$o)),Uu2),Uua)
        <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),Uu2) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(B,type2(B))),Uua) ) ) ) ).

% ATP.lambda_69
tff(fact_8254_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: int] :
          ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adj(int,fun(int,$o)),Uu2),Uua)
        <=> ( aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uu2) = aa(int,int,aa(nat,fun(int,int),bit_se2584673776208193580ke_bit(int),type_len0_len_of(A,type2(A))),Uua) ) ) ) ).

% ATP.lambda_70
tff(fact_8255_ATP_Olambda__71,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_uk(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_71
tff(fact_8256_ATP_Olambda__72,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_72
tff(fact_8257_ATP_Olambda__73,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_he(nat,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu2,Uua))) ) ).

% ATP.lambda_73
tff(fact_8258_ATP_Olambda__74,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),aTP_Lamp_adt(int,fun(nat,$o)),Uu2),Uua)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Uu2),Uua) ) ) ) ).

% ATP.lambda_74
tff(fact_8259_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ng(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,Uu2,Uua)),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_75
tff(fact_8260_ATP_Olambda__76,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_oa(nat,fun(nat,$o)),Uu2),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu2),Uua)
        & ( Uu2 != Uua ) ) ) ).

% ATP.lambda_76
tff(fact_8261_ATP_Olambda__77,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_jx(set(set(A)),fun(set(set(A)),$o),Uu2),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu2)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_77
tff(fact_8262_ATP_Olambda__78,axiom,
    ! [Uu2: set(int),Uua: int] :
      ( aa(int,$o,aTP_Lamp_acc(set(int),fun(int,$o),Uu2),Uua)
    <=> ( member(int,Uua,Uu2)
        & ! [X5: int] :
            ( member(int,X5,Uu2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X5),Uua) ) ) ) ).

% ATP.lambda_78
tff(fact_8263_ATP_Olambda__79,axiom,
    ! [A: $tType,Uu2: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_acd(set(set(A)),fun(set(A),$o),Uu2),Uua)
    <=> ( member(set(A),Uua,Uu2)
        & ! [X5: set(A)] :
            ( member(set(A),X5,Uu2)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X5) ) ) ) ).

% ATP.lambda_79
tff(fact_8264_ATP_Olambda__80,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ij(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_80
tff(fact_8265_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_cn(set(A),fun(set(A),$o),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu2)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_81
tff(fact_8266_ATP_Olambda__82,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_pw(set(A),fun(set(A),$o)),Uu2),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu2),Uua)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_82
tff(fact_8267_ATP_Olambda__83,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hi(nat,fun(nat,nat),Uu2),Uua) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),Uua),Uua) ).

% ATP.lambda_83
tff(fact_8268_ATP_Olambda__84,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_kz(B,fun(A,set(product_prod(B,A))),Uu2),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu2),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_84
tff(fact_8269_ATP_Olambda__85,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_ef(nat,fun(complex,$o),Uu2),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu2) = one_one(complex) ) ) ).

% ATP.lambda_85
tff(fact_8270_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ea(nat,fun(A,$o),Uu2),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu2) = one_one(A) ) ) ) ).

% ATP.lambda_86
tff(fact_8271_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_iq(A,fun(A,$o),Uu2),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu2) ) ) ) ).

% ATP.lambda_87
tff(fact_8272_ATP_Olambda__88,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(B,fun(C,$o)),Uua: fun(A,product_prod(B,C))] :
      ( aa(fun(A,product_prod(B,C)),$o,aTP_Lamp_aey(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),Uu2),Uua)
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(set(A),set(product_prod(B,C)),image(A,product_prod(B,C),Uua),top_top(set(A)))),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu2))) ) ).

% ATP.lambda_88
tff(fact_8273_ATP_Olambda__89,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_nr(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_89
tff(fact_8274_ATP_Olambda__90,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_rg(real,fun(real,$o),Uu2),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu2 ) ) ) ).

% ATP.lambda_90
tff(fact_8275_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_nb(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,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_91
tff(fact_8276_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_na(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu2,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_92
tff(fact_8277_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xl(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_93
tff(fact_8278_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_94
tff(fact_8279_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gs(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_95
tff(fact_8280_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fq(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_96
tff(fact_8281_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qs(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_97
tff(fact_8282_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_nx(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_98
tff(fact_8283_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aq(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_99
tff(fact_8284_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qt(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_100
tff(fact_8285_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fz(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_101
tff(fact_8286_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fy(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_102
tff(fact_8287_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xk(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_103
tff(fact_8288_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bx(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uu2,Uua)),aa(nat,A,Uu2,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_104
tff(fact_8289_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aap(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y5: A] :
                ( aa(A,$o,Uu2,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y5) ) ) ) ) ).

% ATP.lambda_105
tff(fact_8290_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aar(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,$o,Uu2,Uua)
            & ! [Y5: A] :
                ( aa(A,$o,Uu2,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Uua) ) ) ) ) ).

% ATP.lambda_106
tff(fact_8291_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_gv(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_107
tff(fact_8292_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ft(fun(nat,fun(nat,A)),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),Uu2,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_108
tff(fact_8293_ATP_Olambda__109,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_aea(fun(A,B),fun(A,product_prod(B,A)),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu2,Uua)),Uua) ).

% ATP.lambda_109
tff(fact_8294_ATP_Olambda__110,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_za(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu2,Uua)),zero_zero(real)) ) ).

% ATP.lambda_110
tff(fact_8295_ATP_Olambda__111,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_ku(fun(B,A),fun(B,set(A)),Uu2),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(B,A,Uu2,Uua)),bot_bot(set(A))) ).

% ATP.lambda_111
tff(fact_8296_ATP_Olambda__112,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_cy(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_112
tff(fact_8297_ATP_Olambda__113,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_go(fun(A,B),fun(A,$o),Uu2),Uua)
        <=> ( aa(A,B,Uu2,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_113
tff(fact_8298_ATP_Olambda__114,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_yf(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),Uu2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_114
tff(fact_8299_ATP_Olambda__115,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ye(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_xb(fun(nat,real),fun(nat,real),Uu2)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_115
tff(fact_8300_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu2: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_hm(list(A),fun(list(A),list(list(A))),Uu2),Uua) = aa(list(A),list(list(A)),map(A,list(A),aTP_Lamp_hl(list(A),fun(A,list(A)),Uua)),Uu2) ).

% ATP.lambda_116
tff(fact_8301_ATP_Olambda__117,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_yl(fun(A,B),fun(A,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_117
tff(fact_8302_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_rv(A,fun(nat,A),Uu2),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_118
tff(fact_8303_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_rh(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_119
tff(fact_8304_ATP_Olambda__120,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_mi(nat,fun(real,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Uua)),Uu2)) ).

% ATP.lambda_120
tff(fact_8305_ATP_Olambda__121,axiom,
    ! [Uu2: $o,Uua: nat] : aa(nat,nat,aa($o,fun(nat,nat),aTP_Lamp_acp($o,fun(nat,nat)),(Uu2)),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa($o,nat,zero_neq_one_of_bool(nat),(Uu2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% ATP.lambda_121
tff(fact_8306_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_yd(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_122
tff(fact_8307_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_yc(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uua)) ) ).

% ATP.lambda_123
tff(fact_8308_ATP_Olambda__124,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ky(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua)) ).

% ATP.lambda_124
tff(fact_8309_ATP_Olambda__125,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_nq(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua)) ).

% ATP.lambda_125
tff(fact_8310_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_afz(A,fun(set(A),$o),Uu2),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & member(A,Uu2,Uua) ) ) ) ).

% ATP.lambda_126
tff(fact_8311_ATP_Olambda__127,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu2: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_aej(product_prod(A,B),fun(product_prod(A,B),$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uu2)),aa(product_prod(A,B),A,product_fst(A,B),Uua)) ) ) ).

% ATP.lambda_127
tff(fact_8312_ATP_Olambda__128,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu2: product_prod(A,B),Uua: product_prod(A,B)] :
          ( aa(product_prod(A,B),$o,aa(product_prod(A,B),fun(product_prod(A,B),$o),aTP_Lamp_aek(product_prod(A,B),fun(product_prod(A,B),$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),A,product_fst(A,B),Uu2)) ) ) ).

% ATP.lambda_128
tff(fact_8313_ATP_Olambda__129,axiom,
    ! [Uu2: num,Uua: num] : aa(num,int,aa(num,fun(num,int),aTP_Lamp_adx(num,fun(num,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(num,int,numeral_numeral(int),Uu2)),aa(num,int,numeral_numeral(int),Uua)) ).

% ATP.lambda_129
tff(fact_8314_ATP_Olambda__130,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_yg(code_integer,fun(code_integer,num)),Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_130
tff(fact_8315_ATP_Olambda__131,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_yp(code_integer,fun(code_integer,nat)),Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_131
tff(fact_8316_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ic(nat,fun(nat,A),Uu2),Uua) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua),Uu2) ) ).

% ATP.lambda_132
tff(fact_8317_ATP_Olambda__133,axiom,
    ! [A: $tType,B: $tType,Uu2: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_em(list(B),fun(A,list(product_prod(A,B))),Uu2),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)),Uu2) ).

% ATP.lambda_133
tff(fact_8318_ATP_Olambda__134,axiom,
    ! [A: $tType,Uu2: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_afb(set(nat),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu2) ) ).

% ATP.lambda_134
tff(fact_8319_ATP_Olambda__135,axiom,
    ! [Uu2: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_afg(set(nat),fun(nat,$o),Uu2),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu2) ) ).

% ATP.lambda_135
tff(fact_8320_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_pf(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),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_136
tff(fact_8321_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: int,Uua: nat] : aa(nat,int,aa(int,fun(nat,int),aTP_Lamp_aee(int,fun(nat,int)),Uu2),Uua) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Uua),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Uu2)) ) ).

% ATP.lambda_137
tff(fact_8322_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_pe(A,fun(A,product_prod(A,A))),Uu2),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_138
tff(fact_8323_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_nj(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),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_139
tff(fact_8324_ATP_Olambda__140,axiom,
    ! [A: $tType,Uu2: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_lv(A,fun(set(set(A)),set(set(A)))),Uu2),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)),image(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2)),Uua)) ).

% ATP.lambda_140
tff(fact_8325_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_sn(list(A),fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(nat,A,nth(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu2)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% ATP.lambda_141
tff(fact_8326_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_so(list(A),fun(A,$o),Uu2),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% ATP.lambda_142
tff(fact_8327_ATP_Olambda__143,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_kf(set(A),fun(A,$o),Uu2),Uua)
    <=> ( Uu2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_143
tff(fact_8328_ATP_Olambda__144,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_ya(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu2),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_144
tff(fact_8329_ATP_Olambda__145,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_145
tff(fact_8330_ATP_Olambda__146,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_adz(fun(A,B),fun(A,product_prod(A,B)),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_146
tff(fact_8331_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_add(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu2),Uua))) ) ).

% ATP.lambda_147
tff(fact_8332_ATP_Olambda__148,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_xy(fun(nat,A),fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua))) ) ).

% ATP.lambda_148
tff(fact_8333_ATP_Olambda__149,axiom,
    ! [Uu2: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_sj(int,fun(int,int)),Uu2),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu2),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_149
tff(fact_8334_ATP_Olambda__150,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xx(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_150
tff(fact_8335_ATP_Olambda__151,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xq(real,fun(nat,real),Uu2),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu2),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_151
tff(fact_8336_ATP_Olambda__152,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_rd(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),Uu2),set_ord_lessThan(nat,Uua)) ).

% ATP.lambda_152
tff(fact_8337_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_xh(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_153
tff(fact_8338_ATP_Olambda__154,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_nl(nat,fun(list(A),$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu2),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_154
tff(fact_8339_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_ie(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_155
tff(fact_8340_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_if(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_156
tff(fact_8341_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_hw(A,fun(nat,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_157
tff(fact_8342_ATP_Olambda__158,axiom,
    ! [B: $tType,A: $tType,Uu2: list(product_prod(A,B)),Uua: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_afx(list(product_prod(A,B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu2),Uua) = map_add(A,B,Uua,map_of(A,B,Uu2)) ).

% ATP.lambda_158
tff(fact_8343_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu2: list(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_le(list(set(A)),fun(set(A),$o),Uu2),Uua)
    <=> member(set(A),Uua,aa(list(set(A)),set(set(A)),set2(set(A)),Uu2)) ) ).

% ATP.lambda_159
tff(fact_8344_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_mn(list(A),fun(A,$o),Uu2),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_160
tff(fact_8345_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: word(A),Uua: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_di(word(A),fun(word(A),$o)),Uu2),Uua)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uua),Uu2) ) ) ).

% ATP.lambda_161
tff(fact_8346_ATP_Olambda__162,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_an(set(A),fun(set(A),$o),Uu2),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu2) ) ).

% ATP.lambda_162
tff(fact_8347_ATP_Olambda__163,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aj(nat,fun(nat,$o)),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu2) ) ).

% ATP.lambda_163
tff(fact_8348_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_lf(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_164
tff(fact_8349_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dc(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_165
tff(fact_8350_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_je(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_166
tff(fact_8351_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ev(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu2) ) ) ).

% ATP.lambda_167
tff(fact_8352_ATP_Olambda__168,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mf(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu2) ).

% ATP.lambda_168
tff(fact_8353_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_ge(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_169
tff(fact_8354_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_adf(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu2) ) ).

% ATP.lambda_170
tff(fact_8355_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: word(A),Uua: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_dj(word(A),fun(word(A),$o)),Uu2),Uua)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Uua),Uu2) ) ) ).

% ATP.lambda_171
tff(fact_8356_ATP_Olambda__172,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ai(nat,fun(nat,$o)),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu2) ) ).

% ATP.lambda_172
tff(fact_8357_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_lg(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_173
tff(fact_8358_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_zc(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_174
tff(fact_8359_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_dd(A,fun(A,$o)),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_175
tff(fact_8360_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ps(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_176
tff(fact_8361_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bb(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu2) ) ) ).

% ATP.lambda_177
tff(fact_8362_ATP_Olambda__178,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xg(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu2) ).

% ATP.lambda_178
tff(fact_8363_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_fv(A,fun(A,A),Uu2),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu2) ) ).

% ATP.lambda_179
tff(fact_8364_ATP_Olambda__180,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_do(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu2) ).

% ATP.lambda_180
tff(fact_8365_ATP_Olambda__181,axiom,
    ! [Uu2: nat,Uua: real] : aa(real,real,aTP_Lamp_tu(nat,fun(real,real),Uu2),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu2) ).

% ATP.lambda_181
tff(fact_8366_ATP_Olambda__182,axiom,
    ! [Uu2: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_np(code_integer,fun(code_integer,code_integer),Uu2),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu2) ).

% ATP.lambda_182
tff(fact_8367_ATP_Olambda__183,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sr(nat,fun(nat,nat),Uu2),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu2) ).

% ATP.lambda_183
tff(fact_8368_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu2: A,Uua: ref(A)] : aa(ref(A),assn,aa(A,fun(ref(A),assn),aTP_Lamp_jt(A,fun(ref(A),assn)),Uu2),Uua) = sngr_assn(A,Uua,Uu2) ) ).

% ATP.lambda_184
tff(fact_8369_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu2: list(A),Uua: array(A)] : aa(array(A),assn,aa(list(A),fun(array(A),assn),aTP_Lamp_jr(list(A),fun(array(A),assn)),Uu2),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu2) ) ).

% ATP.lambda_185
tff(fact_8370_ATP_Olambda__186,axiom,
    ! [Uu2: real,Uua: real] : aa(real,real,aTP_Lamp_tw(real,fun(real,real),Uu2),Uua) = powr(real,Uua,Uu2) ).

% ATP.lambda_186
tff(fact_8371_ATP_Olambda__187,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_nw(nat,fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu2) ) ).

% ATP.lambda_187
tff(fact_8372_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ny(A,fun(A,$o),Uu2),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu2) ) ) ).

% ATP.lambda_188
tff(fact_8373_ATP_Olambda__189,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ep(B,fun(A,product_prod(A,B))),Uu2),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu2) ).

% ATP.lambda_189
tff(fact_8374_ATP_Olambda__190,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_ady(A,fun(B,product_prod(B,A))),Uu2),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu2) ).

% ATP.lambda_190
tff(fact_8375_ATP_Olambda__191,axiom,
    ! [Uu2: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_hh(nat,fun(nat,nat),Uu2),Uua) = binomial(Uua,Uu2) ).

% ATP.lambda_191
tff(fact_8376_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] : aa(A,list(A),aTP_Lamp_hl(list(A),fun(A,list(A)),Uu2),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu2) ).

% ATP.lambda_192
tff(fact_8377_ATP_Olambda__193,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xp(real,fun(nat,real),Uu2),Uua) = aa(real,real,root(Uua),Uu2) ).

% ATP.lambda_193
tff(fact_8378_ATP_Olambda__194,axiom,
    ! [B: $tType,Uu2: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_acs(set(B),fun(B,$o),Uu2),Uua)
    <=> member(B,Uua,Uu2) ) ).

% ATP.lambda_194
tff(fact_8379_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ir(set(A),fun(A,$o),Uu2),Uua)
        <=> member(A,Uua,Uu2) ) ) ).

% ATP.lambda_195
tff(fact_8380_ATP_Olambda__196,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_a(set(A),fun(A,$o),Uu2),Uua)
    <=> member(A,Uua,Uu2) ) ).

% ATP.lambda_196
tff(fact_8381_ATP_Olambda__197,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_nk(nat,fun(list(A),A),Uu2),Uua) = aa(nat,A,nth(A,Uua),Uu2) ).

% ATP.lambda_197
tff(fact_8382_ATP_Olambda__198,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ah(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua = Uu2 ) ) ).

% ATP.lambda_198
tff(fact_8383_ATP_Olambda__199,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aaj(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_199
tff(fact_8384_ATP_Olambda__200,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aac(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_200
tff(fact_8385_ATP_Olambda__201,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aas(fun(A,real),fun(A,$o),Uu2),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_201
tff(fact_8386_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_cc(fun(A,nat),fun(A,$o),Uu2),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,Uu2,Uua)) ) ).

% ATP.lambda_202
tff(fact_8387_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_yu(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,Uu2,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_203
tff(fact_8388_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu2: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_aez(fun(nat,$o),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_204
tff(fact_8389_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu2: fun(nat,$o),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_afa(fun(nat,$o),fun(product_prod(A,nat),$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)) ) ).

% ATP.lambda_205
tff(fact_8390_ATP_Olambda__206,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_aer(fun(B,C),fun(product_prod(A,B),C),Uu2),Uua) = aa(B,C,Uu2,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_206
tff(fact_8391_ATP_Olambda__207,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ael(fun(A,C),fun(product_prod(A,B),C),Uu2),Uua) = aa(A,C,Uu2,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_207
tff(fact_8392_ATP_Olambda__208,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_io(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> aa(nat,$o,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_208
tff(fact_8393_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qp(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_209
tff(fact_8394_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xf(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_210
tff(fact_8395_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rk(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_211
tff(fact_8396_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gm(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_212
tff(fact_8397_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bv(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_213
tff(fact_8398_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu2: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sw(fun(nat,A),fun(nat,A),Uu2),Uua) = aa(nat,A,Uu2,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_214
tff(fact_8399_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_ada(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),ord_min(A),Uu2),Uua)) ) ).

% ATP.lambda_215
tff(fact_8400_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_oy(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),ord_max(A),Uu2),Uua)) ) ).

% ATP.lambda_216
tff(fact_8401_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_afr(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),sup_sup(A),Uu2),Uua)) ) ).

% ATP.lambda_217
tff(fact_8402_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu2: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_afu(A,fun(option(A),option(A))),Uu2),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu2,aa(A,fun(A,A),inf_inf(A),Uu2),Uua)) ) ).

% ATP.lambda_218
tff(fact_8403_ATP_Olambda__219,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),list(A)),aTP_Lamp_ta(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu2),Uua) = aa(fun(list(A),fun(list(A),list(A))),fun(product_prod(list(A),list(A)),list(A)),product_case_prod(list(A),list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_sz(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu2),Uua)) ) ).

% ATP.lambda_219
tff(fact_8404_ATP_Olambda__220,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,fun(A,C)),Uua: C] : aa(C,fun(product_prod(B,A),C),aTP_Lamp_ox(fun(C,fun(A,C)),fun(C,fun(product_prod(B,A),C)),Uu2),Uua) = aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_ow(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu2),Uua)) ).

% ATP.lambda_220
tff(fact_8405_ATP_Olambda__221,axiom,
    ! [Uu2: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_un(fun(nat,real),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_um(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua)) ).

% ATP.lambda_221
tff(fact_8406_ATP_Olambda__222,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_ty(fun(nat,A),fun(A,A),Uu2),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua)) ) ).

% ATP.lambda_222
tff(fact_8407_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: real,Uua: A] : aa(A,set(A),aTP_Lamp_yt(real,fun(A,set(A)),Uu2),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ys(real,fun(A,fun(A,$o)),Uu2),Uua)) ) ).

% ATP.lambda_223
tff(fact_8408_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_uw(A,fun(A,A),Uu2),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu2)),Uua)),aa(A,A,inverse_inverse(A),Uu2))) ) ).

% ATP.lambda_224
tff(fact_8409_ATP_Olambda__225,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_afn(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu2),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_afm(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu2)) ).

% ATP.lambda_225
tff(fact_8410_ATP_Olambda__226,axiom,
    ! [A: $tType,Uu2: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_ma(fun(nat,set(A)),fun(nat,set(A)),Uu2),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_226
tff(fact_8411_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_id(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_227
tff(fact_8412_ATP_Olambda__228,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_hq(A,fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_228
tff(fact_8413_ATP_Olambda__229,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_mo(list(A),fun(A,$o),Uu2),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu2)) ) ).

% ATP.lambda_229
tff(fact_8414_ATP_Olambda__230,axiom,
    ! [Uu2: real,Uua: nat] : aa(nat,real,aTP_Lamp_xv(real,fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uua)) ).

% ATP.lambda_230
tff(fact_8415_ATP_Olambda__231,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_jz(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu2),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert(product_prod(B,A)),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu2),Uua)) ).

% ATP.lambda_231
tff(fact_8416_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_to(A,fun(A,$o),Uu2),Uua)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),Uua) ) ) ).

% ATP.lambda_232
tff(fact_8417_ATP_Olambda__233,axiom,
    ! [A: $tType,Uu2: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_az(set(A),fun(A,$o),Uu2),Uua)
    <=> ~ member(A,Uua,Uu2) ) ).

% ATP.lambda_233
tff(fact_8418_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu2: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_mt(A,fun(A,$o),Uu2),Uua)
    <=> ( Uua != Uu2 ) ) ).

% ATP.lambda_234
tff(fact_8419_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_qx(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_235
tff(fact_8420_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_qz(fun(nat,A),fun(nat,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_236
tff(fact_8421_ATP_Olambda__237,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_ga(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_237
tff(fact_8422_ATP_Olambda__238,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_gg(fun(B,A),fun(B,real),Uu2),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_238
tff(fact_8423_ATP_Olambda__239,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_wl(fun(A,B),fun(A,real),Uu2),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_239
tff(fact_8424_ATP_Olambda__240,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_si(fun(A,$o),fun(A,B),Uu2),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu2,Uua)) ) ).

% ATP.lambda_240
tff(fact_8425_ATP_Olambda__241,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xn(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_241
tff(fact_8426_ATP_Olambda__242,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ux(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_242
tff(fact_8427_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_tq(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_243
tff(fact_8428_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zb(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_244
tff(fact_8429_ATP_Olambda__245,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wm(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_245
tff(fact_8430_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_ei(fun(A,nat),fun(A,real),Uu2),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_246
tff(fact_8431_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uz(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_247
tff(fact_8432_ATP_Olambda__248,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_jc(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_248
tff(fact_8433_ATP_Olambda__249,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(product_prod(B,B))),Uua: A] : aa(A,set(product_prod(B,B)),aTP_Lamp_el(fun(A,set(product_prod(B,B))),fun(A,set(product_prod(B,B))),Uu2),Uua) = transitive_trancl(B,aa(A,set(product_prod(B,B)),Uu2,Uua)) ).

% ATP.lambda_249
tff(fact_8434_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fo(fun(nat,A),fun(nat,fun(A,A)),Uu2),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu2,Uua)) ) ).

% ATP.lambda_250
tff(fact_8435_ATP_Olambda__251,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wo(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,artanh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_251
tff(fact_8436_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_uo(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcsin,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_252
tff(fact_8437_ATP_Olambda__253,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wp(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu2,Uua)) ).

% ATP.lambda_253
tff(fact_8438_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vn(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,arccos,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_254
tff(fact_8439_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wn(fun(A,B),fun(A,B),Uu2),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_255
tff(fact_8440_ATP_Olambda__256,axiom,
    ! [Uu2: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qy(fun(nat,real),fun(nat,real),Uu2),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu2,Uua)) ).

% ATP.lambda_256
tff(fact_8441_ATP_Olambda__257,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu2: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_im(fun(B,A),fun(B,A),Uu2),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu2,Uua)) ) ).

% ATP.lambda_257
tff(fact_8442_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ub(fun(A,A),fun(A,A),Uu2),Uua) = tanh(A,aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_258
tff(fact_8443_ATP_Olambda__259,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_wr(fun(A,B),fun(A,B),Uu2),Uua) = tanh(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_259
tff(fact_8444_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wb(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,tan(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_260
tff(fact_8445_ATP_Olambda__261,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_hz(fun(A,B),fun(A,B),Uu2),Uua) = exp(B,aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_261
tff(fact_8446_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_wq(fun(A,A),fun(A,A),Uu2),Uua) = aa(A,A,cot(A),aa(A,A,Uu2,Uua)) ) ).

% ATP.lambda_262
tff(fact_8447_ATP_Olambda__263,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: fun(A,B),Uua: A] : aa(A,option(B),aTP_Lamp_kd(fun(A,B),fun(A,option(B)),Uu2),Uua) = aa(B,option(B),some(B),aa(A,B,Uu2,Uua)) ) ).

% ATP.lambda_263
tff(fact_8448_ATP_Olambda__264,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_pp(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu2),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu2,Uua)) ).

% ATP.lambda_264
tff(fact_8449_ATP_Olambda__265,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_aeo(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu2),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_265
tff(fact_8450_ATP_Olambda__266,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_kh(fun(A,set(B)),fun(A,nat),Uu2),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu2,Uua)) ).

% ATP.lambda_266
tff(fact_8451_ATP_Olambda__267,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_vm(fun(real,fun(nat,real)),fun(real,real),Uu2),Uua) = suminf(real,aa(real,fun(nat,real),Uu2,Uua)) ).

% ATP.lambda_267
tff(fact_8452_ATP_Olambda__268,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu2: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_xa(fun(A,fun(nat,B)),fun(A,B),Uu2),Uua) = suminf(B,aa(A,fun(nat,B),Uu2,Uua)) ) ).

% ATP.lambda_268
tff(fact_8453_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vj(fun(A,real),fun(A,real),Uu2),Uua) = aa(real,real,sqrt,aa(A,real,Uu2,Uua)) ) ).

% ATP.lambda_269
tff(fact_8454_ATP_Olambda__270,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_fp(fun(A,B),fun(A,fun(set(B),set(B))),Uu2),Uua) = aa(B,fun(set(B),set(B)),insert(B),aa(A,B,Uu2,Uua)) ).

% ATP.lambda_270
tff(fact_8455_ATP_Olambda__271,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_mj(fun(B,set(A)),fun(B,set(set(A))),Uu2),Uua) = pow(A,aa(B,set(A),Uu2,Uua)) ).

% ATP.lambda_271
tff(fact_8456_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_cz(fun(A,nat),fun(A,nat),Uu2),Uua) = aa(nat,nat,suc,aa(A,nat,Uu2,Uua)) ).

% ATP.lambda_272
tff(fact_8457_ATP_Olambda__273,axiom,
    ! [B: $tType,Uu2: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_lk(fun(B,$o),fun(B,$o),Uu2),Uua)
    <=> ~ aa(B,$o,Uu2,Uua) ) ).

% ATP.lambda_273
tff(fact_8458_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ba(fun(A,$o),fun(A,$o),Uu2),Uua)
    <=> ~ aa(A,$o,Uu2,Uua) ) ).

% ATP.lambda_274
tff(fact_8459_ATP_Olambda__275,axiom,
    ! [A: $tType,Uu2: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abt(list(A),fun(A,$o),Uu2),Uua)
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu2),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu2)) ) ) ).

% ATP.lambda_275
tff(fact_8460_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: set(set(option(A))),Uua: set(option(A))] :
          ( aa(set(option(A)),$o,aTP_Lamp_abv(set(set(option(A))),fun(set(option(A)),$o),Uu2),Uua)
        <=> ? [F7: fun(set(option(A)),option(A))] :
              ( ( Uua = aa(set(set(option(A))),set(option(A)),image(set(option(A)),option(A),F7),Uu2) )
              & ! [X5: set(option(A))] :
                  ( member(set(option(A)),X5,Uu2)
                 => member(option(A),aa(set(option(A)),option(A),F7,X5),X5) ) ) ) ) ).

% ATP.lambda_276
tff(fact_8461_ATP_Olambda__277,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_abw(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu2) )
              & ! [X5: set(A)] :
                  ( member(set(A),X5,Uu2)
                 => member(A,aa(set(A),A,F7,X5),X5) ) ) ) ) ).

% ATP.lambda_277
tff(fact_8462_ATP_Olambda__278,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_abx(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu2) )
              & ! [X5: set(A)] :
                  ( member(set(A),X5,Uu2)
                 => member(A,aa(set(A),A,F7,X5),X5) ) ) ) ) ).

% ATP.lambda_278
tff(fact_8463_ATP_Olambda__279,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu2: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aby(set(set(A)),fun(set(A),$o),Uu2),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F7),Uu2) )
              & ! [X5: set(A)] :
                  ( member(set(A),X5,Uu2)
                 => member(A,aa(set(A),A,F7,X5),X5) ) ) ) ) ).

% ATP.lambda_279
tff(fact_8464_ATP_Olambda__280,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aca(set(A),fun(set(A),$o),Uu2),Uua)
    <=> ? [B9: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B9) )
          & member(set(A),Uu2,pow(A,B9)) ) ) ).

% ATP.lambda_280
tff(fact_8465_ATP_Olambda__281,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ace(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> ! [M5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),M5)
         => ( aa(nat,$o,Uu2,M5)
          <=> aa(nat,$o,Uu2,Uua) ) ) ) ).

% ATP.lambda_281
tff(fact_8466_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu2: set(filter(A)),Uua: filter(A)] :
      ( aa(filter(A),$o,aTP_Lamp_abz(set(filter(A)),fun(filter(A),$o),Uu2),Uua)
    <=> ! [X5: filter(A)] :
          ( member(filter(A),X5,Uu2)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X5) ) ) ).

% ATP.lambda_282
tff(fact_8467_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_abo(set(A),fun(A,$o),Uu2),Uua)
        <=> ! [X5: A] :
              ( member(A,X5,Uu2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X5) ) ) ) ).

% ATP.lambda_283
tff(fact_8468_ATP_Olambda__284,axiom,
    ! [Uu2: set(real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_acb(set(real),fun(real,$o),Uu2),Uua)
    <=> ! [X5: real] :
          ( member(real,X5,Uu2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X5),Uua) ) ) ).

% ATP.lambda_284
tff(fact_8469_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_abp(set(A),fun(A,$o),Uu2),Uua)
        <=> ! [X5: A] :
              ( member(A,X5,Uu2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X5),Uua) ) ) ) ).

% ATP.lambda_285
tff(fact_8470_ATP_Olambda__286,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acg(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),N11)
         => aa(nat,$o,Uu2,N11) ) ) ).

% ATP.lambda_286
tff(fact_8471_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aan(fun(A,$o),fun(A,$o),Uu2),Uua)
        <=> ! [Y5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y5)
             => aa(A,$o,Uu2,Y5) ) ) ) ).

% ATP.lambda_287
tff(fact_8472_ATP_Olambda__288,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acf(fun(nat,$o),fun(nat,$o),Uu2),Uua)
    <=> ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),N11)
         => ~ aa(nat,$o,Uu2,N11) ) ) ).

% ATP.lambda_288
tff(fact_8473_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_acn(A,fun(product_prod(A,B),$o),Uu2),Uua)
    <=> ? [V6: B] : Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu2),V6) ) ).

% ATP.lambda_289
tff(fact_8474_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_abm(fun(B,A),fun(A,$o),Uu2),Uua)
    <=> ? [X5: B] : Uua = aa(B,A,Uu2,X5) ) ).

% ATP.lambda_290
tff(fact_8475_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: product_prod(product_prod($o,A),product_prod($o,B))] :
      ( aa(product_prod(product_prod($o,A),product_prod($o,B)),$o,aTP_Lamp_ack(set(product_prod(A,B)),fun(product_prod(product_prod($o,A),product_prod($o,B)),$o),Uu2),Uua)
    <=> ? [X5: A,Y5: B] :
          ( ( Uua = aa(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B)),aa(product_prod($o,A),fun(product_prod($o,B),product_prod(product_prod($o,A),product_prod($o,B))),product_Pair(product_prod($o,A),product_prod($o,B)),aa(A,product_prod($o,A),aa($o,fun(A,product_prod($o,A)),product_Pair($o,A),$false),X5)),aa(B,product_prod($o,B),aa($o,fun(B,product_prod($o,B)),product_Pair($o,B),$false),Y5)) )
          & member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y5),Uu2) ) ) ).

% ATP.lambda_291
tff(fact_8476_ATP_Olambda__292,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_abs(fun(A,option(B)),fun(product_prod(A,B),$o),Uu2),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),Uu2,A7) = aa(B,option(B),some(B),B7) ) ) ) ).

% ATP.lambda_292
tff(fact_8477_ATP_Olambda__293,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_aci(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu2),Uua)
    <=> ? [X9: set(A),Y8: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X9),Y8) )
          & ( X9 != bot_bot(set(A)) )
          & ! [X5: A] :
              ( member(A,X5,Y8)
             => ? [Xa3: A] :
                  ( member(A,Xa3,X9)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5),Uu2) ) ) ) ) ).

% ATP.lambda_293
tff(fact_8478_ATP_Olambda__294,axiom,
    ! [A: $tType,Uu2: nat,Uua: A] : aa(A,nat,aa(nat,fun(A,nat),aTP_Lamp_am(nat,fun(A,nat)),Uu2),Uua) = aa(nat,nat,suc,Uu2) ).

% ATP.lambda_294
tff(fact_8479_ATP_Olambda__295,axiom,
    ! [Uu2: 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_qc(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu2)),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),Uu2))),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_295
tff(fact_8480_ATP_Olambda__296,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_qd(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu2),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu2)),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),Uu2))),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_296
tff(fact_8481_ATP_Olambda__297,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_qe(num,fun(int,fun(int,product_prod(int,int))),Uu2),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu2)),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),Uu2))),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_297
tff(fact_8482_ATP_Olambda__298,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu2: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_qf(num,fun(A,fun(A,product_prod(A,A))),Uu2),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu2)),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),Uu2))),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_298
tff(fact_8483_ATP_Olambda__299,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qo(set(nat),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(member(nat,Uub,Uu2),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_299
tff(fact_8484_ATP_Olambda__300,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_ly(fun(A,B),fun(set(A),fun(A,B)),Uu2),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu2,Uub),zero_zero(B)) ) ).

% ATP.lambda_300
tff(fact_8485_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_lz(fun(A,B),fun(set(A),fun(A,B)),Uu2),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_301
tff(fact_8486_ATP_Olambda__302,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ak(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_302
tff(fact_8487_ATP_Olambda__303,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gd(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uu2 = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_303
tff(fact_8488_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qj(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_304
tff(fact_8489_ATP_Olambda__305,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_al(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_305
tff(fact_8490_ATP_Olambda__306,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gc(A,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = $ite(Uub = Uu2,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_306
tff(fact_8491_ATP_Olambda__307,axiom,
    ! [Uu2: 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_acv(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu2)),Uub))) ).

% ATP.lambda_307
tff(fact_8492_ATP_Olambda__308,axiom,
    ! [Uu2: 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_act(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu2)),Uub))) ).

% ATP.lambda_308
tff(fact_8493_ATP_Olambda__309,axiom,
    ! [Uu2: 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_acu(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu2),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu2),Uub))) ).

% ATP.lambda_309
tff(fact_8494_ATP_Olambda__310,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ms(fun(A,$o),fun(A,fun(list(A),list(A))),Uu2),Uua),Uub) = $ite(aa(A,$o,Uu2,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).

% ATP.lambda_310
tff(fact_8495_ATP_Olambda__311,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_hr(fun(A,$o),fun(A,fun(set(A),set(A))),Uu2),Uua),Uub) = $ite(aa(A,$o,Uu2,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uua),Uub),Uub) ).

% ATP.lambda_311
tff(fact_8496_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qn(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = $ite(aa(nat,$o,Uu2,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_312
tff(fact_8497_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_mr(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu2),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu2,Uub),zero_zero(A)) ) ).

% ATP.lambda_313
tff(fact_8498_ATP_Olambda__314,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_ce(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),zero_zero(B)) ) ).

% ATP.lambda_314
tff(fact_8499_ATP_Olambda__315,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_gk(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu2,Uub),one_one(B)) ) ).

% ATP.lambda_315
tff(fact_8500_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_ka(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu2),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_jz(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu2) ).

% ATP.lambda_316
tff(fact_8501_ATP_Olambda__317,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_afq(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),Uu2),Uua),Uub) = merges9089515139780605204_merge(A,Uu2,aa(list(A),list(A),mergesort_by_rel(A,Uu2),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu2),Uub)) ).

% ATP.lambda_317
tff(fact_8502_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu2: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ads(fun(A,A),fun(A,fun(nat,A)),Uu2),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),Uu2),Uua) ).

% ATP.lambda_318
tff(fact_8503_ATP_Olambda__319,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu2: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_aep(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu2),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu2,aa(A,D,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_319
tff(fact_8504_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_or(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_320
tff(fact_8505_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_op(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).

% ATP.lambda_321
tff(fact_8506_ATP_Olambda__322,axiom,
    ! [Uu2: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_vl(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu2,Uub),Uua) ).

% ATP.lambda_322
tff(fact_8507_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_323
tff(fact_8508_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fr(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu2,Uub),Uua) ) ).

% ATP.lambda_324
tff(fact_8509_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_iv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu2,Uub),Uua) ) ).

% ATP.lambda_325
tff(fact_8510_ATP_Olambda__326,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_we(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_326
tff(fact_8511_ATP_Olambda__327,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_vd(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_327
tff(fact_8512_ATP_Olambda__328,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_gi(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_328
tff(fact_8513_ATP_Olambda__329,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_bp(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu2),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu2,Uub),Uua) ) ).

% ATP.lambda_329
tff(fact_8514_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,B)),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ke(fun(A,fun(B,B)),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(A,fun(B,B),Uu2,Uub),Uua) ).

% ATP.lambda_330
tff(fact_8515_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ry(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rx(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_331
tff(fact_8516_ATP_Olambda__332,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ru(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rt(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_332
tff(fact_8517_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_rs(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rr(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_333
tff(fact_8518_ATP_Olambda__334,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_rm(nat,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu2,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_334
tff(fact_8519_ATP_Olambda__335,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_acw(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu2),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu2),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_335
tff(fact_8520_ATP_Olambda__336,axiom,
    ! [A: $tType,Uu2: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tn(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu2,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_336
tff(fact_8521_ATP_Olambda__337,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),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_gt(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_337
tff(fact_8522_ATP_Olambda__338,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fs(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),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_fr(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu2),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_338
tff(fact_8523_ATP_Olambda__339,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ve(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_vd(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu2) ) ).

% ATP.lambda_339
tff(fact_8524_ATP_Olambda__340,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_uc(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_340
tff(fact_8525_ATP_Olambda__341,axiom,
    ! [Uu2: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_ud(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ).

% ATP.lambda_341
tff(fact_8526_ATP_Olambda__342,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu2: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ih(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ) ).

% ATP.lambda_342
tff(fact_8527_ATP_Olambda__343,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ey(fun(nat,A),fun(nat,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_343
tff(fact_8528_ATP_Olambda__344,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wx(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = 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,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_344
tff(fact_8529_ATP_Olambda__345,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wu(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = 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,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu2,Uua))),Uub) ) ).

% ATP.lambda_345
tff(fact_8530_ATP_Olambda__346,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),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,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_346
tff(fact_8531_ATP_Olambda__347,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_wv(fun(A,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = 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,Uu2,Uub)),aa(A,A,Uu2,Uua))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),Uua)) ) ).

% ATP.lambda_347
tff(fact_8532_ATP_Olambda__348,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ul(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_348
tff(fact_8533_ATP_Olambda__349,axiom,
    ! [Uu2: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ig(real,fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ).

% ATP.lambda_349
tff(fact_8534_ATP_Olambda__350,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_jh(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_350
tff(fact_8535_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_sa(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu2,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_351
tff(fact_8536_ATP_Olambda__352,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_qa(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu2)) ) ) ) ).

% ATP.lambda_352
tff(fact_8537_ATP_Olambda__353,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_abq(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X5: A,Y5: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Xs6)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5),Uu2) ) ) ) ).

% ATP.lambda_353
tff(fact_8538_ATP_Olambda__354,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_ji(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_354
tff(fact_8539_ATP_Olambda__355,axiom,
    ! [A: $tType,Uu2: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_gb(nat,fun(set(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_355
tff(fact_8540_ATP_Olambda__356,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ar(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2) ) ) ).

% ATP.lambda_356
tff(fact_8541_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_hf(nat,fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_357
tff(fact_8542_ATP_Olambda__358,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_dn(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_358
tff(fact_8543_ATP_Olambda__359,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_dk(set(A),fun(nat,fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu2)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_359
tff(fact_8544_ATP_Olambda__360,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_jg(nat,fun(nat,fun(list(nat),$o)),Uu2),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu2 )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_360
tff(fact_8545_ATP_Olambda__361,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tx(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu2)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_361
tff(fact_8546_ATP_Olambda__362,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_db(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_362
tff(fact_8547_ATP_Olambda__363,axiom,
    ! [A: $tType,Uu2: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_afc(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu2),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu2) ) ).

% ATP.lambda_363
tff(fact_8548_ATP_Olambda__364,axiom,
    ! [Uu2: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_mk(nat,fun(nat,fun(set(nat),$o)),Uu2),Uua),Uub)
    <=> ( member(set(nat),Uub,pow(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu2)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_364
tff(fact_8549_ATP_Olambda__365,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_afd(list(A),fun(set(nat),fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu2))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_365
tff(fact_8550_ATP_Olambda__366,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_it(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,vEBT_set_vebt(Uu2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_366
tff(fact_8551_ATP_Olambda__367,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_iw(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,vEBT_set_vebt(Uu2))
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_367
tff(fact_8552_ATP_Olambda__368,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_mu(fun(A,$o),fun(list(A),fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu2,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_368
tff(fact_8553_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_iz(list(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu2))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_369
tff(fact_8554_ATP_Olambda__370,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ia(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uu2),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_370
tff(fact_8555_ATP_Olambda__371,axiom,
    ! [Uu2: list(vEBT_VEBT),Uua: array(vEBT_VEBTi),Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),assn,aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_hp(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Uua),Uub)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uu2),Uub)) ).

% ATP.lambda_371
tff(fact_8556_ATP_Olambda__372,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu2: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_il(set(A),fun(nat,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,Uu2),Uua)),Uub) ) ) ) ).

% ATP.lambda_372
tff(fact_8557_ATP_Olambda__373,axiom,
    ! [Uu2: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aff(set(nat),fun(set(nat),fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,Uu2)
        & member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_afe(set(nat),fun(nat,fun(nat,$o)),Uu2),Uub))),Uua) ) ) ).

% ATP.lambda_373
tff(fact_8558_ATP_Olambda__374,axiom,
    ! [A: $tType,Uu2: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_hg(set(A),fun(nat,fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu2)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_374
tff(fact_8559_ATP_Olambda__375,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_da(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_375
tff(fact_8560_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_rz(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_376
tff(fact_8561_ATP_Olambda__377,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_tv(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu2),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_377
tff(fact_8562_ATP_Olambda__378,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sd(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uub)) ) ).

% ATP.lambda_378
tff(fact_8563_ATP_Olambda__379,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_hj(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = 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),Uu2),Uub)) ).

% ATP.lambda_379
tff(fact_8564_ATP_Olambda__380,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_lc(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_380
tff(fact_8565_ATP_Olambda__381,axiom,
    ! [A: $tType,Uu2: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ax(A,fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_381
tff(fact_8566_ATP_Olambda__382,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_sf(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_382
tff(fact_8567_ATP_Olambda__383,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: word(A),Uua: word(A),Uub: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_av(word(A),fun(word(A),fun(word(A),$o)),Uu2),Uua),Uub)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uu2),Uub)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uub),Uua) ) ) ) ).

% ATP.lambda_383
tff(fact_8568_ATP_Olambda__384,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dw(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_384
tff(fact_8569_ATP_Olambda__385,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_se(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_385
tff(fact_8570_ATP_Olambda__386,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dz(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_386
tff(fact_8571_ATP_Olambda__387,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dy(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_387
tff(fact_8572_ATP_Olambda__388,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_dx(int,fun(int,fun(int,$o)),Uu2),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu2),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_388
tff(fact_8573_ATP_Olambda__389,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ae(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu2),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uub) ) ) ).

% ATP.lambda_389
tff(fact_8574_ATP_Olambda__390,axiom,
    ! [Uu2: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_af(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu2),Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_390
tff(fact_8575_ATP_Olambda__391,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_afp(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu2) ) ) ).

% ATP.lambda_391
tff(fact_8576_ATP_Olambda__392,axiom,
    ! [Uu2: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_afe(set(nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( member(nat,Uub,Uu2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_392
tff(fact_8577_ATP_Olambda__393,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_lp(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & member(A,Uub,Uua) ) ) ).

% ATP.lambda_393
tff(fact_8578_ATP_Olambda__394,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_cm(set(A),fun(set(A),fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uub),Uu2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uua),Uu2) ) ) ).

% ATP.lambda_394
tff(fact_8579_ATP_Olambda__395,axiom,
    ! [A: $tType,Uu2: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_jd(list(A),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,count_list(A,Uu2),Uub)),aa(A,nat,Uua,Uub)) ).

% ATP.lambda_395
tff(fact_8580_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu2: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aw(set(A),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_396
tff(fact_8581_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_hu(fun(A,$o),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu2,Uub) ) ) ).

% ATP.lambda_397
tff(fact_8582_ATP_Olambda__398,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bl(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uu2 = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_398
tff(fact_8583_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bm(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub = Uu2 )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_399
tff(fact_8584_ATP_Olambda__400,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_by(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_400
tff(fact_8585_ATP_Olambda__401,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ik(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_401
tff(fact_8586_ATP_Olambda__402,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ca(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_402
tff(fact_8587_ATP_Olambda__403,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_in(fun(B,A),fun(set(B),fun(B,$o)),Uu2),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu2,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_403
tff(fact_8588_ATP_Olambda__404,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ju(fun(B,A),fun(set(B),fun(B,$o)),Uu2),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu2,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_404
tff(fact_8589_ATP_Olambda__405,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_nz(set(A),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,Uu2)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_405
tff(fact_8590_ATP_Olambda__406,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa($o,assn,aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_dp(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Uu2),Uua),(Uub)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn((Uub) = vEBT_VEBT_minNull(Uu2))) ).

% ATP.lambda_406
tff(fact_8591_ATP_Olambda__407,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_dt(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn(Uub = vEBT_vebt_mint(Uu2))) ).

% ATP.lambda_407
tff(fact_8592_ATP_Olambda__408,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_du(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn(Uub = vEBT_vebt_maxt(Uu2))) ).

% ATP.lambda_408
tff(fact_8593_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu2: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_bd(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( member(A,Uub,Uu2)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_409
tff(fact_8594_ATP_Olambda__410,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu2) ) ).

% ATP.lambda_410
tff(fact_8595_ATP_Olambda__411,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_oo(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_411
tff(fact_8596_ATP_Olambda__412,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_yq(A,fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_412
tff(fact_8597_ATP_Olambda__413,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu2: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ys(real,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu2) ) ) ).

% ATP.lambda_413
tff(fact_8598_ATP_Olambda__414,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ot(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu2) ) ).

% ATP.lambda_414
tff(fact_8599_ATP_Olambda__415,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ha(A,fun(A,fun(A,A)),Uu2),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),Uu2)),Uua) ) ).

% ATP.lambda_415
tff(fact_8600_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_fx(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_416
tff(fact_8601_ATP_Olambda__417,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_gz(A,fun(A,fun(A,A)),Uu2),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),Uu2)),Uua) ) ).

% ATP.lambda_417
tff(fact_8602_ATP_Olambda__418,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu2: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_fw(A,fun(A,fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),Uub)),Uua) ) ).

% ATP.lambda_418
tff(fact_8603_ATP_Olambda__419,axiom,
    ! [B: $tType,A: $tType,Uu2: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_bs(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu2),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu2) ) ).

% ATP.lambda_419
tff(fact_8604_ATP_Olambda__420,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaz(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub),Uu2) ) ).

% ATP.lambda_420
tff(fact_8605_ATP_Olambda__421,axiom,
    ! [A: $tType,Uu2: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sp(list(list(A)),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu2),Uub)),Uua) ).

% ATP.lambda_421
tff(fact_8606_ATP_Olambda__422,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_eb(nat,fun(complex,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu2) = Uua ) ) ).

% ATP.lambda_422
tff(fact_8607_ATP_Olambda__423,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_ee(complex,fun(nat,fun(complex,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_423
tff(fact_8608_ATP_Olambda__424,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu2: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cj(A,fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_424
tff(fact_8609_ATP_Olambda__425,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_sm(fun(A,B),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_425
tff(fact_8610_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_qu(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_426
tff(fact_8611_ATP_Olambda__427,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_qw(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_427
tff(fact_8612_ATP_Olambda__428,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_um(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_428
tff(fact_8613_ATP_Olambda__429,axiom,
    ! [Uu2: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_rb(fun(nat,real),fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu2,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_429
tff(fact_8614_ATP_Olambda__430,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fc(fun(nat,nat),fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_430
tff(fact_8615_ATP_Olambda__431,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_431
tff(fact_8616_ATP_Olambda__432,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_432
tff(fact_8617_ATP_Olambda__433,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_qv(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_433
tff(fact_8618_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_sc(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_434
tff(fact_8619_ATP_Olambda__435,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ex(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_435
tff(fact_8620_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ez(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_436
tff(fact_8621_ATP_Olambda__437,axiom,
    ! [Uu2: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bt(fun(nat,$o),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> ( aa(nat,$o,Uu2,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_437
tff(fact_8622_ATP_Olambda__438,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_sk(fun(A,B),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(A,B,Uu2,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% ATP.lambda_438
tff(fact_8623_ATP_Olambda__439,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_sl(fun(A,B),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uu2,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% ATP.lambda_439
tff(fact_8624_ATP_Olambda__440,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_ed(fun(B,A),fun(B,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu2,Uua)),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_440
tff(fact_8625_ATP_Olambda__441,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_zp(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_441
tff(fact_8626_ATP_Olambda__442,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zh(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_442
tff(fact_8627_ATP_Olambda__443,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zm(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_443
tff(fact_8628_ATP_Olambda__444,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zw(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_444
tff(fact_8629_ATP_Olambda__445,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_445
tff(fact_8630_ATP_Olambda__446,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_uu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_446
tff(fact_8631_ATP_Olambda__447,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_tp(fun(A,A),fun(fun(A,A),fun(A,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu2,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_447
tff(fact_8632_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aav(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_448
tff(fact_8633_ATP_Olambda__449,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_449
tff(fact_8634_ATP_Olambda__450,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_450
tff(fact_8635_ATP_Olambda__451,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_dv(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_451
tff(fact_8636_ATP_Olambda__452,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ds(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_452
tff(fact_8637_ATP_Olambda__453,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aaf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_453
tff(fact_8638_ATP_Olambda__454,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vy(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_454
tff(fact_8639_ATP_Olambda__455,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_yv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_455
tff(fact_8640_ATP_Olambda__456,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_js(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_456
tff(fact_8641_ATP_Olambda__457,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aat(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu2,Uub)) ).

% ATP.lambda_457
tff(fact_8642_ATP_Olambda__458,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_is(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_458
tff(fact_8643_ATP_Olambda__459,axiom,
    ! [Uu2: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_xm(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(nat,real,Uu2,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_459
tff(fact_8644_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_460
tff(fact_8645_ATP_Olambda__461,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rn(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,Uua,Uub)),aa(nat,A,Uu2,Uub)) ) ).

% ATP.lambda_461
tff(fact_8646_ATP_Olambda__462,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_br(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_462
tff(fact_8647_ATP_Olambda__463,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ck(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uua,Uub)),aa(A,nat,Uu2,Uub)) ).

% ATP.lambda_463
tff(fact_8648_ATP_Olambda__464,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wk(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uua,Uub)),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_464
tff(fact_8649_ATP_Olambda__465,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_lt(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu2,Uub)),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_465
tff(fact_8650_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vx(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_466
tff(fact_8651_ATP_Olambda__467,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ip(fun(A,B),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_467
tff(fact_8652_ATP_Olambda__468,axiom,
    ! [Uu2: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_ua(fun(real,real),fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_468
tff(fact_8653_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vh(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_469
tff(fact_8654_ATP_Olambda__470,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aak(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_470
tff(fact_8655_ATP_Olambda__471,axiom,
    ! [A: $tType,Uu2: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_wg(fun(A,real),fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(real,real,log(aa(A,real,Uu2,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_471
tff(fact_8656_ATP_Olambda__472,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: 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_ws(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu2),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu2,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_472
tff(fact_8657_ATP_Olambda__473,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ld(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_473
tff(fact_8658_ATP_Olambda__474,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_jn(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu2),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),Uu2,Uub)) ).

% ATP.lambda_474
tff(fact_8659_ATP_Olambda__475,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ac(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_475
tff(fact_8660_ATP_Olambda__476,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ad(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,$o,Uu2,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_476
tff(fact_8661_ATP_Olambda__477,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zd(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uu2,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_477
tff(fact_8662_ATP_Olambda__478,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_aei(A,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu2) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_478
tff(fact_8663_ATP_Olambda__479,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_sh(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_479
tff(fact_8664_ATP_Olambda__480,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aah(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_480
tff(fact_8665_ATP_Olambda__481,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ts(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu2,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_481
tff(fact_8666_ATP_Olambda__482,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_ls(fun(A,assn),fun(fun(A,assn),fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu2,Uub)),ex_assn(A,Uua)) ).

% ATP.lambda_482
tff(fact_8667_ATP_Olambda__483,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aco(fun(A,option(B)),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,option(B),Uu2,Uua) = aa(B,option(B),some(B),Uub) ) ) ).

% ATP.lambda_483
tff(fact_8668_ATP_Olambda__484,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu2: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_wf(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu2),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),Uua,Uub)),Uu2) ) ).

% ATP.lambda_484
tff(fact_8669_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu2: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_zq(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_485
tff(fact_8670_ATP_Olambda__486,axiom,
    ! [Uu2: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bu(fun(nat,nat),fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu2,Uub)),Uua) ) ).

% ATP.lambda_486
tff(fact_8671_ATP_Olambda__487,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_kx(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu2),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ) ).

% ATP.lambda_487
tff(fact_8672_ATP_Olambda__488,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zy(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_488
tff(fact_8673_ATP_Olambda__489,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zf(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_489
tff(fact_8674_ATP_Olambda__490,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zu(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_490
tff(fact_8675_ATP_Olambda__491,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ze(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_491
tff(fact_8676_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_qm(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_492
tff(fact_8677_ATP_Olambda__493,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ob(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_493
tff(fact_8678_ATP_Olambda__494,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_494
tff(fact_8679_ATP_Olambda__495,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xe(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_495
tff(fact_8680_ATP_Olambda__496,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_496
tff(fact_8681_ATP_Olambda__497,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zj(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_497
tff(fact_8682_ATP_Olambda__498,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zg(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_498
tff(fact_8683_ATP_Olambda__499,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu2,Uub)),Uua) ) ) ).

% ATP.lambda_499
tff(fact_8684_ATP_Olambda__500,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_hv(fun(A,assn),fun(assn,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu2,Uub)),Uua) ).

% ATP.lambda_500
tff(fact_8685_ATP_Olambda__501,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_501
tff(fact_8686_ATP_Olambda__502,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ri(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_502
tff(fact_8687_ATP_Olambda__503,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xd(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_503
tff(fact_8688_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vt(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu2) ) ).

% ATP.lambda_504
tff(fact_8689_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_505
tff(fact_8690_ATP_Olambda__506,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wj(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_506
tff(fact_8691_ATP_Olambda__507,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_tt(fun(A,A),fun(nat,fun(A,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu2,Uub)),Uua) ) ).

% ATP.lambda_507
tff(fact_8692_ATP_Olambda__508,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_yw(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_508
tff(fact_8693_ATP_Olambda__509,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wh(fun(A,B),fun(nat,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_509
tff(fact_8694_ATP_Olambda__510,axiom,
    ! [Uu2: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_yz(nat,fun(fun(real,real),fun(real,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uua,Uub)),Uu2) ).

% ATP.lambda_510
tff(fact_8695_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu2: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_vu(nat,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu2) ).

% ATP.lambda_511
tff(fact_8696_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lb(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_512
tff(fact_8697_ATP_Olambda__513,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_lr(fun(A,assn),fun(assn,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),sup_sup(assn),aa(A,assn,Uu2,Uub)),Uua) ).

% ATP.lambda_513
tff(fact_8698_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lm(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu2,Uub)),Uua) ).

% ATP.lambda_514
tff(fact_8699_ATP_Olambda__515,axiom,
    ! [A: $tType,Uu2: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_mg(fun(A,assn),fun(assn,fun(A,assn)),Uu2),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),inf_inf(assn),aa(A,assn,Uu2,Uub)),Uua) ).

% ATP.lambda_515
tff(fact_8700_ATP_Olambda__516,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_lx(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_516
tff(fact_8701_ATP_Olambda__517,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_dg(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),Uua) ) ).

% ATP.lambda_517
tff(fact_8702_ATP_Olambda__518,axiom,
    ! [Uu2: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_tz(fun(real,real),fun(real,fun(real,real)),Uu2),Uua),Uub) = powr(real,aa(real,real,Uu2,Uub),Uua) ).

% ATP.lambda_518
tff(fact_8703_ATP_Olambda__519,axiom,
    ! [A: $tType,Uu2: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_aau(real,fun(fun(A,real),fun(A,real)),Uu2),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu2) ).

% ATP.lambda_519
tff(fact_8704_ATP_Olambda__520,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_adi(set(A),fun(fun(B,A),fun(B,$o)),Uu2),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu2) ) ).

% ATP.lambda_520
tff(fact_8705_ATP_Olambda__521,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tj(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ( aa(A,B,Uu2,Uub) = Uua ) ) ) ).

% ATP.lambda_521
tff(fact_8706_ATP_Olambda__522,axiom,
    ! [A: $tType,Uu2: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ay(A,fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( ( Uub != Uu2 )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_522
tff(fact_8707_ATP_Olambda__523,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hk(nat,fun(nat,fun(nat,A)),Uu2),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),Uu2),Uub))) ) ).

% ATP.lambda_523
tff(fact_8708_ATP_Olambda__524,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_rc(real,fun(fun(nat,A),fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ) ).

% ATP.lambda_524
tff(fact_8709_ATP_Olambda__525,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aab(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_525
tff(fact_8710_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu2: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sg(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu2,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_526
tff(fact_8711_ATP_Olambda__527,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_aaq(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu2,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_527
tff(fact_8712_ATP_Olambda__528,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aaw(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))),Uua) ) ) ).

% ATP.lambda_528
tff(fact_8713_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aag(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_529
tff(fact_8714_ATP_Olambda__530,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_aay(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,finite_finite2(A),Uub)
        & ( Uub != bot_bot(set(A)) )
        & ! [X5: A] :
            ( member(A,X5,Uua)
           => ? [Xa3: A] :
                ( member(A,Xa3,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3),Uu2) ) ) ) ) ).

% ATP.lambda_530
tff(fact_8715_ATP_Olambda__531,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A] : aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_td(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu2),Uua),Uub) = case_list(list(A),A,Uua,aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_tc(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_531
tff(fact_8716_ATP_Olambda__532,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu2: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_acy(nat,fun(nat,fun(word(A),$o)),Uu2),Uua),Uub)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less(word(A)),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_532
tff(fact_8717_ATP_Olambda__533,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_acz(nat,fun(nat,fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_533
tff(fact_8718_ATP_Olambda__534,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rp(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),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_534
tff(fact_8719_ATP_Olambda__535,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: list(A),Uub: B] : aa(B,list(A),aa(list(A),fun(B,list(A)),aTP_Lamp_ml(fun(B,A),fun(list(A),fun(B,list(A))),Uu2),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu2,Uub)),nil(A))) ).

% ATP.lambda_535
tff(fact_8720_ATP_Olambda__536,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_nf(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_536
tff(fact_8721_ATP_Olambda__537,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_nc(A,fun(A,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_537
tff(fact_8722_ATP_Olambda__538,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: array(A)] : aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_sx(nat,fun(fun(nat,A),fun(array(A),assn)),Uu2),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu2))) ) ).

% ATP.lambda_538
tff(fact_8723_ATP_Olambda__539,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_df(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_539
tff(fact_8724_ATP_Olambda__540,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_xu(real,fun(real,fun(nat,real)),Uu2),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu2),Uub)) ).

% ATP.lambda_540
tff(fact_8725_ATP_Olambda__541,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_nd(nat,fun(nat,fun(nat,nat)),Uu2),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_541
tff(fact_8726_ATP_Olambda__542,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_pl(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu2),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)),Uu2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).

% ATP.lambda_542
tff(fact_8727_ATP_Olambda__543,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_pm(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu2),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),Uu2),Uub)) ).

% ATP.lambda_543
tff(fact_8728_ATP_Olambda__544,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_zr(real,fun(real,fun(real,$o)),Uu2),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu2,Uua)) ) ).

% ATP.lambda_544
tff(fact_8729_ATP_Olambda__545,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_me(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu2,Uub)) ) ).

% ATP.lambda_545
tff(fact_8730_ATP_Olambda__546,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_md(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu2,Uub)) ) ).

% ATP.lambda_546
tff(fact_8731_ATP_Olambda__547,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zt(B,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu2),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_547
tff(fact_8732_ATP_Olambda__548,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zx(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_548
tff(fact_8733_ATP_Olambda__549,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zv(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_549
tff(fact_8734_ATP_Olambda__550,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zn(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_550
tff(fact_8735_ATP_Olambda__551,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zk(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_551
tff(fact_8736_ATP_Olambda__552,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tl(fun(B,A),fun(A,fun(B,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu2,Uub)) ) ) ).

% ATP.lambda_552
tff(fact_8737_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zo(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_553
tff(fact_8738_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_zi(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_554
tff(fact_8739_ATP_Olambda__555,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tk(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_555
tff(fact_8740_ATP_Olambda__556,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ql(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_556
tff(fact_8741_ATP_Olambda__557,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rj(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_557
tff(fact_8742_ATP_Olambda__558,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xc(A,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_558
tff(fact_8743_ATP_Olambda__559,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu2: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_vs(A,fun(fun(B,A),fun(B,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu2),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_559
tff(fact_8744_ATP_Olambda__560,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wa(fun(A,B),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_560
tff(fact_8745_ATP_Olambda__561,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kv(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_561
tff(fact_8746_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu2: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_xz(fun(A,nat),fun(B,fun(A,B)),Uu2),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu2,Uub)) ) ).

% ATP.lambda_562
tff(fact_8747_ATP_Olambda__563,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_la(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_563
tff(fact_8748_ATP_Olambda__564,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ll(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_564
tff(fact_8749_ATP_Olambda__565,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_lw(B,fun(fun(A,B),fun(A,B)),Uu2),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu2),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_565
tff(fact_8750_ATP_Olambda__566,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_oc(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_566
tff(fact_8751_ATP_Olambda__567,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: 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_aeh(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu2),Uua),Uub) = filtercomap(A,C,Uu2,aa(B,filter(C),Uua,Uub)) ).

% ATP.lambda_567
tff(fact_8752_ATP_Olambda__568,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_po(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu2),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu2,Uub)) ).

% ATP.lambda_568
tff(fact_8753_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_kj(A,fun(fun(B,set(A)),fun(B,set(A))),Uu2),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_569
tff(fact_8754_ATP_Olambda__570,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: 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_adr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu2),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu2),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_570
tff(fact_8755_ATP_Olambda__571,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_iy(fun(A,$o),fun(A,fun($o,$o)),Uu2),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_571
tff(fact_8756_ATP_Olambda__572,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_abr(fun(A,$o),fun(A,fun($o,$o)),Uu2),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu2,Uua) ) ) ).

% ATP.lambda_572
tff(fact_8757_ATP_Olambda__573,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_mp(fun(list(A),A),fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu2,Uua) ) ) ) ).

% ATP.lambda_573
tff(fact_8758_ATP_Olambda__574,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_adv(fun(A,B),fun(A,fun(B,$o)),Uu2),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu2,Uua) ) ) ).

% ATP.lambda_574
tff(fact_8759_ATP_Olambda__575,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hx(A,fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu2),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub))) ) ).

% ATP.lambda_575
tff(fact_8760_ATP_Olambda__576,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu2: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aai(fun(A,B),fun(real,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uub))) ) ) ).

% ATP.lambda_576
tff(fact_8761_ATP_Olambda__577,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_cs(fun(A,nat),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uub))) ) ).

% ATP.lambda_577
tff(fact_8762_ATP_Olambda__578,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cd(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu2,Uub),zero_zero(nat))) ) ).

% ATP.lambda_578
tff(fact_8763_ATP_Olambda__579,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_cw(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu2),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_579
tff(fact_8764_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gp(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_580
tff(fact_8765_ATP_Olambda__581,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_bj(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_581
tff(fact_8766_ATP_Olambda__582,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xi(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_582
tff(fact_8767_ATP_Olambda__583,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_vv(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_583
tff(fact_8768_ATP_Olambda__584,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ra(fun(nat,A),fun(nat,fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_584
tff(fact_8769_ATP_Olambda__585,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_zs(fun(A,$o),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> aa(A,$o,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_585
tff(fact_8770_ATP_Olambda__586,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_wz(fun(A,B),fun(A,fun(A,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_586
tff(fact_8771_ATP_Olambda__587,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_ol(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(product_prod(A,B),C,Uu2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_587
tff(fact_8772_ATP_Olambda__588,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_st(fun(A,$o),fun(list(A),fun(nat,$o)),Uu2),Uua),Uub)
    <=> aa(A,$o,Uu2,aa(nat,A,nth(A,Uua),Uub)) ) ).

% ATP.lambda_588
tff(fact_8773_ATP_Olambda__589,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_wt(B,fun(fun(B,C),fun(B,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu2),Uub)) ) ).

% ATP.lambda_589
tff(fact_8774_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_rl(nat,fun(fun(nat,A),fun(nat,A)),Uu2),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu2)) ) ).

% ATP.lambda_590
tff(fact_8775_ATP_Olambda__591,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu2: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_aeq(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu2),Uua),Uub) = aa(D,fun(B,C),Uu2,aa(A,D,Uua,Uub)) ).

% ATP.lambda_591
tff(fact_8776_ATP_Olambda__592,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_lq(fun(C,A),fun(fun(B,C),fun(B,A)),Uu2),Uua),Uub) = aa(C,A,Uu2,aa(B,C,Uua,Uub)) ).

% ATP.lambda_592
tff(fact_8777_ATP_Olambda__593,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_bf(fun(B,A),fun(fun(C,B),fun(C,A)),Uu2),Uua),Uub) = aa(B,A,Uu2,aa(C,B,Uua,Uub)) ).

% ATP.lambda_593
tff(fact_8778_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_us(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_594
tff(fact_8779_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_yi(fun(A,B),fun(fun(C,A),fun(C,B)),Uu2),Uua),Uub) = aa(A,B,Uu2,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_595
tff(fact_8780_ATP_Olambda__596,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_lu(fun(B,A),fun(fun(A,C),fun(B,C)),Uu2),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu2,Uub)) ).

% ATP.lambda_596
tff(fact_8781_ATP_Olambda__597,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ww(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_597
tff(fact_8782_ATP_Olambda__598,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_yr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_598
tff(fact_8783_ATP_Olambda__599,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( semiring_1(C)
     => ! [Uu2: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ct(fun(A,B),fun(fun(B,C),fun(A,C)),Uu2),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu2,Uub)) ) ).

% ATP.lambda_599
tff(fact_8784_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(C,fun(A,C)),Uua: C,Uub: B] : aa(B,fun(A,C),aa(C,fun(B,fun(A,C)),aTP_Lamp_ow(fun(C,fun(A,C)),fun(C,fun(B,fun(A,C))),Uu2),Uua),Uub) = aa(C,fun(A,C),Uu2,Uua) ).

% ATP.lambda_600
tff(fact_8785_ATP_Olambda__601,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_aem(fun(A,C),fun(A,fun(B,C)),Uu2),Uua),Uub) = aa(A,C,Uu2,Uua) ).

% ATP.lambda_601
tff(fact_8786_ATP_Olambda__602,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_vq(fun(nat,A),fun(A,fun(A,A)),Uu2),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vp(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub)) ) ).

% ATP.lambda_602
tff(fact_8787_ATP_Olambda__603,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu2: A,Uua: set(A),Uub: set(A)] : aa(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_afy(A,fun(set(A),fun(set(A),filter(A))),Uu2),Uua),Uub) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uub),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uu2),bot_bot(set(A))))) ) ).

% ATP.lambda_603
tff(fact_8788_ATP_Olambda__604,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_qr(fun(nat,A),fun(A,fun(nat,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_604
tff(fact_8789_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_gq(fun(A,B),fun(fun(A,B),fun(A,real)),Uu2),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_605
tff(fact_8790_ATP_Olambda__606,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_jj(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uu2,Uub) != Uua ) ) ).

% ATP.lambda_606
tff(fact_8791_ATP_Olambda__607,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_jk(B,fun(fun(A,B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu2 ) ) ).

% ATP.lambda_607
tff(fact_8792_ATP_Olambda__608,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_nu(nat,fun(nat,fun(nat,assn)),Uu2),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu2,Uua)) ).

% ATP.lambda_608
tff(fact_8793_ATP_Olambda__609,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_nt(nat,fun(nat,fun(nat,assn)),Uu2),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu2,Uua)) ).

% ATP.lambda_609
tff(fact_8794_ATP_Olambda__610,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_od(fun(A,B),fun(B,fun(A,$o)),Uu2),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu2,Uub)) ) ) ).

% ATP.lambda_610
tff(fact_8795_ATP_Olambda__611,axiom,
    ! [B: $tType,A: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_cl(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu2),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bn(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub))) ).

% ATP.lambda_611
tff(fact_8796_ATP_Olambda__612,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
      ( aa(A,$o,aa(set(C),fun(A,$o),aTP_Lamp_aev(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [B7: B] : member(C,aa(B,C,aa(A,fun(B,C),Uu2,Uub),B7),Uua) ) ).

% ATP.lambda_612
tff(fact_8797_ATP_Olambda__613,axiom,
    ! [A: $tType,B: $tType,Uu2: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_adb(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Uu2),I3)),aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu2)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_613
tff(fact_8798_ATP_Olambda__614,axiom,
    ! [A: $tType,Uu2: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_afh(list(A),fun(set(nat),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu2),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu2))
          & member(nat,I3,Uua) ) ) ).

% ATP.lambda_614
tff(fact_8799_ATP_Olambda__615,axiom,
    ! [A: $tType,Uu2: nat,Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_abu(nat,fun(list(A),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uua),I3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uu2),I3)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uua)) ) ) ).

% ATP.lambda_615
tff(fact_8800_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afv(set(A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A7) )
              & member(A,A7,Uu2) ) ) ) ).

% ATP.lambda_616
tff(fact_8801_ATP_Olambda__617,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aft(set(A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A7) )
              & member(A,A7,Uu2) ) ) ) ).

% ATP.lambda_617
tff(fact_8802_ATP_Olambda__618,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,set(A)),Uua: list(B),Uub: set(A)] :
      ( aa(set(A),$o,aa(list(B),fun(set(A),$o),aTP_Lamp_aga(fun(B,set(A)),fun(list(B),fun(set(A),$o)),Uu2),Uua),Uub)
    <=> ? [I3: nat] :
          ( ( Uub = aa(B,set(A),Uu2,aa(nat,B,nth(B,Uua),I3)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(list(B),nat,size_size(list(B)),Uua)) ) ) ).

% ATP.lambda_618
tff(fact_8803_ATP_Olambda__619,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_abd(fun(B,A),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu2,L4) )
          & member(B,L4,Uua) ) ) ).

% ATP.lambda_619
tff(fact_8804_ATP_Olambda__620,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_abl(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X5: B] :
          ( ( Uub = aa(B,A,Uu2,X5) )
          & aa(B,$o,Uua,X5) ) ) ).

% ATP.lambda_620
tff(fact_8805_ATP_Olambda__621,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_abh(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [X5: A] :
          ( ( Uub = aa(A,B,Uua,X5) )
          & aa(A,$o,Uu2,X5) ) ) ).

% ATP.lambda_621
tff(fact_8806_ATP_Olambda__622,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_abe(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu2),Uua),Uub)
    <=> ! [X5: A] :
          ( member(A,X5,Uu2)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X5) ) ) ).

% ATP.lambda_622
tff(fact_8807_ATP_Olambda__623,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_iu(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [X5: A] :
          ( member(A,X5,Uu2)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X5) ) ) ).

% ATP.lambda_623
tff(fact_8808_ATP_Olambda__624,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_aeu(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),Uu2),Uua),Uub)
    <=> ? [P6: product_prod(A,B)] :
          ( ( Uub = P6 )
          & member(C,aa(B,C,aa(A,fun(B,C),Uu2,aa(product_prod(A,B),A,product_fst(A,B),P6)),aa(product_prod(A,B),B,product_snd(A,B),P6)),Uua) ) ) ).

% ATP.lambda_624
tff(fact_8809_ATP_Olambda__625,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aal(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [N5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),set_or7035219750837199246ssThan(nat,Uub,N5)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_625
tff(fact_8810_ATP_Olambda__626,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_ix(fun(B,A),fun(set(B),fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [X5: B] :
          ( member(B,X5,Uua)
          & ( Uub = aa(B,A,Uu2,X5) ) ) ) ).

% ATP.lambda_626
tff(fact_8811_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_aax(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu2),Uua),Uub)
        <=> ! [A7: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A7)
             => ! [B7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A7),B7)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),Uu2),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7)) ) ) ) ) ).

% ATP.lambda_627
tff(fact_8812_ATP_Olambda__628,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_abb(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu2),Uua),Uub)
    <=> ? [Y5: A] :
          ( aa(A,$o,Uu2,Y5)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y5) ) ) ).

% ATP.lambda_628
tff(fact_8813_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu2: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adu(fun(A,A),fun(A,fun(A,$o)),Uu2),Uua),Uub)
    <=> ? [N5: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N5),Uu2),Uua) ) ).

% ATP.lambda_629
tff(fact_8814_ATP_Olambda__630,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(B,fun(B,$o)),Uub: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_acm(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu2),Uua),Uub)
    <=> ? [A7: B,B7: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu2,A7)),aa(B,A,Uu2,B7)) )
          & aa(B,$o,aa(B,fun(B,$o),Uua,A7),B7) ) ) ).

% ATP.lambda_630
tff(fact_8815_ATP_Olambda__631,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aeg(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),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)) )
          | ? [U3: list(A),Aa3: A,B7: A,Va4: list(A),W4: list(A)] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B7),Uu2)
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va4)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B7),W4)) ) ) ) ) ).

% ATP.lambda_631
tff(fact_8816_ATP_Olambda__632,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_afw(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A7),B7) )
              & member(A,A7,Uu2)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_632
tff(fact_8817_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu2: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_afs(set(A),fun(set(A),fun(A,$o)),Uu2),Uua),Uub)
        <=> ? [A7: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A7),B7) )
              & member(A,A7,Uu2)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_633
tff(fact_8818_ATP_Olambda__634,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_abf(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu2),Uua),Uub)
    <=> ? [Us3: list(A),Z2: A,Z8: 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),Z2),Vs3)) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Z8),Uu2)
          & ( 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),Z8),Vs3)) ) ) ) ).

% ATP.lambda_634
tff(fact_8819_ATP_Olambda__635,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu2: 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_om(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu2,Uub),Uuc),Uua) ).

% ATP.lambda_635
tff(fact_8820_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rx(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_636
tff(fact_8821_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rr(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_637
tff(fact_8822_ATP_Olambda__638,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_rt(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu2),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_638
tff(fact_8823_ATP_Olambda__639,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_afi(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu2),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu2) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu2)),Uub)),Uuc),Uuc) ).

% ATP.lambda_639
tff(fact_8824_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_aef(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu2),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image(A,B,Uu2),Uua)),the_inv_into(A,B,Uua,Uu2,Uuc),Uub) ).

% ATP.lambda_640
tff(fact_8825_ATP_Olambda__641,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,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_641
tff(fact_8826_ATP_Olambda__642,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_hb(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu2,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_642
tff(fact_8827_ATP_Olambda__643,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_hc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_643
tff(fact_8828_ATP_Olambda__644,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fg(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu2),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_644
tff(fact_8829_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType,Uu2: 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_adq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu2,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_645
tff(fact_8830_ATP_Olambda__646,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ro(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu2,Uuc)) ) ).

% ATP.lambda_646
tff(fact_8831_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_gw(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_647
tff(fact_8832_ATP_Olambda__648,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_dm(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_648
tff(fact_8833_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_gy(A,fun(fun(A,B),fun(B,fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uu2,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_649
tff(fact_8834_ATP_Olambda__650,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,option(A)),Uua: B,Uub: A,Uuc: B] :
      aa(B,option(A),aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_ht(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),Uu2),Uua),Uub),Uuc) = $ite(Uuc = Uua,aa(A,option(A),some(A),Uub),aa(B,option(A),Uu2,Uuc)) ).

% ATP.lambda_650
tff(fact_8835_ATP_Olambda__651,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_lj(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu2),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu2,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_651
tff(fact_8836_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_mc(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_652
tff(fact_8837_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_mb(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu2,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_653
tff(fact_8838_ATP_Olambda__654,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: 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_afl(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu2),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_afk(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu2) ).

% ATP.lambda_654
tff(fact_8839_ATP_Olambda__655,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu2: 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_pv(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu2),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_pu(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu2) ).

% ATP.lambda_655
tff(fact_8840_ATP_Olambda__656,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: 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_jl(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),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,Uu2,aa(nat,B,nth(B,Uub),Uuc))) ) ).

% ATP.lambda_656
tff(fact_8841_ATP_Olambda__657,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ni(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_nh(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_657
tff(fact_8842_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A,Uuc: A] :
          aa(A,fun(list(A),list(A)),aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_tc(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu2),Uua),Uub),Uuc) = case_list(list(A),A,
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu2,Uub)),aa(A,B,Uu2,Uuc)),Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),
            aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_tb(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu2),Uua)) ) ).

% ATP.lambda_658
tff(fact_8843_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fi(nat,fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu2,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_659
tff(fact_8844_ATP_Olambda__660,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,fun(B,assn)),Uua: $o,Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa($o,fun(A,fun(B,assn)),aTP_Lamp_jp(fun(A,fun(B,assn)),fun($o,fun(A,fun(B,assn))),Uu2),(Uua)),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Uu2,Uub),Uuc)),pure_assn((Uua))) ).

% ATP.lambda_660
tff(fact_8845_ATP_Olambda__661,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_gj(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_gi(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_661
tff(fact_8846_ATP_Olambda__662,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_bq(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_bp(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_662
tff(fact_8847_ATP_Olambda__663,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fe(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fd(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_663
tff(fact_8848_ATP_Olambda__664,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_fb(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fa(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_664
tff(fact_8849_ATP_Olambda__665,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ui(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_665
tff(fact_8850_ATP_Olambda__666,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ug(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_666
tff(fact_8851_ATP_Olambda__667,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_ue(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uub),Uua)),Uuc)) ).

% ATP.lambda_667
tff(fact_8852_ATP_Olambda__668,axiom,
    ! [Uu2: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_uf(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu2,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),minus_minus(real),Uua),Uub)),Uuc)) ).

% ATP.lambda_668
tff(fact_8853_ATP_Olambda__669,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu2: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_nh(A,fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu2)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_669
tff(fact_8854_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_yb(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu2),Uua),Uub),Uuc) = 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,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu2,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_670
tff(fact_8855_ATP_Olambda__671,axiom,
    ! [A: $tType,Uu2: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_sy(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu2,Uub)),aa(A,nat,Uu2,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_671
tff(fact_8856_ATP_Olambda__672,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_px(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu2,Uub) = aa(B,option(B),some(B),Uuc) )
        & aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).

% ATP.lambda_672
tff(fact_8857_ATP_Olambda__673,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aam(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_673
tff(fact_8858_ATP_Olambda__674,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_aaa(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_674
tff(fact_8859_ATP_Olambda__675,axiom,
    ! [A: $tType,Uu2: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_acj(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu2),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),X5: A,Y5: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Xs6)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y5),Uu2) ) ) ) ).

% ATP.lambda_675
tff(fact_8860_ATP_Olambda__676,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cr(A,fun(nat,fun(A,fun(nat,A))),Uu2),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),Uu2),Uuc)) ) ).

% ATP.lambda_676
tff(fact_8861_ATP_Olambda__677,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_bg(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image(B,A,Uu2),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_677
tff(fact_8862_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType,Uu2: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bn(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_678
tff(fact_8863_ATP_Olambda__679,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_be(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_679
tff(fact_8864_ATP_Olambda__680,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_bk(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_680
tff(fact_8865_ATP_Olambda__681,axiom,
    ! [A: $tType,C: $tType,Uu2: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ch(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_681
tff(fact_8866_ATP_Olambda__682,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cu(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu2)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_682
tff(fact_8867_ATP_Olambda__683,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_adh(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu2) ) ) ).

% ATP.lambda_683
tff(fact_8868_ATP_Olambda__684,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu2: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_cq(A,fun(nat,fun(A,fun(nat,A))),Uu2),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),Uu2),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_684
tff(fact_8869_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: list(A),Uuc: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_sz(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu2),Uua),Uub),Uuc) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),linorder_sort_key(A,B,Uu2),Uua)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),aa(list(A),list(A),linorder_sort_key(A,B,Uu2),Uuc))) ) ).

% ATP.lambda_685
tff(fact_8870_ATP_Olambda__686,axiom,
    ! [A: $tType,B: $tType,Uu2: 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_tf(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu2),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),Uu2),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_686
tff(fact_8871_ATP_Olambda__687,axiom,
    ! [A: $tType,B: $tType,Uu2: 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_te(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu2),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),Uu2)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_687
tff(fact_8872_ATP_Olambda__688,axiom,
    ! [A: $tType,B: $tType,Uu2: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_pc(A,fun(B,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( ( Uu2 = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_688
tff(fact_8873_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_bh(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu2,Uuc)) ) ) ).

% ATP.lambda_689
tff(fact_8874_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_kq(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc),aa(C,set(product_prod(A,B)),Uu2,Uua)) ) ).

% ATP.lambda_690
tff(fact_8875_ATP_Olambda__691,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_cb(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_691
tff(fact_8876_ATP_Olambda__692,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bz(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu2)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_692
tff(fact_8877_ATP_Olambda__693,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT] :
      aa(vEBT_VEBT,assn,aa(vEBT_VEBTi,fun(vEBT_VEBT,assn),aa(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn)),aTP_Lamp_ib(nat,fun(set(nat),fun(vEBT_VEBTi,fun(vEBT_VEBT,assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uuc),Uub)),
        pure_assn(( ( Uua = vEBT_set_vebt(Uuc) )
          & vEBT_invar_vebt(Uuc,Uu2) ))) ).

% ATP.lambda_693
tff(fact_8878_ATP_Olambda__694,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Uu2),Uua),Uub),(Uuc)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn((Uuc) = aa(nat,$o,vEBT_vebt_member(Uu2),Uub))) ).

% ATP.lambda_694
tff(fact_8879_ATP_Olambda__695,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_at(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn(Uuc = vEBT_vebt_succ(Uu2,Uub))) ).

% ATP.lambda_695
tff(fact_8880_ATP_Olambda__696,axiom,
    ! [Uu2: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_au(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu2),Uua)),pure_assn(Uuc = vEBT_vebt_pred(Uu2,Uub))) ).

% ATP.lambda_696
tff(fact_8881_ATP_Olambda__697,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu2: nat,Uua: list(A),Uub: array(A),Uuc: A] : aa(A,assn,aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_ek(nat,fun(list(A),fun(array(A),fun(A,assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),Uua)),pure_assn(Uuc = aa(nat,A,nth(A,Uua),Uu2))) ) ).

% ATP.lambda_697
tff(fact_8882_ATP_Olambda__698,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_yh(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu2,Uub)),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub))))) ) ).

% ATP.lambda_698
tff(fact_8883_ATP_Olambda__699,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu2: 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_ja(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu2,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_699
tff(fact_8884_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_vp(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(A,A,aa(A,fun(A,A),minus_minus(A),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(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_700
tff(fact_8885_ATP_Olambda__701,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu2,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_701
tff(fact_8886_ATP_Olambda__702,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fm(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_702
tff(fact_8887_ATP_Olambda__703,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aad(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu2,Uuc),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),Uub),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_703
tff(fact_8888_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu2: 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_jb(fun(B,A),fun(A,fun(B,fun(A,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu2,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_704
tff(fact_8889_ATP_Olambda__705,axiom,
    ! [Uu2: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_fd(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu2),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu2,Uuc)),aa(nat,nat,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ).

% ATP.lambda_705
tff(fact_8890_ATP_Olambda__706,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_fa(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu2,Uuc)),aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),Uuc))) ) ).

% ATP.lambda_706
tff(fact_8891_ATP_Olambda__707,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu2: 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_pq(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu2),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu2,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_707
tff(fact_8892_ATP_Olambda__708,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu2: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_mq(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu2,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_708
tff(fact_8893_ATP_Olambda__709,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_gh(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bn(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_709
tff(fact_8894_ATP_Olambda__710,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_bo(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu2),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_bn(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_710
tff(fact_8895_ATP_Olambda__711,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_va(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu2,Uua))) ) ).

% ATP.lambda_711
tff(fact_8896_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_up(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_712
tff(fact_8897_ATP_Olambda__713,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_vo(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),minus_minus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu2,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_713
tff(fact_8898_ATP_Olambda__714,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_tb(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu2),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(list(A),list(A))),list(A),aa(fun(list(A),fun(product_prod(list(A),list(A)),list(A))),fun(product_prod(list(A),product_prod(list(A),list(A))),list(A)),product_case_prod(list(A),product_prod(list(A),list(A)),list(A)),aTP_Lamp_ta(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu2)),linorder_part(A,B,Uu2,aa(A,B,Uu2,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Uua)) ) ).

% ATP.lambda_714
tff(fact_8899_ATP_Olambda__715,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ym(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu2),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu2,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_715
tff(fact_8900_ATP_Olambda__716,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_dr(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu2)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_716
tff(fact_8901_ATP_Olambda__717,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_dq(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu2)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_717
tff(fact_8902_ATP_Olambda__718,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu2: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_aae(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu2,Uuc))),Uub)) ) ) ).

% ATP.lambda_718
tff(fact_8903_ATP_Olambda__719,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yn(A,A)))))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yn(A,A))))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_yn(A,A)))))) ) ).

% ATP.lambda_719
tff(fact_8904_ATP_Olambda__720,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_yj(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu2,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uua)))) ) ).

% ATP.lambda_720
tff(fact_8905_ATP_Olambda__721,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_yk(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu2),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,Uu2,Uuc)),aa(A,B,Uu2,Uub))),aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),minus_minus(A),Uuc),Uub)))) ) ).

% ATP.lambda_721
tff(fact_8906_ATP_Olambda__722,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_cv(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cu(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_722
tff(fact_8907_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType,Uu2: $o,Uua: fun(A,fun(B,assn)),Uub: A,Uuc: B] : aa(B,assn,aa(A,fun(B,assn),aa(fun(A,fun(B,assn)),fun(A,fun(B,assn)),aTP_Lamp_jq($o,fun(fun(A,fun(B,assn)),fun(A,fun(B,assn))),(Uu2)),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((Uu2))),aa(B,assn,aa(A,fun(B,assn),Uua,Uub),Uuc)) ).

% ATP.lambda_723
tff(fact_8908_ATP_Olambda__724,axiom,
    ! [B: $tType,A: $tType,Uu2: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: A] : aa(A,assn,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_en(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Uu2),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu2),aa(A,assn,aa(B,fun(A,assn),Uua,Uub),Uuc)) ).

% ATP.lambda_724
tff(fact_8909_ATP_Olambda__725,axiom,
    ! [A: $tType,B: $tType,Uu2: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_pb($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu2)),Uua),Uub),Uuc)
    <=> ( (Uu2)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_725
tff(fact_8910_ATP_Olambda__726,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_gl(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ch(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_726
tff(fact_8911_ATP_Olambda__727,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_gn(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cu(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_727
tff(fact_8912_ATP_Olambda__728,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ci(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu2),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ch(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu2),Uub),Uuc))) ) ).

% ATP.lambda_728
tff(fact_8913_ATP_Olambda__729,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu2: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_cx(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu2),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_cu(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu2),Uua),Uuc))) ) ).

% ATP.lambda_729
tff(fact_8914_ATP_Olambda__730,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu2: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_yy(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu2),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(A,fun(nat,A)),Uu2),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_730
tff(fact_8915_ATP_Olambda__731,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun($o,fun($o,B)),Uub: $o,Uuc: $o] : aa($o,A,aa($o,fun($o,A),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_et(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),Uu2),Uua),(Uub)),(Uuc)) = aa(B,A,Uu2,aa($o,B,aa($o,fun($o,B),Uua,(Uub)),(Uuc))) ).

% ATP.lambda_731
tff(fact_8916_ATP_Olambda__732,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu2: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_on(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(B,A,Uu2,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_732
tff(fact_8917_ATP_Olambda__733,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_gr(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_733
tff(fact_8918_ATP_Olambda__734,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_dh(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_734
tff(fact_8919_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu2: 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_gx(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_735
tff(fact_8920_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: 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_fu(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu2),Uua),Uub),Uuc) = aa(nat,A,Uu2,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_736
tff(fact_8921_ATP_Olambda__737,axiom,
    ! [A: $tType,Uu2: fun(A,$o),Uua: nat,Uub: list(A),Uuc: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aa(nat,fun(list(A),fun(nat,$o)),aTP_Lamp_ss(fun(A,$o),fun(nat,fun(list(A),fun(nat,$o))),Uu2),Uua),Uub),Uuc)
    <=> aa(A,$o,Uu2,aa(nat,A,nth(A,take(A,Uua,Uub)),Uuc)) ) ).

% ATP.lambda_737
tff(fact_8922_ATP_Olambda__738,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu2: 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_ou(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu2),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu2,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_738
tff(fact_8923_ATP_Olambda__739,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu2: 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_ov(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu2),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu2,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_739
tff(fact_8924_ATP_Olambda__740,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu2: fun(A,B),Uua: B,Uub: A,Uuc: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_th(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Uu2),Uua),Uub),Uuc) = aa(fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_case_prod(list(A),list(A),product_prod(list(A),product_prod(list(A),list(A)))),aa(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),aa(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))),aa(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))))),aTP_Lamp_tg(fun(A,B),fun(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))))),Uu2),Uua),Uub),Uuc)) ) ).

% ATP.lambda_740
tff(fact_8925_ATP_Olambda__741,axiom,
    ! [A: $tType,B: $tType,Uu2: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_pj(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu2),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_pi(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc)) ).

% ATP.lambda_741
tff(fact_8926_ATP_Olambda__742,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),$o),aa(A,fun(B,fun(product_prod(A,B),$o)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o))),aTP_Lamp_ph(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),$o)))),Uu2),Uua),Uub),Uuc) = aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_pg(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc)) ).

% ATP.lambda_742
tff(fact_8927_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu2: fun(B,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_uy(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu2),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu2,Uua)))) ) ).

% ATP.lambda_743
tff(fact_8928_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType,Uu2: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_aao(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu2),Uua),Uub),Uuc)
    <=> ! [X5: A] :
          ( ( member(A,X5,Uu2)
           => member(B,aa(A,B,Uuc,X5),Uua) )
          & ( ~ member(A,X5,Uu2)
           => ( aa(A,B,Uuc,X5) = Uub ) ) ) ) ).

% ATP.lambda_744
tff(fact_8929_ATP_Olambda__745,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_afj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [Y5: C] :
          ( member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y5),Uu2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y5),Uuc),Uua) ) ) ).

% ATP.lambda_745
tff(fact_8930_ATP_Olambda__746,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_abc(fun(B,A),fun(B,fun(set(B),fun(A,$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [L4: B] :
          ( ( Uuc = aa(B,A,Uu2,L4) )
          & ( ( L4 = Uua )
            | member(B,L4,Uub) ) ) ) ).

% ATP.lambda_746
tff(fact_8931_ATP_Olambda__747,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu2: A,Uua: A,Uub: fun(A,B),Uuc: B] :
          ( aa(B,$o,aa(fun(A,B),fun(B,$o),aa(A,fun(fun(A,B),fun(B,$o)),aTP_Lamp_abn(A,fun(A,fun(fun(A,B),fun(B,$o))),Uu2),Uua),Uub),Uuc)
        <=> ? [I3: A] :
              ( ( Uuc = aa(A,B,Uub,I3) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu2),I3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I3),Uua) ) ) ) ).

% ATP.lambda_747
tff(fact_8932_ATP_Olambda__748,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_abg(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu2),Uua),Uub),Uuc)
    <=> ? [X5: A,Y5: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X5),Y5) )
          & aa(A,$o,Uu2,X5)
          & aa(B,$o,Uua,Y5) ) ) ).

% ATP.lambda_748
tff(fact_8933_ATP_Olambda__749,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu2: 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_pu(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu2),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert(product_prod(C,B)),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu2),Uuc)),Uud),Uud) ).

% ATP.lambda_749
tff(fact_8934_ATP_Olambda__750,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: 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_afk(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu2),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert(product_prod(A,C)),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu2),Uuc)),Uud),Uud) ).

% ATP.lambda_750
tff(fact_8935_ATP_Olambda__751,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_ut(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu2,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_751
tff(fact_8936_ATP_Olambda__752,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_vg(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu2),Uua),Uub),Uuc),Uud) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_vf(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud)),Uu2) ) ).

% ATP.lambda_752
tff(fact_8937_ATP_Olambda__753,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fk(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_fj(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uud))) ) ).

% ATP.lambda_753
tff(fact_8938_ATP_Olambda__754,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu2: fun(code_integer,product_prod(A,A)),Uua: A,Uub: code_integer,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_qh(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),Uu2),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),code_T4080844693773952564amples(A,Uu2,Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),one_one(code_integer)))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uud),nil(A)))) ) ).

% ATP.lambda_754
tff(fact_8939_ATP_Olambda__755,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu2: fun(code_integer,product_prod(A,A)),Uua: code_integer,Uub: A,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_qi(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),Uu2),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),code_T4080844693773952564amples(A,Uu2,Uub,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uua),one_one(code_integer)))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uud),nil(A)))) ) ).

% ATP.lambda_755
tff(fact_8940_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: fun(C,assn),Uud: C] : aa(C,assn,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_aa(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,Uu2,Uua),Uub)),aa(C,assn,Uuc,Uud)) ).

% ATP.lambda_756
tff(fact_8941_ATP_Olambda__757,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),aTP_Lamp_eh(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uud = aa(nat,option(nat),some(nat),Uuc) )
          = vEBT_VEBT_min_in_set(Uua,Uuc) ))) ).

% ATP.lambda_757
tff(fact_8942_ATP_Olambda__758,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),aTP_Lamp_ec(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uud = aa(nat,option(nat),some(nat),Uuc) )
          = vEBT_VEBT_max_in_set(Uua,Uuc) ))) ).

% ATP.lambda_758
tff(fact_8943_ATP_Olambda__759,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn))),aTP_Lamp_nm(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun($o,assn)))),Uu2),Uua),Uub),Uuc),(Uud)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),pure_assn((Uud) = member(nat,Uuc,Uua))) ).

% ATP.lambda_759
tff(fact_8944_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu2: fun(A,fun(C,A)),Uua: fun(A,fun(B,fun(C,B))),Uub: A,Uuc: B,Uud: C] : aa(C,product_prod(A,B),aa(B,fun(C,product_prod(A,B)),aa(A,fun(B,fun(C,product_prod(A,B))),aa(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B)))),aTP_Lamp_aen(fun(A,fun(C,A)),fun(fun(A,fun(B,fun(C,B))),fun(A,fun(B,fun(C,product_prod(A,B))))),Uu2),Uua),Uub),Uuc),Uud) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,aa(A,fun(C,A),Uu2,Uub),Uud)),aa(C,B,aa(B,fun(C,B),aa(A,fun(B,fun(C,B)),Uua,Uub),Uuc),Uud)) ).

% ATP.lambda_760
tff(fact_8945_ATP_Olambda__761,axiom,
    ! [Uu2: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_uh(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu2),Uua),Uub),Uuc),Uud) = aa(real,real,aa(real,fun(real,real),minus_minus(real),aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_ug(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uuc))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu2),Uuc)))))) ).

% ATP.lambda_761
tff(fact_8946_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fn(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu2),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fm(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_762
tff(fact_8947_ATP_Olambda__763,axiom,
    ! [A: $tType,Uu2: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o))),aTP_Lamp_py(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu2),transitive_trancl(A,Uub))
          | ( Uuc = Uu2 ) )
        & ( 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_763
tff(fact_8948_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu2: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_fj(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu2),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,Uu2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_764
tff(fact_8949_ATP_Olambda__765,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu2: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_zz(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu2),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu2,Uud),Uua)) ) ) ).

% ATP.lambda_765
tff(fact_8950_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu2: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_er(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),list_update(A,Uua,Uu2,Uuc))),pure_assn(Uud = Uub)) ) ).

% ATP.lambda_766
tff(fact_8951_ATP_Olambda__767,axiom,
    ! [B: $tType,A: $tType,Uu2: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: nat,Uud: list(A)] : aa(list(A),assn,aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_eo(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu2),aa(list(A),assn,vEBT_List_list_assn(B,A,Uua,replicate(B,Uuc,Uub)),Uud)) ).

% ATP.lambda_767
tff(fact_8952_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: nat,Uud: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_eu(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Uu2),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uud),aa(B,assn,aa(A,fun(B,assn),Uu2,aa(nat,A,nth(A,Uua),Uuc)),aa(nat,B,nth(B,Uub),Uuc))) ).

% ATP.lambda_768
tff(fact_8953_ATP_Olambda__769,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_oj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),Uuc) ),
        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub),
        $ite(
          ( ( Uuc = Uud )
          & ( Uuc = Uue ) ),
          vEBT_Node(none(product_prod(nat,nat)),Uu2,Uua,Uub),
          $let(
            xn: nat,
            xn:= 
              $ite(Uuc = Uud,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),the2(nat,vEBT_vebt_mint(Uub))),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),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),the2(nat,vEBT_vebt_mint(Uub)))))),Uuc),
            $let(
              minn: nat,
              minn:= 
                $ite(Uuc = Uud,xn,Uud),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,Uua,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Uub,h),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Uue,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),the2(nat,maxs))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),the2(nat,maxs)))))) ),
                                  Uue ))),Uu2,newlist,sn) ),
                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Uue,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),the2(nat,vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Uue))),Uu2,newlist,Uub) ) ) ),
                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub) ) ) ) ) ) ) ).

% ATP.lambda_769
tff(fact_8954_ATP_Olambda__770,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,option(nat),aa(nat,fun(nat,option(nat)),aa(nat,fun(nat,fun(nat,option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat))))),aTP_Lamp_qg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,option(nat)))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),
        aa(nat,option(nat),some(nat),Uud),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Uuc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Uub,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),the2(nat,sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% ATP.lambda_770
tff(fact_8955_ATP_Olambda__771,axiom,
    ! [Uu2: option(product_prod(nat,nat)),Uua: nat,Uub: list(vEBT_VEBT),Uuc: vEBT_VEBT,Uud: nat,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),vEBT_VEBT,aa(nat,fun(product_prod(nat,nat),vEBT_VEBT),aa(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT)))),aTP_Lamp_ok(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(product_prod(nat,nat),vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),one_one(nat)),vEBT_Node(Uu2,Uua,Uub,Uuc),aa(product_prod(nat,nat),vEBT_VEBT,aa(fun(nat,fun(nat,vEBT_VEBT)),fun(product_prod(nat,nat),vEBT_VEBT),product_case_prod(nat,nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_oj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uua),Uub),Uuc),Uud)),Uue)) ).

% ATP.lambda_771
tff(fact_8956_ATP_Olambda__772,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat] :
      aa(nat,vEBT_VEBT,aa(nat,fun(nat,vEBT_VEBT),aa(nat,fun(nat,fun(nat,vEBT_VEBT)),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT)))),aTP_Lamp_oz(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,vEBT_VEBT))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uud,Uuc),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uua))
            & ~ ( ( Uuc = Uud )
                | ( Uuc = Uue ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uud),Uuc,Uud)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Uue))),Uu2,list_update(vEBT_VEBT,Uua,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),h)),vEBT_vebt_insert(Uub,h),Uub)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uu2,Uua,Uub) ) ) ) ).

% ATP.lambda_772
tff(fact_8957_ATP_Olambda__773,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_no(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uue = aa(nat,option(nat),some(nat),Uud) )
          = vEBT_is_succ_in_set(Uua,Uuc,Uud) ))) ).

% ATP.lambda_773
tff(fact_8958_ATP_Olambda__774,axiom,
    ! [Uu2: nat,Uua: set(nat),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: option(nat)] :
      aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(nat,fun(nat,fun(option(nat),assn)),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))),aa(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn)))),aTP_Lamp_nn(nat,fun(set(nat),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(option(nat),assn))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,vEBT_Intf_vebt_assn(Uu2,Uua),Uub)),
        pure_assn(( ( Uue = aa(nat,option(nat),some(nat),Uud) )
          = vEBT_is_pred_in_set(Uua,Uuc,Uud) ))) ).

% ATP.lambda_774
tff(fact_8959_ATP_Olambda__775,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu2: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_vf(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_vd(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),Uu2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_775
tff(fact_8960_ATP_Olambda__776,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_uv(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu2,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_776
tff(fact_8961_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,Uu2: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_pi(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc),Uud),Uue)
    <=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud),Uu2)
        | ( ( Uub = Uud )
          & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),Uua) ) ) ) ).

% ATP.lambda_777
tff(fact_8962_ATP_Olambda__778,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu2,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu2,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu2,Uub)))) ) ).

% ATP.lambda_778
tff(fact_8963_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu2: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_vc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu2,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_779
tff(fact_8964_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType,Uu2: fun(A,$o),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aa(A,fun(B,fun(A,fun(B,$o))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o)))),aTP_Lamp_pg(fun(A,$o),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,$o))))),Uu2),Uua),Uub),Uuc),Uud),Uue)
    <=> ( ( Uub = Uud )
        & aa(A,$o,Uu2,Uud)
        & member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue),aa(A,set(product_prod(B,B)),Uua,Uud)) ) ) ).

% ATP.lambda_780
tff(fact_8965_ATP_Olambda__781,axiom,
    ! [Uu2: fun(a,b),Uua: b,Uub: a,Uuc: list(a),Uud: list(a),Uue: list(a)] :
      aa(list(a),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))),aa(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))),aa(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))),aa(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))))),aTP_Lamp_tg(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = $let(
        x2: b,
        x2:= aa(a,b,Uu2,Uub),
        $ite(
          aa(b,$o,aa(b,fun(b,$o),ord_less(b),x2),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),Uue)),
          $ite(aa(b,$o,aa(b,fun(b,$o),ord_less(b),Uua),x2),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),aa(list(a),list(a),aa(a,fun(list(a),list(a)),cons(a),Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_781
tff(fact_8966_ATP_Olambda__782,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_es(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),Uu2),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu2,aa(vEBT_VEBTi,B,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,B),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_782
tff(fact_8967_ATP_Olambda__783,axiom,
    ! [Uu2: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_aba(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu2))
        & ! [I3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I3)),X_13)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I3) ) )
        & $ite(
            Uue = Uuf,
            ! [X5: vEBT_VEBT] :
              ( member(vEBT_VEBT,X5,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X9: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X5),X9) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X5: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X5),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu2))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X5)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X5)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X5),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_783
tff(fact_8968_ATP_Olambda__784,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(A,fun(B,assn)),Uua: A,Uub: list(A),Uuc: B,Uud: list(B),Uue: nat,Uuf: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_fh(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uuf),aa(B,assn,aa(A,fun(B,assn),Uu2,aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),Uud)),Uue))) ).

% ATP.lambda_784
tff(fact_8969_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType,Uu2: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_pd($o,fun(A,fun(B,$o)),(Uu2)),Uua),Uub)
    <=> (Uu2) ) ).

% ATP.lambda_785
tff(fact_8970_ATP_Olambda__786,axiom,
    ! [A: $tType,B: $tType,Uu2: multiset(B),Uua: A] : aa(A,multiset(B),aTP_Lamp_kk(multiset(B),fun(A,multiset(B)),Uu2),Uua) = Uu2 ).

% ATP.lambda_786
tff(fact_8971_ATP_Olambda__787,axiom,
    ! [A: $tType,Uu2: assn,Uua: A] : aa(A,assn,aTP_Lamp_hs(assn,fun(A,assn),Uu2),Uua) = Uu2 ).

% ATP.lambda_787
tff(fact_8972_ATP_Olambda__788,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu2: word(B),Uua: A] : aa(A,word(B),aTP_Lamp_adc(word(B),fun(A,word(B)),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_788
tff(fact_8973_ATP_Olambda__789,axiom,
    ! [A: $tType,Uu2: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_li($o,fun(A,$o),(Uu2)),Uua)
    <=> (Uu2) ) ).

% ATP.lambda_789
tff(fact_8974_ATP_Olambda__790,axiom,
    ! [B: $tType,A: $tType,Uu2: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_kp(set(A),fun(B,set(A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_790
tff(fact_8975_ATP_Olambda__791,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu2: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_aes(fun(B,C),fun(A,fun(B,C)),Uu2),Uua) = Uu2 ).

% ATP.lambda_791
tff(fact_8976_ATP_Olambda__792,axiom,
    ! [A: $tType,B: $tType,Uu2: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_kc(fun(B,B),fun(A,fun(B,B)),Uu2),Uua) = Uu2 ).

% ATP.lambda_792
tff(fact_8977_ATP_Olambda__793,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu2: fun(A,fun(C,C)),Uua: B] : aa(B,fun(A,fun(C,C)),aTP_Lamp_pz(fun(A,fun(C,C)),fun(B,fun(A,fun(C,C))),Uu2),Uua) = Uu2 ).

% ATP.lambda_793
tff(fact_8978_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_kb(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_794
tff(fact_8979_ATP_Olambda__795,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_kn(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_795
tff(fact_8980_ATP_Olambda__796,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_uq(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_796
tff(fact_8981_ATP_Olambda__797,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_ko(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_797
tff(fact_8982_ATP_Olambda__798,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_ao(B,fun(A,B),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_798
tff(fact_8983_ATP_Olambda__799,axiom,
    ! [A: $tType,B: $tType,Uu2: B,Uua: A] : aa(A,B,aTP_Lamp_cf(B,fun(A,B),Uu2),Uua) = Uu2 ).

% ATP.lambda_799
tff(fact_8984_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_lh(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_800
tff(fact_8985_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_qq(A,fun(nat,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_801
tff(fact_8986_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu2: A,Uua: A] : aa(A,A,aTP_Lamp_tr(A,fun(A,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_802
tff(fact_8987_ATP_Olambda__803,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_kt(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_803
tff(fact_8988_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu2: A,Uua: B] : aa(B,A,aTP_Lamp_vw(A,fun(B,A),Uu2),Uua) = Uu2 ) ).

% ATP.lambda_804
tff(fact_8989_ATP_Olambda__805,axiom,
    ! [A: $tType,Uu2: A,Uua: nat] : aa(nat,A,aTP_Lamp_sv(A,fun(nat,A),Uu2),Uua) = Uu2 ).

% ATP.lambda_805
tff(fact_8990_ATP_Olambda__806,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cg(A,fun(B,A)),Uu2),Uua) = Uu2 ).

% ATP.lambda_806
tff(fact_8991_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType,Uu2: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_aet(B,fun(A,A)),Uu2),Uua) = Uua ).

% ATP.lambda_807
tff(fact_8992_ATP_Olambda__808,axiom,
    ! [B: $tType,A: $tType,Uu2: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_pk(A,fun(B,$o)),Uu2),Uua)
    <=> $true ) ).

% ATP.lambda_808
tff(fact_8993_ATP_Olambda__809,axiom,
    ! [Uu2: complex] : aa(complex,complex,aTP_Lamp_eg(complex,complex),Uu2) = Uu2 ).

% ATP.lambda_809
tff(fact_8994_ATP_Olambda__810,axiom,
    ! [Uu2: nat] : aa(nat,nat,aTP_Lamp_de(nat,nat),Uu2) = Uu2 ).

% ATP.lambda_810
tff(fact_8995_ATP_Olambda__811,axiom,
    ! [Uu2: int] : aa(int,int,aTP_Lamp_ne(int,int),Uu2) = Uu2 ).

% ATP.lambda_811
tff(fact_8996_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_yn(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_812
tff(fact_8997_ATP_Olambda__813,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_jo(A,A),Uu2) = Uu2 ) ).

% ATP.lambda_813
tff(fact_8998_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu2: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu2) = Uu2 ).

% ATP.lambda_814
tff(fact_8999_ATP_Olambda__815,axiom,
    ! [C: $tType,B: $tType,Uu2: C] : aa(C,set(B),aTP_Lamp_acq(C,set(B)),Uu2) = bot_bot(set(B)) ).

% ATP.lambda_815
tff(fact_9000_ATP_Olambda__816,axiom,
    ! [B: $tType,A: $tType,Uu2: B] : aa(B,set(A),aTP_Lamp_ks(B,set(A)),Uu2) = bot_bot(set(A)) ).

% ATP.lambda_816
tff(fact_9001_ATP_Olambda__817,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_kl(B,A),Uu2) = bot_bot(A) ) ).

% ATP.lambda_817
tff(fact_9002_ATP_Olambda__818,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_km(B,A),Uu2) = bot_bot(A) ) ).

% ATP.lambda_818
tff(fact_9003_ATP_Olambda__819,axiom,
    ! [A: $tType,D: $tType,Uu2: A] : aa(A,set(D),aTP_Lamp_acr(A,set(D)),Uu2) = bot_bot(set(D)) ).

% ATP.lambda_819
tff(fact_9004_ATP_Olambda__820,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_qk(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_820
tff(fact_9005_ATP_Olambda__821,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu2: nat] : aa(nat,A,aTP_Lamp_oi(nat,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_821
tff(fact_9006_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_ag(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_822
tff(fact_9007_ATP_Olambda__823,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu2: B] : aa(B,A,aTP_Lamp_jf(B,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_823
tff(fact_9008_ATP_Olambda__824,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_ur(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_824
tff(fact_9009_ATP_Olambda__825,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu2: A] : aa(A,A,aTP_Lamp_bi(A,A),Uu2) = zero_zero(A) ) ).

% ATP.lambda_825
tff(fact_9010_ATP_Olambda__826,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu2: A] : aa(A,B,aTP_Lamp_tm(A,B),Uu2) = zero_zero(B) ) ).

% ATP.lambda_826
tff(fact_9011_ATP_Olambda__827,axiom,
    ! [B: $tType,A: $tType,Uu2: B] : aa(B,option(A),aTP_Lamp_hy(B,option(A)),Uu2) = none(A) ).

% ATP.lambda_827
tff(fact_9012_ATP_Olambda__828,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,option(B),aTP_Lamp_cp(A,option(B)),Uu2) = none(B) ).

% ATP.lambda_828
tff(fact_9013_ATP_Olambda__829,axiom,
    ! [A: $tType,B: $tType,Uu2: A] : aa(A,B,aTP_Lamp_afo(A,B),Uu2) = undefined(B) ).

% ATP.lambda_829
tff(fact_9014_ATP_Olambda__830,axiom,
    ! [Uu2: real] :
      ( aa(real,$o,aTP_Lamp_kr(real,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_830
tff(fact_9015_ATP_Olambda__831,axiom,
    ! [Uu2: nat] :
      ( aa(nat,$o,aTP_Lamp_ach(nat,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_831
tff(fact_9016_ATP_Olambda__832,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_bc(A,$o),Uu2)
    <=> $false ) ).

% ATP.lambda_832
tff(fact_9017_ATP_Olambda__833,axiom,
    ! [A: $tType,Uu2: A] :
      ( aa(A,$o,aTP_Lamp_lo(A,$o),Uu2)
    <=> $true ) ).

% ATP.lambda_833
tff(fact_9018_ATP_Olambda__834,axiom,
    ! [A: $tType,Uu2: A] : aa(A,fun(nat,nat),aTP_Lamp_ap(A,fun(nat,nat)),Uu2) = suc ).

% ATP.lambda_834

% Type constructors (1188)
tff(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
    typerep(vEBT_VEBTi) ).

tff(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep_1,axiom,
    typerep(vEBT_VEBT) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_2,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(heap_Time_Heap(A17)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
    typerep(code_integer) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_4,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(heap_ext(A17)) ) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_5,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_6,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(product_prod(A17,A18)) ) ).

tff(tcon_Numeral__Type_Onum1___Typerep_Otyperep_7,axiom,
    typerep(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum0___Typerep_Otyperep_8,axiom,
    typerep(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_9,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_10,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_11,axiom,
    ! [A17: $tType] :
      ( type_len0(A17)
     => type_len0(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_12,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_13,axiom,
    ! [A17: $tType] :
      ( type_len0(A17)
     => type_len0(numeral_bit0(A17)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_14,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(multiset(A17)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_15,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_16,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_17,axiom,
    typerep(complex) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_18,axiom,
    typerep(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_19,axiom,
    bounded_lattice_top(assn) ).

tff(tcon_String_Oliteral___Typerep_Otyperep_20,axiom,
    typerep(literal) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_21,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_22,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__top_23,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounded_lattice_top(option(A17)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_24,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(filter(A17)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_25,axiom,
    ! [A17: $tType] : bounded_lattice_top(filter(A17)) ).

tff(tcon_Sum__Type_Osum___Typerep_Otyperep_26,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(sum_sum(A17,A18)) ) ).

tff(tcon_String_Ochar___Typerep_Otyperep_27,axiom,
    typerep(char) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_28,axiom,
    ! [A17: $tType] : typerep(array(A17)) ).

tff(tcon_Word_Oword___Typerep_Otyperep_29,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(word(A17)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_30,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_31,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(list(A17)) ) ).

tff(tcon_Heap_Oref___Typerep_Otyperep_32,axiom,
    ! [A17: $tType] : typerep(ref(A17)) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_33,axiom,
    typerep($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_34,axiom,
    bounded_lattice_top($o) ).

tff(tcon_Set_Oset___Typerep_Otyperep_35,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => typerep(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_36,axiom,
    ! [A17: $tType] : bounded_lattice_top(set(A17)) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_37,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_38,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_39,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_40,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_41,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => typerep(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top_42,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice_top(fun(A17,A18)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => condit1219197933456340205attice(fun(A17,A18)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( counta3822494911875563373attice(A18)
     => counta3822494911875563373attice(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple592849572758109894attice(A18)
     => comple592849572758109894attice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4967611905675639751up_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounde4346867609351753570nf_top(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple6319245703460814977attice(fun(A17,A18)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A17: $tType,A18: $tType] :
      ( boolea8198339166811842893lgebra(A18)
     => boolea8198339166811842893lgebra(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A17: $tType,A18: $tType] :
      ( comple6319245703460814977attice(A18)
     => comple9053668089753744459l_ccpo(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_sup(A18)
     => semilattice_sup(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A17: $tType,A18: $tType] :
      ( semilattice_inf(A18)
     => semilattice_inf(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( distrib_lattice(A18)
     => distrib_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bounded_lattice(A18)
     => bounded_lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_top(A18)
     => order_top(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order_bot(A18)
     => order_bot(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( preorder(A18)
     => preorder(fun(A17,A18)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(fun(A17,A18)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A17: $tType,A18: $tType] :
      ( lattice(A18)
     => lattice(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A17: $tType,A18: $tType] :
      ( order(A18)
     => order(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ord(A18)
     => ord(fun(A17,A18)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A17: $tType,A18: $tType] :
      ( bot(A18)
     => bot(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( uminus(A18)
     => uminus(fun(A17,A18)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A17: $tType,A18: $tType] :
      ( minus(A18)
     => minus(fun(A17,A18)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_43,axiom,
    condit1219197933456340205attice(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(int) ).

tff(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___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___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_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_44,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_45,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_46,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___Generic__set__bit_Oset__bit,axiom,
    generic_set_set_bit(int) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_47,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_48,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Lattices_Olattice_49,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_50,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_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_51,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_52,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___Groups_Otimes,axiom,
    times(int) ).

tff(tcon_Int_Oint___Groups_Ominus_53,axiom,
    minus(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_54,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_55,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_56,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_57,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_58,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_59,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_60,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_61,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_62,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_63,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_64,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_65,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_66,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_67,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_68,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_69,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_70,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_71,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_72,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_73,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_74,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_75,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_76,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_77,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_78,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_79,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_80,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_81,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_82,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_83,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_84,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_85,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_86,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_87,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_88,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_89,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_90,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_91,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_92,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_93,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_94,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_95,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_96,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_97,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_98,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_99,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_100,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_101,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_102,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_103,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_104,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_105,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_106,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_107,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_108,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_109,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_110,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_111,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_112,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_113,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_114,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_115,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_116,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_117,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_118,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_119,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_120,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_121,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_122,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_123,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_124,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_125,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_126,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_127,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_128,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_129,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_130,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_131,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Orderings_Obot_132,axiom,
    bot(nat) ).

tff(tcon_Nat_Onat___Groups_Otimes_133,axiom,
    times(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_134,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_135,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_136,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_137,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_138,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_139,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_140,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_141,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_142,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_143,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_144,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_145,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_146,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_147,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Otimes_148,axiom,
    times(num) ).

tff(tcon_Num_Onum___Groups_Oplus_149,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_150,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_151,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_152,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_153,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_154,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_155,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_156,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_157,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_158,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_159,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_160,axiom,
    quickc6926020345158392990erm_of(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_161,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_162,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_163,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_164,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_165,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_166,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_167,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_168,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_169,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_170,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_171,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_172,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_173,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_174,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_175,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_176,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_177,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_178,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_179,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_180,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_181,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_182,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_183,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_184,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_185,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_186,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_187,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_188,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_189,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_190,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_191,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_192,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_193,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_194,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_195,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_196,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_197,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_198,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_199,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_200,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_201,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_202,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_203,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_204,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_205,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_206,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_207,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_208,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_209,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_210,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_211,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_212,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_213,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_214,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_215,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_216,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_217,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_218,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_219,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_220,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_221,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_222,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_223,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_224,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_225,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_226,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_227,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Otimes_228,axiom,
    times(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_229,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_230,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_231,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_232,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_233,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_234,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_235,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_236,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_237,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(set(A17)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_238,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_239,axiom,
    ! [A17: $tType] : comple592849572758109894attice(set(A17)) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_240,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_241,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_242,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(set(A17)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(set(A17)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_244,axiom,
    ! [A17: $tType] : boolea8198339166811842893lgebra(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_245,axiom,
    ! [A17: $tType] : bounded_lattice_bot(set(A17)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_247,axiom,
    ! [A17: $tType] : semilattice_sup(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_248,axiom,
    ! [A17: $tType] : semilattice_inf(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_249,axiom,
    ! [A17: $tType] : distrib_lattice(set(A17)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
    ! [A17: $tType] : bounded_lattice(set(A17)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_251,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(set(A17)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_252,axiom,
    ! [A17: $tType] : order_top(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_253,axiom,
    ! [A17: $tType] : order_bot(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_254,axiom,
    ! [A17: $tType] : preorder(set(A17)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_255,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(set(A17)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_256,axiom,
    ! [A17: $tType] : lattice(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oorder_257,axiom,
    ! [A17: $tType] : order(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Oord_258,axiom,
    ! [A17: $tType] : ord(set(A17)) ).

tff(tcon_Set_Oset___Orderings_Obot_259,axiom,
    ! [A17: $tType] : bot(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ouminus_260,axiom,
    ! [A17: $tType] : uminus(set(A17)) ).

tff(tcon_Set_Oset___Groups_Ominus_261,axiom,
    ! [A17: $tType] : minus(set(A17)) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_263,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_264,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_265,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_266,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_267,axiom,
    quickc6926020345158392990erm_of($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_268,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_269,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_270,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_271,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_272,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_273,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_274,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_275,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_276,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_277,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_278,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_279,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_280,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_281,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_282,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_283,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_284,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_285,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_286,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_287,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_288,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_289,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_290,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_291,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_292,axiom,
    heap($o) ).

tff(tcon_Heap_Oref___Quickcheck__Narrowing_Opartial__term__of_293,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(ref(A17)) ) ).

tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_294,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(ref(A17)) ) ).

tff(tcon_Heap_Oref___Heap_Oheap_295,axiom,
    ! [A17: $tType] : heap(ref(A17)) ).

tff(tcon_Heap_Oref___Nat_Osize_296,axiom,
    ! [A17: $tType] : size(ref(A17)) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_297,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(list(A17)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_298,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(list(A17)) ) ).

tff(tcon_List_Olist___Heap_Oheap_299,axiom,
    ! [A17: $tType] :
      ( heap(A17)
     => heap(list(A17)) ) ).

tff(tcon_List_Olist___Nat_Osize_300,axiom,
    ! [A17: $tType] : size(list(A17)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_301,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_302,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_303,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
    condit5016429287641298734tinuum(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_304,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_305,axiom,
    semiri6575147826004484403cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
    real_V4412858255891104859lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
    real_V5355595471888546746vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_306,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_307,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_308,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_309,axiom,
    linord2810124833399127020strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
    real_V822414075346904944vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_310,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_311,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_312,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_313,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_314,axiom,
    quickc6926020345158392990erm_of(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_315,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_316,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_317,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_318,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_319,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_320,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_321,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_322,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_323,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_324,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_325,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_326,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_327,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_328,axiom,
    archim2362893244070406136eiling(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
    real_V4867850818363320053vector(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_329,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_330,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_331,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_332,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_333,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_334,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_335,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_336,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_337,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_338,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_339,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_340,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_341,axiom,
    linordered_semiring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
    real_Vector_banach(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring__0_342,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_343,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_344,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_345,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_346,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_347,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_348,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_349,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_350,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_351,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_352,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_353,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_354,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_355,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_356,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_357,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_358,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_359,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_360,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_361,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_362,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_363,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_364,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_365,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_366,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_367,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_368,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_369,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_370,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_371,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_372,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_373,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_374,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_375,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_376,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_377,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_378,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_379,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_380,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_381,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_382,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_383,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_384,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_385,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_386,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_387,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_388,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_389,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_390,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_391,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_392,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_393,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_394,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_395,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Otimes_396,axiom,
    times(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_397,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_398,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_399,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_400,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_401,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_402,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_403,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_404,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_405,axiom,
    dvd(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_406,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => bit_se359711467146920520ations(word(A17)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_407,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(word(A17)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_408,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => bit_bi6583157726757044596ension(word(A17)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_409,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => bit_ri3973907225187159222ations(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_410,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => cancel2418104881723323429up_add(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_411,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => cancel1802427076303600483id_add(word(A17)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_412,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => bit_semiring_bits(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_413,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => cancel_semigroup_add(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_414,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => ab_semigroup_mult(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_415,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_1_cancel(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_416,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_monoid_mult(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_417,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => ab_semigroup_add(word(A17)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_418,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => generic_set_set_bit(word(A17)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_419,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(word(A17)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_420,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_parity(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_421,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_monoid_add(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_422,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_modulo(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_423,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_semiring_1(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_424,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_semiring_0(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_425,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semigroup_mult(word(A17)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_426,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_numeral(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_427,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semigroup_add(word(A17)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_428,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => wellorder(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_429,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => ab_group_add(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_430,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => zero_neq_one(word(A17)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_431,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => preorder(word(A17)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_432,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => linorder(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_433,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => monoid_mult(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_434,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_ring_1(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_435,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => monoid_add(word(A17)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_436,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => finite_finite(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_437,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_1(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_438,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => semiring_0(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_439,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => group_add(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_440,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => mult_zero(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_441,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => comm_ring(word(A17)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_442,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => order(word(A17)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_443,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => neg_numeral(word(A17)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_444,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => ord(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_445,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => uminus(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_446,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => ring_1(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Otimes_447,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => times(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_448,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => minus(word(A17)) ) ).

tff(tcon_Word_Oword___Power_Opower_449,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => power(word(A17)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_450,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => numeral(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_451,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => zero(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_452,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => plus(word(A17)) ) ).

tff(tcon_Word_Oword___Groups_Oone_453,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => one(word(A17)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_454,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => dvd(word(A17)) ) ).

tff(tcon_Word_Oword___Nat_Osize_455,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => size(word(A17)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_456,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(array(A17)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_457,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(array(A17)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_458,axiom,
    ! [A17: $tType] : heap(array(A17)) ).

tff(tcon_Heap_Oarray___Nat_Osize_459,axiom,
    ! [A17: $tType] : size(array(A17)) ).

tff(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_460,axiom,
    quickc6926020345158392990erm_of(char) ).

tff(tcon_String_Ochar___Lattices_Osemilattice__sup_461,axiom,
    semilattice_sup(char) ).

tff(tcon_String_Ochar___Lattices_Osemilattice__inf_462,axiom,
    semilattice_inf(char) ).

tff(tcon_String_Ochar___Lattices_Odistrib__lattice_463,axiom,
    distrib_lattice(char) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_464,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Orderings_Opreorder_465,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_466,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_467,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Lattices_Olattice_468,axiom,
    lattice(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_469,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_470,axiom,
    ord(char) ).

tff(tcon_String_Ochar___Heap_Oheap_471,axiom,
    heap(char) ).

tff(tcon_String_Ochar___Nat_Osize_472,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Quickcheck__Narrowing_Opartial__term__of_473,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => quickc6926020345158392990erm_of(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_474,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_475,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Heap_Oheap_476,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( heap(A17)
        & heap(A18) )
     => heap(sum_sum(A17,A18)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_477,axiom,
    ! [A17: $tType,A18: $tType] : size(sum_sum(A17,A18)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_478,axiom,
    ! [A17: $tType] : condit1219197933456340205attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_479,axiom,
    ! [A17: $tType] : counta3822494911875563373attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_480,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(filter(A17)) ) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_481,axiom,
    ! [A17: $tType] : bounde4967611905675639751up_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_482,axiom,
    ! [A17: $tType] : bounde4346867609351753570nf_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_483,axiom,
    ! [A17: $tType] : comple6319245703460814977attice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_484,axiom,
    ! [A17: $tType] : bounded_lattice_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_485,axiom,
    ! [A17: $tType] : comple9053668089753744459l_ccpo(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_486,axiom,
    ! [A17: $tType] : semilattice_sup(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_487,axiom,
    ! [A17: $tType] : semilattice_inf(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_488,axiom,
    ! [A17: $tType] : distrib_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_489,axiom,
    ! [A17: $tType] : bounded_lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_490,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(filter(A17)) ) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_491,axiom,
    ! [A17: $tType] : order_top(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_492,axiom,
    ! [A17: $tType] : order_bot(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_493,axiom,
    ! [A17: $tType] : preorder(filter(A17)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_494,axiom,
    ! [A17: $tType] : lattice(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_495,axiom,
    ! [A17: $tType] : order(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_496,axiom,
    ! [A17: $tType] : ord(filter(A17)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_497,axiom,
    ! [A17: $tType] : bot(filter(A17)) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_498,axiom,
    ! [A17: $tType] :
      ( comple5582772986160207858norder(A17)
     => condit6923001295902523014norder(option(A17)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_499,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => condit1219197933456340205attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Countable__Complete__Lattices_Ocountable__complete__lattice_500,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => counta3822494911875563373attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_501,axiom,
    ! [A17: $tType] :
      ( comple592849572758109894attice(A17)
     => comple592849572758109894attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_502,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_503,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => bounde4967611905675639751up_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_504,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounde4346867609351753570nf_top(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
    ! [A17: $tType] :
      ( comple5582772986160207858norder(A17)
     => comple5582772986160207858norder(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_505,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple6319245703460814977attice(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice__bot_506,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => bounded_lattice_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Complete__Partial__Order_Occpo_507,axiom,
    ! [A17: $tType] :
      ( comple6319245703460814977attice(A17)
     => comple9053668089753744459l_ccpo(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__sup_508,axiom,
    ! [A17: $tType] :
      ( semilattice_sup(A17)
     => semilattice_sup(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Osemilattice__inf_509,axiom,
    ! [A17: $tType] :
      ( semilattice_inf(A17)
     => semilattice_inf(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Odistrib__lattice_510,axiom,
    ! [A17: $tType] :
      ( distrib_lattice(A17)
     => distrib_lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Obounded__lattice_511,axiom,
    ! [A17: $tType] :
      ( bounded_lattice_top(A17)
     => bounded_lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_512,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_513,axiom,
    ! [A17: $tType] :
      ( wellorder(A17)
     => wellorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_514,axiom,
    ! [A17: $tType] :
      ( order_top(A17)
     => order_top(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_515,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => order_bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_516,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => preorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_517,axiom,
    ! [A17: $tType] :
      ( linorder(A17)
     => linorder(option(A17)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_518,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(option(A17)) ) ).

tff(tcon_Option_Ooption___Lattices_Olattice_519,axiom,
    ! [A17: $tType] :
      ( lattice(A17)
     => lattice(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_520,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => order(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_521,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ord(option(A17)) ) ).

tff(tcon_Option_Ooption___Orderings_Obot_522,axiom,
    ! [A17: $tType] :
      ( order(A17)
     => bot(option(A17)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_523,axiom,
    ! [A17: $tType] :
      ( heap(A17)
     => heap(option(A17)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_524,axiom,
    ! [A17: $tType] : size(option(A17)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_525,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_526,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_527,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_528,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_529,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_530,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_531,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_532,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_533,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_534,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_535,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_536,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_537,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_538,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_539,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_540,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_541,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_542,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_543,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_544,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_545,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_546,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_547,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_548,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_549,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_550,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_551,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_552,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_553,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_554,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_555,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_556,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_557,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_558,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_559,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_560,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_561,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_562,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_563,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Otimes_564,axiom,
    times(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_565,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_566,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_567,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_568,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_569,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_570,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_571,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_572,axiom,
    size(uint32) ).

tff(tcon_String_Oliteral___Quickcheck__Narrowing_Opartial__term__of_573,axiom,
    quickc6926020345158392990erm_of(literal) ).

tff(tcon_String_Oliteral___Code__Evaluation_Oterm__of_574,axiom,
    code_term_of(literal) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_575,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_576,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_577,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_578,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_579,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_580,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_581,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_582,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Heap_Oheap_583,axiom,
    heap(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_584,axiom,
    size(literal) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__sup__bot_585,axiom,
    bounde4967611905675639751up_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__semilattice__inf__top_586,axiom,
    bounde4346867609351753570nf_top(assn) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_587,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice__bot_588,axiom,
    bounded_lattice_bot(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_589,axiom,
    semilattice_sup(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_590,axiom,
    semilattice_inf(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Odistrib__lattice_591,axiom,
    distrib_lattice(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Obounded__lattice_592,axiom,
    bounded_lattice(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_593,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_594,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_595,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_596,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_597,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_598,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_599,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Lattices_Olattice_600,axiom,
    lattice(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_601,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_602,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Obot_603,axiom,
    bot(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_604,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Otimes_605,axiom,
    times(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_606,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_607,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_608,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_609,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_610,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_611,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_612,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_613,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_614,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_615,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_616,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_617,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_618,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_619,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_620,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_621,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_622,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_623,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_624,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_625,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_626,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_627,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_628,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_629,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_630,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_631,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_632,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_633,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_634,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_635,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_636,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_637,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_638,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_639,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_640,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_641,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_642,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_643,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_644,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_645,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_646,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_647,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_648,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_649,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_650,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_651,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_652,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_653,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_654,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_655,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_656,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_657,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_658,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_659,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_660,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_661,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_662,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_663,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_664,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_665,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_666,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_667,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_668,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_669,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_670,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Otimes_671,axiom,
    times(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_672,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_673,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_674,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_675,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_676,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_677,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_678,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_679,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_680,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_681,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_682,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_683,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_684,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_685,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_686,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_687,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_688,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_689,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_690,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_691,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_692,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_693,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_694,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_695,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_696,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_697,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_698,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_699,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_700,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_701,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_702,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_703,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_704,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_705,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_706,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_707,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_708,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_709,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_710,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_711,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_712,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_713,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_714,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_715,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_716,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_717,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_718,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_719,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_720,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_721,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_722,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_723,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_724,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_725,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_726,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_727,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_728,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_729,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_730,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_731,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_732,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Otimes_733,axiom,
    times(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_734,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_735,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_736,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_737,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_738,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_739,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_740,axiom,
    dvd(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_741,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_742,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ordere6658533253407199908up_add(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_743,axiom,
    ! [A17: $tType] : cancel2418104881723323429up_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_744,axiom,
    ! [A17: $tType] : cancel1802427076303600483id_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_745,axiom,
    ! [A17: $tType] : cancel_semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_746,axiom,
    ! [A17: $tType] : comm_monoid_diff(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_747,axiom,
    ! [A17: $tType] : ab_semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_748,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_749,axiom,
    ! [A17: $tType] : comm_monoid_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_750,axiom,
    ! [A17: $tType] : semigroup_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_751,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => preorder(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_752,axiom,
    ! [A17: $tType] : monoid_add(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_753,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => order(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_754,axiom,
    ! [A17: $tType] :
      ( preorder(A17)
     => ord(multiset(A17)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_755,axiom,
    ! [A17: $tType] : minus(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_756,axiom,
    ! [A17: $tType] : zero(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_757,axiom,
    ! [A17: $tType] : plus(multiset(A17)) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_758,axiom,
    ! [A17: $tType] : size(multiset(A17)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_759,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel2418104881723323429up_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_760,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel1802427076303600483id_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_761,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel_semigroup_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_762,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_semigroup_mult(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_763,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_1_cancel(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_764,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_monoid_mult(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_765,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_semigroup_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_766,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_767,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_monoid_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_768,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_semiring_1(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_769,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_semiring_0(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_770,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semigroup_mult(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_771,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_numeral(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_772,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semigroup_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_773,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => wellorder(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_774,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_group_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_775,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => zero_neq_one(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_776,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => preorder(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_777,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => linorder(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_778,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => monoid_mult(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_779,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_ring_1(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_780,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => monoid_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_781,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Cardinality_Ocard2,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => card2(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_782,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_1(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_783,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_0(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_784,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => group_add(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen,axiom,
    ! [A17: $tType] :
      ( type_len(A17)
     => type_len(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_785,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => mult_zero(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_786,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_ring(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_787,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => order(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_788,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => neg_numeral(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_789,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ord(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_790,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => uminus(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_791,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ring_1(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Otimes_792,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => times(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_793,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => minus(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_794,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => power(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_795,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => numeral(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_796,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => zero(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_797,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => plus(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_798,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => one(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_799,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => dvd(numeral_bit0(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_800,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel2418104881723323429up_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_801,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel1802427076303600483id_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_802,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => cancel_semigroup_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_803,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_semigroup_mult(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_804,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_1_cancel(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_805,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_monoid_mult(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_806,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_semigroup_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_807,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_808,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_monoid_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_809,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_semiring_1(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_810,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_semiring_0(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_811,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semigroup_mult(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_812,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_numeral(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_813,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semigroup_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_814,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => wellorder(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_815,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ab_group_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_816,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => zero_neq_one(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_817,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => preorder(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_818,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => linorder(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_819,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => monoid_mult(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_820,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_ring_1(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_821,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => monoid_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_822,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => finite_finite(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Cardinality_Ocard2_823,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => card2(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_824,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_1(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_825,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => semiring_0(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_826,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => group_add(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_827,axiom,
    ! [A17: $tType] :
      ( type_len0(A17)
     => type_len(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_828,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => mult_zero(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_829,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => comm_ring(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_830,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => order(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_831,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => neg_numeral(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_832,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ord(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_833,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => uminus(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_834,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => ring_1(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Otimes_835,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => times(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_836,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => minus(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_837,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => power(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_838,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => numeral(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_839,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => zero(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_840,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => plus(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_841,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => one(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_842,axiom,
    ! [A17: $tType] :
      ( finite_finite(A17)
     => dvd(numeral_bit1(A17)) ) ).

tff(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_843,axiom,
    quickc6926020345158392990erm_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_844,axiom,
    code_term_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_845,axiom,
    quickc6926020345158392990erm_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_846,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_847,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_848,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_849,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_850,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_851,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_852,axiom,
    code_term_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_853,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_854,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_855,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_856,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_857,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_858,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_859,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_860,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_861,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_862,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_863,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_864,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_865,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_866,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_867,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_868,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_869,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_870,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_871,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Otimes_872,axiom,
    times(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_873,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_874,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_875,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_876,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_877,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_878,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_879,axiom,
    dvd(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_880,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topolo4958980785337419405_space(A17)
        & topolo4958980785337419405_space(A18) )
     => topolo4958980785337419405_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_881,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => quickc6926020345158392990erm_of(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_882,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( topological_t2_space(A17)
        & topological_t2_space(A18) )
     => topological_t2_space(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_883,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( typerep(A17)
        & typerep(A18) )
     => code_term_of(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_884,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( finite_finite(A17)
        & finite_finite(A18) )
     => finite_finite(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_885,axiom,
    ! [A17: $tType,A18: $tType] :
      ( ( heap(A17)
        & heap(A18) )
     => heap(product_prod(A17,A18)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_886,axiom,
    ! [A17: $tType,A18: $tType] : size(product_prod(A17,A18)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_887,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_888,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_889,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_890,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_891,axiom,
    quickc6926020345158392990erm_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_892,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_893,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_894,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_895,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_896,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_897,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_898,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_899,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_900,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_901,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_902,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_903,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_904,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_905,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_906,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_907,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_908,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_909,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_910,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_911,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_912,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_913,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_914,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_915,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_916,axiom,
    heap(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_917,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(heap_ext(A17)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_918,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(heap_ext(A17)) ) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_919,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_920,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_921,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_922,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_923,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_924,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_925,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_926,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_927,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_928,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_929,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_930,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_931,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_932,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_933,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_934,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_935,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_936,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_937,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_938,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_939,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_940,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_941,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_942,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_943,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_944,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_945,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_946,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_947,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_948,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_949,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_950,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_951,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_952,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_953,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_954,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_955,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_956,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_957,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_958,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_959,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_960,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_961,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_962,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_963,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_964,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_965,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_966,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_967,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_968,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_969,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_970,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_971,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_972,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_973,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_974,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_975,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_976,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_977,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_978,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_979,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_980,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_981,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_982,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_983,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_984,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_985,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_986,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_987,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_988,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_989,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_990,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_991,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_992,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_993,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_994,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_995,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_996,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_997,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_998,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_999,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1000,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1001,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1002,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1003,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Otimes_1004,axiom,
    times(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1005,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1006,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1007,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1008,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1009,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1010,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1011,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1012,axiom,
    dvd(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1013,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A17)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1014,axiom,
    ! [A17: $tType] :
      ( typerep(A17)
     => code_term_of(heap_Time_Heap(A17)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1015,axiom,
    ! [A17: $tType] : size(heap_Time_Heap(A17)) ).

tff(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1016,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1017,axiom,
    code_term_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1018,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1019,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1020,axiom,
    code_term_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1021,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1022,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (8)
tff(help_fconj_3_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fconj,(P2)),(Q2))
      | (Q2) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fconj,(P2)),(Q2))
      | (P2) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ (P2)
      | ~ (Q2)
      | aa($o,$o,aa($o,fun($o,$o),fconj,(P2)),(Q2)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ aa($o,$o,aa($o,fun($o,$o),fdisj,(P2)),(Q2))
      | (P2)
      | (Q2) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q2: $o,P2: $o] :
      ( ~ (Q2)
      | aa($o,$o,aa($o,fun($o,$o),fdisj,(P2)),(Q2)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P2: $o,Q2: $o] :
      ( ~ (P2)
      | aa($o,$o,aa($o,fun($o,$o),fdisj,(P2)),(Q2)) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ( X10 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X10: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y)
      | ( X10 = Y ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (6)
tff(conj_0,hypothesis,
    s = vEBT_set_vebt(t) ).

tff(conj_1,hypothesis,
    vEBT_invar_vebt(t,n) ).

tff(conj_2,hypothesis,
    aa(product_prod(heap_ext(product_unit),set(nat)),$o,rep_assn(aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(t,x)),r)),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),a2),b2)) ).

tff(conj_3,hypothesis,
    member(nat,xa,vEBT_set_vebt(t)) ).

tff(conj_4,hypothesis,
    ~ member(nat,xa,vEBT_set_vebt(vEBT_vebt_delete(t,x))) ).

tff(conj_5,conjecture,
    xa = x ).